Finite Group Databasebeta

← #33055419

Subgroups — #33055419

Order 18688.

754 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619038C211yesmaximal, characteristic, complemented
2#22618952C211yesmaximal, characteristic, complemented
3C4×#14556615C211yesmaximal, characteristic
4#22608752C211yesmaximal, characteristic, complemented
5#22608745C211yesmaximal, characteristic, complemented
6#22608740C211yesmaximal, characteristic, complemented
7#22608730C211yesmaximal, characteristic, complemented
8C8×#11924016C211yesmaximal, characteristic, complemented
9D8×#10596831C211yesmaximal, characteristic, complemented
10C8×#11924013C211yesmaximal, characteristic, complemented
11#22608680C211yesmaximal, characteristic, complemented
12#22608310C211yesmaximal, characteristic, complemented
13D4⋊C8×D73C211yesmaximal, characteristic, complemented
14#22608210C211yesmaximal, characteristic, complemented
15C73×#1580C211yesmaximal, characteristic, complemented
16C2×#14556615C411yescharacteristic
17C4×#11924027C2211yescharacteristic
18C4×D4×D73C2211yescharacteristic
19C8×Dic146C2211yescharacteristic, complemented
20#17988711C2211yescharacteristic, complemented
21#17988699C411yescharacteristic
22#17988696C411yescharacteristic
23#17988686C411yescharacteristic
24#17988684C411yescharacteristic
25C4×#11924017C2211yescharacteristic
26C4×#11924016C2211yescharacteristic
27C4×#11924013C2211yescharacteristic
28C8×D292C2211yescharacteristic
29D4×#10596831C2211yescharacteristic
30D4×#10596831C2211yescharacteristic, complemented
31#17988652C2211yescharacteristic
32C8×C22⋊D73—21no—
33C8×C22⋊D73—21no—
34#17988650—21no—
35#17988650—21no—
36#17988639C2211yescharacteristic
37#17988638C2211yescharacteristic
38#17988637—21no—
39#17988628C411yescharacteristic
40C2×#14556520—21no—
41C4⋊C8×D73C2211yescharacteristic
42#17988590C411yescharacteristic
43#17988585C411yescharacteristic
44C2×#14556503—21no—
45C22⋊C8×D73—21no—
46C4×#11924011C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988340C2211yescharacteristic, complemented
49#17988339C2211yescharacteristic, complemented
50#17988337C2211yescharacteristic, complemented
51#17988319C2211yescharacteristic, complemented
52#17988317C2211yescharacteristic, complemented
53#17988316C2211yescharacteristic
54D4⋊#10596831C2211yescharacteristic
55D4⋊#10596831C2211yescharacteristic, complemented
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic, complemented
58C8×#10596831C2211yescharacteristic, complemented
59C8×#10596831C2211yescharacteristic, complemented
60C4×D8×C73C2211yescharacteristic
61C8×D4×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic, complemented
63#334×C73C2211yescharacteristic, complemented
64D4⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic, complemented
66#322×C73C2211yescharacteristic, complemented
67C2×#11924027C2×C411yescharacteristic
68C2×D4×D73C2×C411yescharacteristic
69C22×C4×D73—21no—
70#14556615C818yescomplemented
71C2×#11924017C2×C411yescharacteristic
72C2×#11924016C2×C411yescharacteristic
73C2×#11924013C2×C411yescharacteristic
74C2×#11924011C2×C411yescharacteristic
75C2×#11924011C2×C411yescharacteristic
76C22×#10596831—21no—
77C22×#10596831—21no—
78C2×C8×D73—21no—
79C2×C8×D73—21no—
80C2×C8×D73—21no—
81C2×C8×D73—21no—
82C4×Dic146C2311yescharacteristic
83#14556556C2×C411yescharacteristic
84C4×D292C2311yescharacteristic
85D4×Dic73C2×C411yescharacteristic
86D4×Dic73C2×C411yescharacteristic
87#14556553C2×C411yescharacteristic
88C4×C22⋊D73—21no—
89C4×C22⋊D73—21no—
90#14556551—21no—
91#14556551—21no—
92#14556549C2×C411yescharacteristic
93#14556547—21no—
94C4⋊C4×D73C2×C411yescharacteristic
95C22⋊C4×D73—21no—
96C2×C4×Dic73—21no—
97C42×D73C2311yescharacteristic
98#14556524C2×C411yescharacteristic
99#14556523C2×C411yescharacteristic
100#14556522C2×C411yescharacteristic
101C4⋊#10596831C2311yescharacteristic
102C4⋊#10596831C2311yescharacteristic
103#14556520—21no—
104#14556520—21no—
105#14556520—21no—
106#14556520D411yescharacteristic
107#14556520D411yescharacteristic
108#14556516C2×C411yescharacteristic
109#14556514C2×C411yescharacteristic
110#14556513C2×C411yescharacteristic
111#14556512C2×C411yescharacteristic
112#14556512C2×C411yescharacteristic
113#14556511C2×C411yescharacteristic
114C22⋊#10596831—21no—
115C22⋊#10596831—21no—
116#14556503—21no—
117#14556503—21no—
118#14556503—21no—
119#14556503D411yescharacteristic
120#14556503D411yescharacteristic
121#14556501C2×C411yescharacteristic
122#14556501C2×C411yescharacteristic
123C4×#10596831C2311yescharacteristic
124C4×#10596831C2311yescharacteristic
125C4×#10596831C2311yescharacteristic
126C8×Dic73—21no—
127C8×Dic73—21no—
128C8×Dic73—21no—
129C8×Dic73—21no—
130C2×D8×C73C2×C411yescharacteristic
131C22×C8×C73—21no—
132C22×C8×C73—21no—
133C4×D4×C73C2311yescharacteristic
134C4×D4×C73C2311yescharacteristic
135#107×C73C2×C411yescharacteristic
136C4⋊C8×C73C2311yescharacteristic
137C4⋊C8×C73C2311yescharacteristic
138#102×C73C2×C411yescharacteristic
139#102×C73C2×C411yescharacteristic
140C22⋊C8×C73—21no—
141C22⋊C8×C73—21no—
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145C23×D73—21no—
146#11924027—22no—
147#11924027C2×C814yescomplemented
148C2×Dic146C22×C411yescharacteristic
149C2×D292C22×C411yescharacteristic
150C2×C22⋊D73—21no—
151C2×C22⋊D73—21no—
152D4×D73—22no—
153D4×D73C2×C814yescomplemented
154C22×Dic73—21no—
155C22×Dic73—21no—
156C22×Dic73—21no—
157C2×C4×D73—21no—
158C2×C4×D73—21no—
159C2×C4×D73—21no—
160C2×C4×D73—81no—
161C2×C4×D73—81no—
162C2×C4×D73C22×C411yescharacteristic
163C2×C4×D73C2×D411yescharacteristic
164C2×C4×D73#4111yescharacteristic
165#11924017C2×C814yescomplemented
166#11924016C2×C814yescomplemented
167#11924013C2×C814yescomplemented
168#11924011—22no—
169#11924011C2×C814yescomplemented
170C2×#10596831—21no—
171C2×#10596831—21no—
172C2×#10596831—21no—
173C2×#10596831—81no—
174C2×#10596831—81no—
175C2×#10596831C22×C411yescharacteristic
176C2×#10596831C22×C411yescharacteristic
177C2×#10596831C2×D411yescharacteristic
178C2×#10596831C2×D411yescharacteristic
179C8×D73—42no—
180C8×D73—81no—
181C8×D73—81no—
182C8×D73—81no—
183C4⋊Dic73C22×C411yescharacteristic
184C4⋊Dic73C22×C411yescharacteristic
185#11924007—21no—
186#11924007—21no—
187#11924007—21no—
188C22⋊Dic73—21no—
189C22⋊Dic73—21no—
190#11924005—21no—
191#11924005—21no—
192#11924005—21no—
193C4×Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73—21no—
196C4×Dic73C22×C411yescharacteristic
197C4×Dic73C2×D411yescharacteristic
198C4×Dic73#4111yescharacteristic
199C2×D4×C73C22×C411yescharacteristic
200C2×D4×C73C22×C411yescharacteristic
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203D8×C73C2×C814yescomplemented
204C2×C8×C73—21no—
205C2×C8×C73—21no—
206C2×C8×C73—21no—
207C2×C8×C73—81no—
208C2×C8×C73—81no—
209C2×C8×C73C22×C411yescharacteristic
210C2×C8×C73C22×C411yescharacteristic
211C2×C8×C73C2×D411yescharacteristic
212C2×C8×C73C2×D411yescharacteristic
213C4⋊C4×C73C22×C411yescharacteristic
214C4⋊C4×C73C22×C411yescharacteristic
215C22⋊C4×C73—21no—
216C22⋊C4×C73—21no—
217C42×C73C2411yescharacteristic
218C23×C73—21no—
219C23×C73—21no—
220C22×D73—21no—
221C22×D73—24no—
222C22×D73—81no—
223C22×D73—81no—
224C22×D73C4×D411yescharacteristic
225D4×C73—21no—
226D4×C73—21no—
227D4×C73C22×C812yes—
228D4×C73C22×C812yes—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73C23×C411yescharacteristic
236C2×C4×C73C22×D411yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C22⋊D73—24no—
239C22⋊D73—24no—
240C2×Dic73—21no—
241C2×Dic73—24no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—81no—
245C2×Dic73—81no—
246C2×Dic73C4×D411yescharacteristic
247C2×Dic73C4×D411yescharacteristic
248C2×Dic73C4×D411yescharacteristic
249D292—21no—
250D292C22×C812yes—
251C4×D73—21no—
252C4×D73—21no—
253C4×D73—41no—
254C4×D73—41no—
255C4×D73—41no—
256C4×D73—81no—
257C4×D73—81no—
258C4×D73C22×C812yes—
259C4×D73#13112yes—
260Dic146—21no—
261Dic146C22×C812yes—
262C8×C73—21no—
263C8×C73—21no—
264C8×C73—21no—
265C8×C73—41no—
266C8×C73—41no—
267C8×C73—41no—
268C8×C73C22×C812yes—
269#10596831—22no—
270#10596831—21no—
271#10596831—21no—
272#10596831—21no—
273#10596831—41no—
274#10596831—41no—
275#10596831C22×C812yes—
276C22×C73—22no—
277C22×C73—22no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73—41no—
282C22×C73C2×C4×D411yescharacteristic
283C2×D73—21no—
284C2×D73—21no—
285C2×D73—22no—
286C2×D73—41no—
287C2×D73—41no—
288C2×D73—84no—
289C2×D73C8×D412yes—
290C4×C73—21no—
291C4×C73—41no—
292C4×C73—41no—
293C4×C73C23×C811yescharacteristic
294C4×C73C2×#13111yescharacteristic
295C4×C73#57611yescharacteristic
296C4×C73#57711yescharacteristic
297Dic73—22no—
298Dic73—21no—
299Dic73—21no—
300Dic73—41no—
301Dic73—41no—
302Dic73C8×D412yes—
303#17852—731nomaximal, Sylow
304C2×C73—42no—
305C2×C73—42no—
306C2×C73C2×C8×D411yescharacteristic
307C2×C73#294611yescharacteristic
308C2×C73#295611yescharacteristic
309D73—22no—
310D73—42no—
311#2988—731no—
312#2972—731no—
313C4×AGL(1,ℤ/8)—731no—
314#1580—731no—
315#1580D7311yescharacteristic, complemented
316SD16⋊C8—731no—
317SD16⋊C8—731no—
318C8×SD16—731no—
319C8×SD16—731no—
320C8×D8—731no—
321C8×D8—731no—
322#1565—731no—
323#1476—731no—
324C2×D4⋊C8—731no—
325C8×M16—731no—
326C73#1785211yescharacteristic, complemented, Sylow
327C2×AGL(1,ℤ/8)—731no—
328C4×#41—731no—
329C2×C4×D4—731no—
330C8×Q8—731no—
331C8×Q8—731no—
332#442—731no—
333#442—731no—
334#440—731no—
335#440—731no—
336C4×SD16—731no—
337C4×SD16—731no—
338C4×D8—731no—
339C4×D8C2×D7311yescharacteristic
340C8×D4—731no—
341C8×D4—731no—
342C8×D4—731no—
343C8×D4—731no—
344C8×D4—1461no—
345C8×D4—1461no—
346C8×D4—1461no—
347C8×D4—1461no—
348C8×D4C2×D7311yescharacteristic
349C8×D4C2×D7311yescharacteristic, complemented
350#431—731no—
351#431—731no—
352#431—1461no—
353#428—731no—
354C2×C4⋊C8—731no—
355C2×C4⋊C8—1461no—
356#418—731no—
357#417—731no—
358C2×C22⋊C8—1461no—
359C2×C22⋊C8—1461no—
360C4×M16—731no—
361C2×C4×C8—731no—
362#335—731no—
363#335—731no—
364#334—731no—
365#334C2×D7311yescharacteristic, complemented
366Q8⋊C8—731no—
367Q8⋊C8—731no—
368D4⋊C8—731no—
369D4⋊C8—731no—
370D4⋊C8—731no—
371D4⋊C8—731no—
372D4⋊C8C2×D7311yescharacteristic
373D4⋊C8C2×D7311yescharacteristic, complemented
374#322—731no—
375#322C2×D7311yescharacteristic, complemented
376C82—731no—
377C82—731no—
378C2×#41—731no—
379C22×D4—731no—
380C23×C4—1461no—
381AGL(1,ℤ/8)—738no—
382C2×SD16—731no—
383C2×SD16—731no—
384C2×D8—731no—
385C2×D8C4×D7311yescharacteristic
386C2×M16—731no—
387C2×M16—731no—
388C22×C8—21no—
389C22×C8—21no—
390C22×C8—1461no—
391C22×C8—1461no—
392C22×C8—1461no—
393C22×C8—1461no—
394C22×C8—1461no—
395C22×C8—1461no—
396C4×Q8—731no—
397C4×Q8—731no—
398C4×D4—731no—
399C4×D4—731no—
400C4×D4—731no—
401C4×D4—731no—
402C4×D4—1461no—
403C4×D4—1461no—
404C4×D4—1461no—
405C4×D4—1461no—
406C4×D4C22×D7311yescharacteristic
407C4×D4C22×D7311yescharacteristic
408#117—731no—
409#117—1461no—
410C2×C4⋊C4—731no—
411C2×C22⋊C4—1461no—
412C2×C42—731no—
413C2×C42—1461no—
414#107—731no—
415#107C4×D7311yescharacteristic
416#106—731no—
417#106—731no—
418C4⋊C8—731no—
419C4⋊C8—731no—
420C4⋊C8—731no—
421C4⋊C8—731no—
422C4⋊C8—1461no—
423C4⋊C8—1461no—
424C4⋊C8—1461no—
425C4⋊C8C22×D7311yescharacteristic
426C4⋊C8C22×D7311yescharacteristic
427#103—731no—
428#103—731no—
429#102—731no—
430#102—731no—
431#102—731no—
432#102—731no—
433#102C4×D7311yescharacteristic
434#102C4×D7311yescharacteristic
435C22⋊C8—21no—
436C22⋊C8—21no—
437C22⋊C8—731no—
438C22⋊C8—731no—
439C22⋊C8—1461no—
440C22⋊C8—1461no—
441C22⋊C8—1461no—
442C22⋊C8—1461no—
443C22⋊C8—1461no—
444#97—731no—
445#97—731no—
446C4×C8—731no—
447C4×C8—731no—
448C4×C8—731no—
449C4×C8—1461no—
450C4×C8—1461no—
451C4×C8—1461no—
452C4×C8—1461no—
453C4×C8C22×D7311yescharacteristic
454C4×C8C22×D7311yescharacteristic
455C4×C8C22×D7311yescharacteristic
456C24—1461no—
457#41—734no—
458#41—1462no—
459C2×Q8—731no—
460C2×D4—734no—
461C2×D4—731no—
462C2×D4—1462no—
463C2×D4—1461no—
464C2×D4—1461no—
465C2×D4C2×C4×D7311yescharacteristic
466C2×D4C2×C4×D7311yescharacteristic
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—5841no—
470C22×C4—5841no—
471C22×C4—731no—
472C22×C4—731no—
473C22×C4—731no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480SD16—734no—
481SD16—734no—
482D8—734no—
483D8C8×D7314yescomplemented
484M16—734no—
485M16—1462no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—81no—
490C2×C8—81no—
491C2×C8—2922no—
492C2×C8—5841no—
493C2×C8—5841no—
494C2×C8—5841no—
495C2×C8—5841no—
496C2×C8—5841no—
497C2×C8—731no—
498C2×C8—731no—
499C2×C8—731no—
500C2×C8—731no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8C2×C4×D7311yescharacteristic
505C2×C8C2×C4×D7311yescharacteristic
506C2×C8D4×D7311yescharacteristic
507C2×C8D4×D7311yescharacteristic
508C4⋊C4—731no—
509C4⋊C4—731no—
510C4⋊C4—1461no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4C2×C4×D7311yescharacteristic
514C4⋊C4C2×C4×D7311yescharacteristic
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—1461no—
518C22⋊C4—1461no—
519C22⋊C4—1461no—
520C22⋊C4—1461no—
521C22⋊C4—1461no—
522C42—731no—
523C42—731no—
524C42—731no—
525C42—1461no—
526C42—1461no—
527C42—1461no—
528C42C23×D7311yescharacteristic
529C23—21no—
530C23—21no—
531C23—5841no—
532C23—5841no—
533C23—731no—
534C23—1464no—
535C23—1461no—
536Q8—732no—
537Q8—1461no—
538D4—21no—
539D4—21no—
540D4—732no—
541D4—1464no—
542D4—1464no—
543D4—1461no—
544D4C2×C8×D7312yes—
545D4C2×C8×D7312yes—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—5841no—
556C2×C4—5841no—
557C2×C4—5841no—
558C2×C4—5841no—
559C2×C4—731no—
560C2×C4—732no—
561C2×C4—732no—
562C2×C4—731no—
563C2×C4—731no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1464no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4C22×C4×D7311yescharacteristic
571C2×C4C2×D4×D7311yescharacteristic
572C2×C4#41×D7311yescharacteristic
573C8—21no—
574C8—21no—
575C8—21no—
576C8—41no—
577C8—41no—
578C8—41no—
579C8—2921no—
580C8—2921no—
581C8—732no—
582C8—1462no—
583C8—1461no—
584C8—1461no—
585C8—1461no—
586C8C2×C8×D7312yes—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—2921no—
594C22—2921no—
595C22—5844no—
596C22—732no—
597C22—1462no—
598C22—1461no—
599C22—1461no—
600C22C4×D4×D7311yescharacteristic
601C4—21no—
602C4—41no—
603C4—41no—
604C4—2921no—
605C4—2921no—
606C4—732no—
607C4—1461no—
608C4—1461no—
609C4—1462no—
610C4C22×C8×D7311yescharacteristic
611C4#131×D7311yescharacteristic
612C4#1798939111yescharacteristic
613C4#1798939711yescharacteristic
614C2—42no—
615C2—42no—
616C2—2922no—
617C2—1462no—
618C2C8×D4×D7311yescharacteristic
619C2#2261876011yescharacteristic
620C2#2261883911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC4×C73C23×C8
FrattiniC42C23×D73
FittingC73×#1580C2
socleC22×C73C2×C4×D4
prime radicalC2×#14556615C4
cosocleC42×C73C24
hypercenter#1580D73
hypocenterC73#17852
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1580

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22619038C2semidirect
2. #22618952C2semidirect
4. #22608752C2semidirect
5. #22608745C2semidirect
6. #22608740C2semidirect
7. #22608730C2semidirect
8. C8×#11924016C2semidirect
9. D8×#10596831C2semidirect
10. C8×#11924013C2semidirect
11. #22608680C2semidirect
12. #22608310C2semidirect
13. D4⋊C8×D73C2semidirect
14. #22608210C2semidirect
15. C73×#1580C2semidirect
19. C8×Dic146C22semidirect
20. #17988711C22semidirect
30. D4×#10596831C22semidirect
48. #17988340C22semidirect
49. #17988339C22semidirect
50. #17988337C22semidirect
51. #17988319C22semidirect
52. #17988317C22semidirect
55. D4⋊#10596831C22semidirect
57. #17988309C22semidirect
58. C8×#10596831C22semidirect
59. C8×#10596831C22semidirect
62. C8×D4×C73C22semidirect
63. #334×C73C22semidirect
65. D4⋊C8×C73C22semidirect
66. #322×C73C22semidirect
70. #14556615C8semidirect
147. #11924027C2×C8semidirect
153. D4×D73C2×C8semidirect
165. #11924017C2×C8semidirect
166. #11924016C2×C8semidirect
167. #11924013C2×C8semidirect
169. #11924011C2×C8semidirect
203. D8×C73C2×C8semidirect
315. #1580D73semidirect
326. C73#17852semidirect
349. C8×D4C2×D73semidirect
365. #334C2×D73semidirect
373. D4⋊C8C2×D73semidirect
375. #322C2×D73semidirect
483. D8C8×D73semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226190382
#226189522
C4×#145566152
#226087522
#226087452
#226087402
#226087302
C8×#119240162
D8×#105968312
C8×#119240132
#226086802
#226083102
D4⋊C8×D732
#226082102
C73×#15802
#1785273