Finite Group Databasebeta

← #33055949

Subgroups — #33055949

Order 18688.

720 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619976C211yesmaximal, characteristic
2#22618903C211yesmaximal, characteristic, complemented
3C8×D4×D73C211yesmaximal, characteristic, complemented
4#22608798C211yesmaximal, characteristic, complemented
5#22608796C211yesmaximal, characteristic, complemented
6#22608787C211yesmaximal, characteristic, complemented
7#22608785C211yesmaximal, characteristic, complemented
8C8×#11924017C211yesmaximal, characteristic, complemented
9C8×#11924016C211yesmaximal, characteristic, complemented
10C8×#11924013C211yesmaximal, characteristic, complemented
11#22608715C211yesmaximal, characteristic, complemented
12#22608372C211yesmaximal, characteristic, complemented
13#22608341C211yesmaximal, characteristic, complemented
14C8×#11924011C211yesmaximal, characteristic, complemented
15C8×D8×C73C211yesmaximal, characteristic, complemented
16C22×C8×D73—21no—
17#17989119C2211yescharacteristic
18C4×D4×D73C2211yescharacteristic
19#17988924C411yescharacteristic
20#17988904C411yescharacteristic
21#17988890C411yescharacteristic
22#17988889C411yescharacteristic
23#17988801C411yescharacteristic
24#17988800C411yescharacteristic
25#17988739C411yescharacteristic
26#17988719C411yescharacteristic
27#17988714C2211yescharacteristic, complemented
28C8×Dic146C2211yescharacteristic, complemented
29C4×#11924017C2211yescharacteristic
30C4×#11924016C2211yescharacteristic
31C4×#11924013C2211yescharacteristic
32#17988665C2211yescharacteristic, complemented
33#17988661—21no—
34C8×D292C2211yescharacteristic
35D4×#10596831C2211yescharacteristic
36#17988652C2211yescharacteristic
37C8×C22⋊D73—21no—
38C8×C22⋊D73—21no—
39#17988650—21no—
40#17988645—21no—
41#17988643C2211yescharacteristic
42#17988638C2211yescharacteristic
43C4⋊C8×D73C2211yescharacteristic
44C22⋊C8×D73—21no—
45#17988542—21no—
46C4×#11924011C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988339C2211yescharacteristic, complemented
49#17988338C2211yescharacteristic, complemented
50#17988337C2211yescharacteristic, complemented
51#17988319C2211yescharacteristic, complemented
52#17988317C2211yescharacteristic, complemented
53#17988316C2211yescharacteristic
54D4⋊#10596831C2211yescharacteristic
55D4⋊#10596831C2211yescharacteristic, complemented
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic, complemented
58#17988308C2211yescharacteristic, complemented
59C8×#10596831C2211yescharacteristic, complemented
60C4×D8×C73C2211yescharacteristic
61C8×D4×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic, complemented
63#335×C73C2211yescharacteristic, complemented
64D4⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic, complemented
66C82×C73C2211yescharacteristic, complemented
67C2×D4×D73C2×C411yescharacteristic
68C22×C4×D73—21no—
69C2×#11924017C2×C411yescharacteristic
70C2×#11924016C2×C411yescharacteristic
71C2×#11924013C2×C411yescharacteristic
72C2×#11924012—21no—
73C2×#11924011—21no—
74C22×#10596831—21no—
75C2×C8×D73—21no—
76C2×C8×D73—21no—
77C2×C8×D73—21no—
78C2×C8×D73—44no—
79C2×C8×D73D411yescharacteristic
80C2×C8×D73D411yescharacteristic
81#14556584—21no—
82#14556582C2×C411yescharacteristic
83#14556578C2×C411yescharacteristic
84#14556573—21no—
85#14556571C2×C411yescharacteristic
86#14556562C2×C411yescharacteristic
87C4×Dic146C2311yescharacteristic
88C4×D292C2311yescharacteristic
89D4×Dic73C2×C411yescharacteristic
90#14556553C2×C411yescharacteristic
91C4×C22⋊D73—21no—
92C4×C22⋊D73—21no—
93#14556551—21no—
94#14556550—21no—
95C4⋊C4×D73C2×C411yescharacteristic
96C22⋊C4×D73—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#14556520C2×C411yescharacteristic
107#14556520C2×C411yescharacteristic
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#14556503C2×C411yescharacteristic
120#14556503C2×C411yescharacteristic
121#14556501—21no—
122#14556501—21no—
123#14556501—21no—
124C4×#10596831C2311yescharacteristic
125C4×#10596831C2311yescharacteristic
126C4×#10596831C2311yescharacteristic
127C8×Dic73—21no—
128C8×Dic73D411yescharacteristic
129C8×Dic73D411yescharacteristic
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—
146C2×Dic146C22×C411yescharacteristic
147C2×D292C22×C411yescharacteristic
148C2×C22⋊D73—21no—
149C2×C22⋊D73—21no—
150D4×D73—22no—
151D4×D73—22no—
152C22×Dic73—21no—
153C2×C4×D73—21no—
154C2×C4×D73—21no—
155C2×C4×D73—21no—
156C2×C4×D73—42no—
157C2×C4×D73—42no—
158C2×C4×D73C22×C411yescharacteristic
159C2×C4×D73C2×D411yescharacteristic
160C2×C4×D73#4111yescharacteristic
161#11924017—22no—
162#11924016—22no—
163#11924013—22no—
164#11924012—42no—
165#11924011—24no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—21no—
169C2×#10596831—42no—
170C2×#10596831C22×C411yescharacteristic
171C2×#10596831C22×C411yescharacteristic
172C2×#10596831C2×D411yescharacteristic
173C2×#10596831C2×D411yescharacteristic
174C8×D73—22no—
175C8×D73—41no—
176C8×D73—42no—
177C8×D73—42no—
178C8×D73—42no—
179C4⋊Dic73—21no—
180C4⋊Dic73—21no—
181C4⋊Dic73C22×C411yescharacteristic
182C4⋊Dic73C22×C411yescharacteristic
183#11924007—21no—
184#11924007—21no—
185#11924007—21no—
186C22⋊Dic73—21no—
187C22⋊Dic73—21no—
188C22⋊Dic73—21no—
189C22⋊Dic73—21no—
190#11924005—21no—
191#11924005—21no—
192#11924005—21no—
193C4×Dic73—21no—
194C4×Dic73C22×C411yescharacteristic
195C4×Dic73C2×D411yescharacteristic
196C4×Dic73#4111yescharacteristic
197C2×D4×C73C22×C411yescharacteristic
198C2×D4×C73C22×C411yescharacteristic
199C22×C4×C73—21no—
200C22×C4×C73—21no—
201D8×C73—22no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73—42no—
206C2×C8×C73—42no—
207C2×C8×C73C22×C411yescharacteristic
208C2×C8×C73C22×C411yescharacteristic
209C2×C8×C73C2×D411yescharacteristic
210C2×C8×C73C2×D411yescharacteristic
211C4⋊C4×C73C22×C411yescharacteristic
212C4⋊C4×C73C22×C411yescharacteristic
213C22⋊C4×C73—21no—
214C22⋊C4×C73—21no—
215C42×C73C2411yescharacteristic
216C23×C73—21no—
217C23×C73—21no—
218C22×D73—24no—
219C22×D73—21no—
220C22×D73—42no—
221C22×D73—42no—
222C22×D73C4×D411yescharacteristic
223D4×C73—21no—
224D4×C73—21no—
225D4×C73—21no—
226D4×C73#13112yes—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73C23×C411yescharacteristic
234C2×C4×C73C22×D411yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C22⋊D73—24no—
237C22⋊D73—42no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—42no—
242C2×Dic73C4×D411yescharacteristic
243C2×Dic73C4×D411yescharacteristic
244C2×Dic73C4×D411yescharacteristic
245D292—21no—
246D292#13112yes—
247C4×D73—22no—
248C4×D73—21no—
249C4×D73—21no—
250C4×D73—21no—
251C4×D73—21no—
252C4×D73—42no—
253C4×D73—42no—
254C4×D73—41no—
255C4×D73—41no—
256Dic146—21no—
257Dic146—21no—
258C8×C73—22no—
259C8×C73—21no—
260C8×C73—21no—
261C8×C73—21no—
262C8×C73—41no—
263C8×C73—41no—
264C8×C73AGL(1,ℤ/8)12yes—
265#10596831—22no—
266#10596831—21no—
267#10596831—21no—
268#10596831—21no—
269#10596831—21no—
270#10596831—41no—
271#10596831—41no—
272C22×C73—22no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73—41no—
278C22×C73C2×C4×D411yescharacteristic
279C2×D73—21no—
280C2×D73—21no—
281C2×D73—22no—
282C2×D73—48no—
283C2×D73—41no—
284C2×D73—41no—
285C2×D73#44312yes—
286C4×C73—21no—
287C4×C73—41no—
288C4×C73—41no—
289C4×C73C2×#13111yescharacteristic
290C4×C73C2×#13111yescharacteristic
291C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
292C4×C73#57511yescharacteristic
293Dic73—21no—
294Dic73—21no—
295Dic73—21no—
296Dic73—41no—
297Dic73—41no—
298Dic73—41no—
299#17910—731nomaximal, Sylow
300C2×C73—42no—
301C2×C73—81no—
302C2×C73#292911yescharacteristic
303C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C73C2×#44311yescharacteristic
305D73—22no—
306D73—42no—
307#3105—731no—
308#2966—731no—
309C2×C8×D4—731no—
310#1586—731no—
311#1586—731no—
312#1585—731no—
313#1585—731no—
314C8×SD16—731no—
315C8×SD16—731no—
316C8×D8—731no—
317C8×D8D7311yescharacteristic, complemented
318#1573—731no—
319#1490—731no—
320#1483—731no—
321C8×M16—731no—
322C73#1791011yescharacteristic, complemented, Sylow
323C23×C8—1461no—
324#529—731no—
325C2×C4×D4—731no—
326#484—731no—
327#481—731no—
328#478—731no—
329#478—731no—
330#463—731no—
331#463—731no—
332#450—731no—
333#447—731no—
334#446—731no—
335C8×Q8—731no—
336C4×SD16—731no—
337C4×SD16—731no—
338C4×D8—731no—
339C4×D8C2×D7311yescharacteristic
340#436—731no—
341#435—1461no—
342C8×D4—731no—
343C8×D4—731no—
344C8×D4—731no—
345C8×D4—1461no—
346C8×D4—1461no—
347C8×D4—1461no—
348C8×D4C2×D7311yescharacteristic
349C8×D4C2×D7311yescharacteristic, complemented
350#433—1461no—
351#432—731no—
352#431—731no—
353C2×C4⋊C8—731no—
354C2×C22⋊C8—1461no—
355#405—1461no—
356C4×M16—731no—
357C2×C4×C8—731no—
358#335—731no—
359#335C2×D7311yescharacteristic, complemented
360#334—731no—
361#334—731no—
362Q8⋊C8—731no—
363Q8⋊C8—731no—
364D4⋊C8—731no—
365D4⋊C8—731no—
366D4⋊C8—731no—
367D4⋊C8—731no—
368D4⋊C8C2×D7311yescharacteristic
369D4⋊C8C2×D7311yescharacteristic, complemented
370#322—731no—
371#322—731no—
372C82—731no—
373C82C2×D7311yescharacteristic, complemented
374C22×D4—731no—
375C23×C4—1461no—
376C2×SD16—731no—
377C2×SD16—731no—
378C2×D8—731no—
379C2×D8C4×D7311yescharacteristic
380C2×M16—1461no—
381C2×M16—1461no—
382C22×C8—21no—
383C22×C8—21no—
384C22×C8—2924no—
385C22×C8—731no—
386C22×C8—731no—
387C22×C8—1461no—
388C22×C8—1461no—
389C22×C8—1461no—
390C22×C8—1461no—
391#126—1461no—
392#125—731no—
393#124—731no—
394#123—1461no—
395C22⋊Q8—731no—
396#121—731no—
397C4×Q8—731no—
398C4×D4—731no—
399C4×D4—731no—
400C4×D4—731no—
401C4×D4—1461no—
402C4×D4—1461no—
403C4×D4—1461no—
404C4×D4C22×D7311yescharacteristic
405C4×D4C22×D7311yescharacteristic
406#117—1461no—
407C2×C4⋊C4—731no—
408C2×C22⋊C4—1461no—
409C2×C42—731no—
410#107—731no—
411#107C4×D7311yescharacteristic
412#106—731no—
413#106—731no—
414C4⋊C8—731no—
415C4⋊C8—731no—
416C4⋊C8—731no—
417C4⋊C8—731no—
418C4⋊C8—1461no—
419C4⋊C8—1461no—
420C4⋊C8—1461no—
421C4⋊C8C22×D7311yescharacteristic
422C4⋊C8C22×D7311yescharacteristic
423#103—731no—
424#103—731no—
425#102—731no—
426#102—731no—
427#102—731no—
428#102—731no—
429#102C4×D7311yescharacteristic
430#102C4×D7311yescharacteristic
431C22⋊C8—21no—
432C22⋊C8—21no—
433C22⋊C8—731no—
434C22⋊C8—731no—
435C22⋊C8—1461no—
436C22⋊C8—1461no—
437C22⋊C8—1461no—
438C22⋊C8—1461no—
439C22⋊C8—1461no—
440#97—1461no—
441#97—1461no—
442#97—1461no—
443C4×C8—731no—
444C4×C8—731no—
445C4×C8—731no—
446C4×C8—731no—
447C4×C8—731no—
448C4×C8—1461no—
449C4×C8C22×D7311yescharacteristic
450C4×C8C22×D7311yescharacteristic
451C4×C8C22×D7311yescharacteristic
452C24—1461no—
453C2×Q8—731no—
454C2×D4—731no—
455C2×D4—1461no—
456C2×D4—1462no—
457C2×D4—1461no—
458C2×D4—1462no—
459C2×D4C2×C4×D7311yescharacteristic
460C2×D4C2×C4×D7311yescharacteristic
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—2922no—
464C22×C4—2922no—
465C22×C4—731no—
466C22×C4—731no—
467C22×C4—731no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4—1461no—
471C22×C4—1461no—
472SD16—1462no—
473SD16—1462no—
474D8—22no—
475D8—1462no—
476M16—2922no—
477M16—1464no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—2921no—
484C2×C8—2922no—
485C2×C8—2922no—
486C2×C8—2922no—
487C2×C8—2922no—
488C2×C8—731no—
489C2×C8—731no—
490C2×C8—731no—
491C2×C8—731no—
492C2×C8—1461no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8—1462no—
496C2×C8C2×C4×D7311yescharacteristic
497C2×C8C2×C4×D7311yescharacteristic
498C2×C8D4×D7311yescharacteristic
499C2×C8D4×D7311yescharacteristic
500C4⋊C4—731no—
501C4⋊C4—731no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4—1461no—
506C4⋊C4—1461no—
507C4⋊C4C2×C4×D7311yescharacteristic
508C4⋊C4C2×C4×D7311yescharacteristic
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—1461no—
512C22⋊C4—1461no—
513C22⋊C4—1461no—
514C22⋊C4—1461no—
515C22⋊C4—1461no—
516C22⋊C4—1461no—
517C22⋊C4—1461no—
518C42—731no—
519C42—731no—
520C42—731no—
521C42—1461no—
522C42C23×D7311yescharacteristic
523C23—21no—
524C23—21no—
525C23—2922no—
526C23—2922no—
527C23—731no—
528C23—1464no—
529C23—1461no—
530Q8—1461no—
531Q8—1461no—
532D4—21no—
533D4—21no—
534D4—21no—
535D4—2922no—
536D4—732no—
537D4—1464no—
538D4—1461no—
539D4#1455659712yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—2922no—
547C2×C4—2922no—
548C2×C4—2922no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—731no—
552C2×C4—731no—
553C2×C4—731no—
554C2×C4—1462no—
555C2×C4—1461no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4C22×C4×D7311yescharacteristic
563C2×C4C2×D4×D7311yescharacteristic
564C2×C4#41×D7311yescharacteristic
565C8—22no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—41no—
570C8—41no—
571C8—2921no—
572C8—2921no—
573C8—1462no—
574C8—1461no—
575C8—1461no—
576C8—1461no—
577C8—1461no—
578C8#1455661512yes—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—2928no—
586C22—2921no—
587C22—2921no—
588C22—732no—
589C22—1462no—
590C22—1461no—
591C22—1461no—
592C22C4×D4×D7311yescharacteristic
593C4—21no—
594C4—41no—
595C4—41no—
596C4—2921no—
597C4—2921no—
598C4—2921no—
599C4—1461no—
600C4—1461no—
601C4—1461no—
602C4#131×D7311yescharacteristic
603C4C2×#1455659711yescharacteristic
604C4C2×#1455661511yescharacteristic
605C4#1798938411yescharacteristic
606C2—42no—
607C2—81no—
608C2—2922no—
609C2—1462no—
610C2#2261862011yescharacteristic
611C2C4×#1455661511yescharacteristic
612C2#443×D7311yescharacteristic
subgroupG/H
centerC2×C8D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC8×D8×C73C2
socleC22×C73C2×C4×D4
prime radical#17988719C4
cosocleC42×C73C24
hypercenterC8×D8D73
hypocenterC73#17910
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ C8×D8

Complements

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

NH ≅ G/Ntype
2. #22618903C2semidirect
3. C8×D4×D73C2semidirect
4. #22608798C2semidirect
5. #22608796C2semidirect
6. #22608787C2semidirect
7. #22608785C2semidirect
8. C8×#11924017C2semidirect
9. C8×#11924016C2semidirect
10. C8×#11924013C2semidirect
11. #22608715C2semidirect
12. #22608372C2semidirect
13. #22608341C2semidirect
14. C8×#11924011C2semidirect
15. C8×D8×C73C2semidirect
27. #17988714C22semidirect
28. C8×Dic146C22semidirect
32. #17988665C22semidirect
48. #17988339C22semidirect
49. #17988338C22semidirect
50. #17988337C22semidirect
51. #17988319C22semidirect
52. #17988317C22semidirect
55. D4⋊#10596831C22semidirect
57. #17988309C22semidirect
58. #17988308C22semidirect
59. C8×#10596831C22semidirect
62. C8×D4×C73C22semidirect
63. #335×C73C22semidirect
65. D4⋊C8×C73C22semidirect
66. C82×C73C22semidirect
317. C8×D8D73semidirect
322. C73#17910semidirect
349. C8×D4C2×D73semidirect
359. #335C2×D73semidirect
369. D4⋊C8C2×D73semidirect
373. C82C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226199762
#226189032
C8×D4×D732
#226087982
#226087962
#226087872
#226087852
C8×#119240172
C8×#119240162
C8×#119240132
#226087152
#226083722
#226083412
C8×#119240112
C8×D8×C732
#1791073