Finite Group Databasebeta

← #33057281

Subgroups — #33057281

Order 18688.

656 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619929C211yesmaximal, characteristic
2#22618994C211yesmaximal, characteristic, complemented
3#22618964C211yesmaximal, characteristic, complemented
4#22608842C211yesmaximal, characteristic, complemented
5#22608817C211yesmaximal, characteristic, complemented
6#22608795C211yesmaximal, characteristic, complemented
7#22608795C211yesmaximal, characteristic, complemented
8#22608761C211yesmaximal, characteristic, complemented
9#22608759C211yesmaximal, characteristic, complemented
10#22608752C211yesmaximal, characteristic, complemented
11#22608710C211yesmaximal, characteristic, complemented
12#22608398C211yesmaximal, characteristic, complemented
13#22608320C211yesmaximal, characteristic, complemented
14#22608230C211yesmaximal, characteristic, complemented
15C8×D8×C73C211yesmaximal, characteristic, complemented
16#17989134C2211yescharacteristic
17C4×D4×D73C2211yescharacteristic
18#17988902C411yescharacteristic
19#17988899C411yescharacteristic
20#17988866C411yescharacteristic
21#17988865C411yescharacteristic
22#17988781C411yescharacteristic
23#17988780C411yescharacteristic
24#17988722C411yescharacteristic
25#17988719C411yescharacteristic
26C4×D584C2211yescharacteristic
27C4×#11924013C2211yescharacteristic
28C4×#11924013C2211yescharacteristic
29#17988666C2211yescharacteristic
30#17988666C2211yescharacteristic, complemented
31#17988665C2211yescharacteristic, complemented
32#17988664C2211yescharacteristic
33#17988662—21no—
34#17988658—21no—
35#17988655—21no—
36D4×#10596831C2211yescharacteristic
37#17988652C2211yescharacteristic, complemented
38C8×C22⋊D73—21no—
39#17988643C2211yescharacteristic
40#17988639C2211yescharacteristic
41#17988638C2211yescharacteristic
42#17988638C2211yescharacteristic
43#17988638C2211yescharacteristic
44#17988553—21no—
45#17988552—21no—
46#17988549—21no—
47C2×#14556503—21no—
48#17988340C2211yescharacteristic, complemented
49#17988337C2211yescharacteristic, complemented
50#17988337C2211yescharacteristic, complemented
51#17988316C2211yescharacteristic
52#17988316C2211yescharacteristic, complemented
53D4⋊#10596831C2211yescharacteristic
54D4⋊#10596831C2211yescharacteristic, complemented
55#17988314C2211yescharacteristic
56#17988314C2211yescharacteristic, complemented
57#17988309C2211yescharacteristic, complemented
58#17988309C2211yescharacteristic, complemented
59#17988309C2211yescharacteristic, 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×#11924028—21no—
68C2×D4×D73C2×C411yescharacteristic
69C22×C4×D73—21no—
70C2×D584C2×C411yescharacteristic
71C2×#11924013C2×C411yescharacteristic
72C2×#11924013C2×C411yescharacteristic
73C2×#11924012—21no—
74C2×#11924011—21no—
75C2×#11924011—21no—
76C2×#11924011—21no—
77C22×#10596831—21no—
78C2×C8×D73—21no—
79#14556590C2×C411yescharacteristic
80#14556587C2×C411yescharacteristic
81#14556577—21no—
82#14556566C2×C411yescharacteristic
83#14556564—21no—
84#14556562C2×C411yescharacteristic
85C4×D292C2311yescharacteristic
86C4×D292C2311yescharacteristic
87D4×Dic73C2×C411yescharacteristic
88#14556553C2×C411yescharacteristic
89C4×C22⋊D73—21no—
90C4×C22⋊D73—21no—
91#14556551—21no—
92C4⋊C4×D73C2×C411yescharacteristic
93C22⋊C4×D73—21no—
94C42×D73C2311yescharacteristic
95#14556525C2×C411yescharacteristic
96#14556524C2×C411yescharacteristic
97#14556524C2×C411yescharacteristic
98C4⋊#10596831C2311yescharacteristic
99C4⋊#10596831C2311yescharacteristic
100#14556520—21no—
101#14556520C2×C411yescharacteristic
102#14556520C2×C411yescharacteristic
103#14556520D411yescharacteristic
104#14556520D411yescharacteristic
105#14556513C2×C411yescharacteristic
106#14556513C2×C411yescharacteristic
107#14556512C2×C411yescharacteristic
108#14556512C2×C411yescharacteristic
109#14556511C2×C411yescharacteristic
110#14556511C2×C411yescharacteristic
111C22⋊#10596831—21no—
112C22⋊#10596831—21no—
113#14556503—21no—
114#14556503—21no—
115#14556503—21no—
116#14556503—21no—
117#14556503—21no—
118#14556503—21no—
119#14556503—21no—
120#14556503C2×C411yescharacteristic
121#14556503C2×C411yescharacteristic
122#14556503D411yescharacteristic
123#14556503D411yescharacteristic
124#14556501—21no—
125C4×#10596831C2311yescharacteristic
126C4×#10596831C2311yescharacteristic
127C4×#10596831C2311yescharacteristic
128C8×Dic73—21no—
129C2×D8×C73C2×C411yescharacteristic
130C22×C8×C73—21no—
131C22×C8×C73—21no—
132C4×D4×C73C2311yescharacteristic
133C4×D4×C73C2311yescharacteristic
134#107×C73C2×C411yescharacteristic
135C4⋊C8×C73C2311yescharacteristic
136C4⋊C8×C73C2311yescharacteristic
137#102×C73C2×C411yescharacteristic
138#102×C73C2×C411yescharacteristic
139C22⋊C8×C73—21no—
140C22⋊C8×C73—21no—
141C4×C8×C73C2311yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C23×D73—21no—
145#11924028—81no—
146#11924028—81no—
147C2×Dic146—21no—
148C2×D292—21no—
149C2×D292C22×C411yescharacteristic
150C2×D292C22×C411yescharacteristic
151C2×C22⋊D73—21no—
152C2×C22⋊D73—21no—
153C2×C22⋊D73—21no—
154C2×C22⋊D73—21no—
155D4×D73—22no—
156D4×D73—41no—
157C22×Dic73—21no—
158C2×C4×D73—21no—
159C2×C4×D73—21no—
160C2×C4×D73—21no—
161C2×C4×D73—21no—
162C2×C4×D73—81no—
163C2×C4×D73—81no—
164C2×C4×D73C22×C411yescharacteristic
165C2×C4×D73C2×D411yescharacteristic
166C2×C4×D73#4111yescharacteristic
167D584—41no—
168#11924013—22no—
169#11924013—41no—
170#11924012—81no—
171#11924011—81no—
172#11924011—81no—
173#11924011—81no—
174C2×#10596831—21no—
175C2×#10596831—21no—
176C2×#10596831—21no—
177C2×#10596831—81no—
178C2×#10596831C22×C411yescharacteristic
179C2×#10596831C22×C411yescharacteristic
180C2×#10596831C2×D411yescharacteristic
181C2×#10596831C2×D411yescharacteristic
182C8×D73—81no—
183C4⋊Dic73C22×C411yescharacteristic
184C4⋊Dic73C22×C411yescharacteristic
185#11924007—21no—
186#11924007—21no—
187C22⋊Dic73—21no—
188C22⋊Dic73—21no—
189#11924005—21no—
190#11924005—21no—
191#11924005—21no—
192#11924005—21no—
193C4×Dic73C22×C411yescharacteristic
194C4×Dic73C2×D411yescharacteristic
195C4×Dic73#4111yescharacteristic
196C2×D4×C73C22×C411yescharacteristic
197C2×D4×C73C22×C411yescharacteristic
198C22×C4×C73—21no—
199C22×C4×C73—21no—
200D8×C73—22no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—81no—
205C2×C8×C73—81no—
206C2×C8×C73C22×C411yescharacteristic
207C2×C8×C73C22×C411yescharacteristic
208C2×C8×C73C2×D411yescharacteristic
209C2×C8×C73C2×D411yescharacteristic
210C4⋊C4×C73C22×C411yescharacteristic
211C4⋊C4×C73C22×C411yescharacteristic
212C22⋊C4×C73—21no—
213C22⋊C4×C73—21no—
214C42×C73C2411yescharacteristic
215C23×C73—21no—
216C23×C73—21no—
217C22×D73—21no—
218C22×D73—21no—
219C22×D73—42no—
220C22×D73—42no—
221C22×D73—81no—
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—42no—
237C22⋊D73—42no—
238C22⋊D73—81no—
239C22⋊D73—81no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—81no—
243C2×Dic73C4×D411yescharacteristic
244C2×Dic73C4×D411yescharacteristic
245C2×Dic73C4×D411yescharacteristic
246D292—21no—
247D292—21no—
248D292—21no—
249D292—21no—
250D292—81no—
251C4×D73—21no—
252C4×D73—21no—
253C4×D73—21no—
254C4×D73—21no—
255C4×D73—41no—
256C4×D73—41no—
257C4×D73—41no—
258C4×D73—41no—
259C4×D73—41no—
260C4×D73—81no—
261C4×D73—81no—
262Dic146—81no—
263C8×C73—21no—
264C8×C73—21no—
265C8×C73—21no—
266C8×C73—21no—
267C8×C73—41no—
268C8×C73—41no—
269C8×C73—41no—
270#10596831—21no—
271#10596831—21no—
272#10596831—21no—
273#10596831—21no—
274#10596831—41no—
275#10596831—41no—
276#10596831—41no—
277C22×C73—22no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73—41no—
282C22×C73—41no—
283C22×C73C2×C4×D411yescharacteristic
284C2×D73—21no—
285C2×D73—21no—
286C2×D73—21no—
287C2×D73—41no—
288C2×D73—41no—
289C2×D73—41no—
290C2×D73—41no—
291C2×D73—41no—
292C2×D73—41no—
293C2×D73—84no—
294C4×C73—21no—
295C4×C73—41no—
296C4×C73—41no—
297C4×C73C2×#13111yescharacteristic
298C4×C73#56811yescharacteristic
299C4×C73#57611yescharacteristic
300C4×C73#57611yescharacteristic
301Dic73—21no—
302Dic73—21no—
303Dic73—21no—
304Dic73—41no—
305Dic73—41no—
306#18026—731nomaximal, Sylow
307C2×C73—42no—
308C2×C73—81no—
309C2×C73#293311yescharacteristic
310C2×C73#294811yescharacteristic
311C2×C73#295511yescharacteristic
312D73—41no—
313D73—81no—
314D73—81no—
315#3100—731no—
316#2979—731no—
317#2974—731no—
318#1594—731no—
319#1589—731no—
320#1586—731no—
321#1586—731no—
322#1581—731no—
323#1581—731no—
324#1580—731no—
325C8×D8D7311yescharacteristic, complemented
326#1572—731no—
327#1496—731no—
328#1478—731no—
329#1454—731no—
330C73#1802611yescharacteristic, complemented, Sylow
331#532—731no—
332C2×C4×D4—731no—
333#480—731no—
334#480—731no—
335#474—731no—
336#474—731no—
337#459—731no—
338#459—731no—
339#447—731no—
340#447—731no—
341C4×D8—731no—
342C4×D8—731no—
343C4×D8—731no—
344C4×D8C2×D7311yescharacteristic
345#436—731no—
346#436—731no—
347#436—731no—
348#436—731no—
349#436—1461no—
350#435—1461no—
351#435—1461no—
352C8×D4—731no—
353C8×D4—731no—
354C8×D4—1461no—
355C8×D4C2×D7311yescharacteristic
356C8×D4C2×D7311yescharacteristic, complemented
357#432—731no—
358#431—731no—
359#431—731no—
360#431—731no—
361#431—731no—
362#408—1461no—
363#408—1461no—
364#407—1461no—
365C2×C22⋊C8—1461no—
366#335—731no—
367#335C2×D7311yescharacteristic, complemented
368#334—731no—
369#334—731no—
370D4⋊C8—731no—
371D4⋊C8—731no—
372D4⋊C8—731no—
373D4⋊C8—731no—
374D4⋊C8—731no—
375D4⋊C8—731no—
376D4⋊C8C2×D7311yescharacteristic
377D4⋊C8C2×D7311yescharacteristic, complemented
378#322—731no—
379#322—731no—
380#322—731no—
381C82C2×D7311yescharacteristic, complemented
382C2×#41—1461no—
383C22×D4—731no—
384C23×C4—1461no—
385C2×D8—731no—
386C2×D8—731no—
387C2×D8—731no—
388C2×D8C4×D7311yescharacteristic
389C2×M16—1461no—
390C2×M16—1461no—
391C2×M16—1461no—
392C2×M16—1461no—
393C22×C8—21no—
394C22×C8—21no—
395C22×C8—1461no—
396C22×C8—1461no—
397C4⋊Q8—731no—
398#127—731no—
399#124—1461no—
400#121—731no—
401#121—731no—
402#121—1461no—
403C4×D4—731no—
404C4×D4—731no—
405C4×D4—731no—
406C4×D4—731no—
407C4×D4—1461no—
408C4×D4—1461no—
409C4×D4—1461no—
410C4×D4C22×D7311yescharacteristic
411C4×D4C22×D7311yescharacteristic
412C2×C4⋊C4—731no—
413C2×C22⋊C4—1461no—
414C2×C42—731no—
415#107—731no—
416#107—731no—
417#107—731no—
418#107C4×D7311yescharacteristic
419C4⋊C8—731no—
420C4⋊C8—731no—
421C4⋊C8—731no—
422C4⋊C8—731no—
423C4⋊C8—731no—
424C4⋊C8—731no—
425C4⋊C8—1461no—
426C4⋊C8C22×D7311yescharacteristic
427C4⋊C8C22×D7311yescharacteristic
428#102—731no—
429#102—731no—
430#102—731no—
431#102—731no—
432#102—731no—
433#102—731no—
434#102C4×D7311yescharacteristic
435#102C4×D7311yescharacteristic
436C22⋊C8—21no—
437C22⋊C8—21no—
438C22⋊C8—731no—
439C22⋊C8—731no—
440C22⋊C8—731no—
441C22⋊C8—731no—
442C22⋊C8—1461no—
443C22⋊C8—1461no—
444C22⋊C8—1461no—
445C22⋊C8—1461no—
446C22⋊C8—1461no—
447C22⋊C8—1461no—
448C22⋊C8—1461no—
449C22⋊C8—1461no—
450C22⋊C8—1461no—
451#97—1461no—
452C4×C8—731no—
453C4×C8—731no—
454C4×C8—731no—
455C4×C8—1461no—
456C4×C8C22×D7311yescharacteristic
457C4×C8C22×D7311yescharacteristic
458C4×C8C22×D7311yescharacteristic
459C24—1461no—
460#41—5841no—
461#41—5841no—
462C2×Q8—1461no—
463C2×D4—2921no—
464C2×D4—731no—
465C2×D4—731no—
466C2×D4—1461no—
467C2×D4—1461no—
468C2×D4—1461no—
469C2×D4—1461no—
470C2×D4—1462no—
471C2×D4—1461no—
472C2×D4C2×C4×D7311yescharacteristic
473C2×D4C2×C4×D7311yescharacteristic
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—5841no—
477C22×C4—5841no—
478C22×C4—731no—
479C22×C4—731no—
480C22×C4—731no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4—1461no—
484C22×C4—1461no—
485C22×C4—1461no—
486D8—22no—
487D8—2921no—
488D8—2921no—
489D8—1462no—
490M16—5841no—
491M16—5841no—
492M16—5841no—
493M16—5841no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—81no—
498C2×C8—81no—
499C2×C8—5841no—
500C2×C8—5841no—
501C2×C8—731no—
502C2×C8—731no—
503C2×C8—731no—
504C2×C8—731no—
505C2×C8—1461no—
506C2×C8—1461no—
507C2×C8—1461no—
508C2×C8C2×C4×D7311yescharacteristic
509C2×C8C2×C4×D7311yescharacteristic
510C2×C8D4×D7311yescharacteristic
511C2×C8D4×D7311yescharacteristic
512C4⋊C4—731no—
513C4⋊C4—731no—
514C4⋊C4—1461no—
515C4⋊C4—1461no—
516C4⋊C4C2×C4×D7311yescharacteristic
517C4⋊C4C2×C4×D7311yescharacteristic
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—1461no—
521C22⋊C4—1461no—
522C22⋊C4—1461no—
523C22⋊C4—1461no—
524C22⋊C4—1461no—
525C22⋊C4—1461no—
526C42—731no—
527C42—731no—
528C42—731no—
529C42C23×D7311yescharacteristic
530C23—21no—
531C23—21no—
532C23—2922no—
533C23—2922no—
534C23—5841no—
535C23—731no—
536C23—1461no—
537C23—1461no—
538Q8—5841no—
539D4—21no—
540D4—21no—
541D4—21no—
542D4—2922no—
543D4—2922no—
544D4—5841no—
545D4—5841no—
546D4—5841no—
547D4—1461no—
548D4—1461no—
549D4—1461no—
550D4—1461no—
551D4#1455659712yes—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—5841no—
564C2×C4—5841no—
565C2×C4—5841no—
566C2×C4—731no—
567C2×C4—731no—
568C2×C4—731no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4—1461no—
573C2×C4—1461no—
574C2×C4—1461no—
575C2×C4C22×C4×D7311yescharacteristic
576C2×C4C2×D4×D7311yescharacteristic
577C2×C4#41×D7311yescharacteristic
578C8—21no—
579C8—21no—
580C8—21no—
581C8—21no—
582C8—41no—
583C8—41no—
584C8—41no—
585C8—2921no—
586C8—2921no—
587C8—2921no—
588C8—1461no—
589C8—1461no—
590C8—1461no—
591C8—1461no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—2921no—
599C22—2921no—
600C22—2921no—
601C22—2921no—
602C22—2921no—
603C22—2921no—
604C22—5844no—
605C22—1461no—
606C22—1461no—
607C22—1461no—
608C22C4×D4×D7311yescharacteristic
609C4—21no—
610C4—41no—
611C4—41no—
612C4—2921no—
613C4—2921no—
614C4—1461no—
615C4—1461no—
616C4—1461no—
617C4C2×#1455659711yescharacteristic
618C4#1798935211yescharacteristic
619C4#1798938811yescharacteristic
620C4#1798938911yescharacteristic
621C2—42no—
622C2—81no—
623C2—2921no—
624C2—5841no—
625C2—5841no—
626C2#2261865111yescharacteristic
627C2#2261877811yescharacteristic
628C2#2261882911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC8×D8×C73C2
socleC22×C73C2×C4×D4
prime radical#22619929C2
cosocleC42×C73C24
hypercenterC8×D8D73
hypocenterC73#18026
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×D8

Complements

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

NH ≅ G/Ntype
2. #22618994C2semidirect
3. #22618964C2semidirect
4. #22608842C2semidirect
5. #22608817C2semidirect
6. #22608795C2semidirect
7. #22608795C2semidirect
8. #22608761C2semidirect
9. #22608759C2semidirect
10. #22608752C2semidirect
11. #22608710C2semidirect
12. #22608398C2semidirect
13. #22608320C2semidirect
14. #22608230C2semidirect
15. C8×D8×C73C2semidirect
30. #17988666C22semidirect
31. #17988665C22semidirect
37. #17988652C22semidirect
48. #17988340C22semidirect
49. #17988337C22semidirect
50. #17988337C22semidirect
52. #17988316C22semidirect
54. D4⋊#10596831C22semidirect
56. #17988314C22semidirect
57. #17988309C22semidirect
58. #17988309C22semidirect
59. #17988309C22semidirect
62. C8×D4×C73C22semidirect
63. #335×C73C22semidirect
65. D4⋊C8×C73C22semidirect
66. C82×C73C22semidirect
325. C8×D8D73semidirect
330. C73#18026semidirect
356. C8×D4C2×D73semidirect
367. #335C2×D73semidirect
377. D4⋊C8C2×D73semidirect
381. C82C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226199292
#226189942
#226189642
#226088422
#226088172
#226087952
#226087612
#226087592
#226087522
#226087102
#226083982
#226083202
#226082302
C8×D8×C732
#1802673