Finite Group Databasebeta

← #34019252

Subgroups — #34019252

Order 18688.

600 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626857C211yesmaximal, characteristic
2#22626840C211yesmaximal, characteristic, complemented
3#22625571C211yesmaximal, characteristic, complemented
4#22625569C211yesmaximal, characteristic, complemented
5#22625557C211yesmaximal, characteristic, complemented
6#22625556C211yesmaximal, characteristic, complemented
7#22625548C211yesmaximal, characteristic, complemented
8#22625546C211yesmaximal, characteristic, complemented
9#22625514C211yesmaximal, characteristic, complemented
10SD16×#10596833C211yesmaximal, characteristic, complemented
11#22625507C211yesmaximal, characteristic, complemented
12#22625279C211yesmaximal, characteristic, complemented
13#22625222C211yesmaximal, characteristic
14#22625136C211yesmaximal, characteristic, complemented
15#22619995C211yesmaximal, characteristic
16#17989639—21no—
17#17989638C2211yescharacteristic
18#17989638C2211yescharacteristic
19#17989638C2211yescharacteristic
20#17989637C2211yescharacteristic
21#17989636—21no—
22#17989636C2211yescharacteristic
23Q8×#10596833C2211yescharacteristic
24#17989618—21no—
25#17989618—21no—
26D4×#10596833C2211yescharacteristic
27D4×#10596833C2211yescharacteristic
28#17989611—21no—
29#17989611—21no—
30#17989607—21no—
31#17989604C2211yescharacteristic
32#17989603C2211yescharacteristic
33#17989590C2211yescharacteristic
34C2×#14556658—21no—
35C2×C4×#10596833C2211yescharacteristic
36#17989451C2211yescharacteristic
37#17989450C2211yescharacteristic
38#17989449C2211yescharacteristic
39#17989448C2211yescharacteristic
40#17989437C2211yescharacteristic
41#17989437C2211yescharacteristic
42Q8⋊#10596833C2211yescharacteristic
43Q8⋊#10596833C2211yescharacteristic
44Q8⋊#10596833C2211yescharacteristic
45Q8⋊#10596833C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic
47D4⋊#10596833C2211yescharacteristic
48#17989429C2211yescharacteristic
49#17989429C2211yescharacteristic
50C8×#10596833C2211yescharacteristic
51C8×#10596833C2211yescharacteristic
52#17989139C2211yescharacteristic
53#41×Dic73C2211yescharacteristic
54#17988929C411yescharacteristic
55#17988925C411yescharacteristic
56#17988870C411yescharacteristic
57#17988865C411yescharacteristic
58#17988795C411yescharacteristic
59#17988790C411yescharacteristic
60#17988749C411yescharacteristic
61Q16×Dic73C2211yescharacteristic
62#17988679C2211yescharacteristic
63#17988678C2211yescharacteristic
64SD16×Dic73C2211yescharacteristic
65#17988637C2211yescharacteristic
66#452×C73C411yescharacteristic
67C2×#11924037—21no—
68C22×#10596833—21no—
69C22×#10596833—21no—
70C22×#10596833—21no—
71#14556658—21no—
72#14556658—21no—
73#14556658—21no—
74#14556658—21no—
75#14556658—21no—
76#14556658D411yescharacteristic
77#14556658D411yescharacteristic
78C4⋊#10596833—21no—
79C4⋊#10596833—21no—
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84C22⋊#10596833—21no—
85C22⋊#10596833—21no—
86C22⋊#10596833—21no—
87C22⋊#10596833D411yescharacteristic
88C22⋊#10596833D411yescharacteristic
89#14556646—21no—
90#14556646—21no—
91#14556646—21no—
92#14556646—21no—
93C4×#10596833—21no—
94C4×#10596833—21no—
95C4×#10596833—21no—
96C4×#10596833—21no—
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C2×#11924018C2×C411yescharacteristic
102C2×#11924016C2×C411yescharacteristic
103#14556590C2×C411yescharacteristic
104#14556589C2×C411yescharacteristic
105#14556581—21no—
106#14556570C2×C411yescharacteristic
107#14556568—21no—
108Q8×Dic73C2×C411yescharacteristic
109Q8×Dic73C2311yescharacteristic
110Q8×Dic73C2311yescharacteristic
111#14556556—21no—
112#14556556C2311yescharacteristic
113D4×Dic73—21no—
114D4×Dic73—21no—
115D4×Dic73C2×C411yescharacteristic
116D4×Dic73C2311yescharacteristic
117#14556550—21no—
118#14556550C2×C411yescharacteristic
119#14556547—21no—
120C2×C4×Dic73—21no—
121C2×C4×Dic73C2311yescharacteristic
122#14556525C2×C411yescharacteristic
123#14556524C2×C411yescharacteristic
124#14556523C2×C411yescharacteristic
125#14556522C2×C411yescharacteristic
126#14556520C2×C411yescharacteristic
127#14556520C2×C411yescharacteristic
128#14556516C2×C411yescharacteristic
129#14556516C2×C411yescharacteristic
130#14556515C2×C411yescharacteristic
131#14556515C2×C411yescharacteristic
132#14556515C2×C411yescharacteristic
133#14556512C2×C411yescharacteristic
134C22⋊#10596831C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137C2×#41×C73C2×C411yescharacteristic
138C2×Q16×C73C2×C411yescharacteristic
139C2×SD16×C73C2×C411yescharacteristic
140C22⋊Q8×C73C2×C411yescharacteristic
141#103×C73C2×C411yescharacteristic
142#102×C73C2×C411yescharacteristic
143C22⋊C8×C73C2×C411yescharacteristic
144#11924037—81no—
145C2×#10596833—24no—
146C2×#10596833—21no—
147C2×#10596833—21no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—81no—
153C2×#10596833—81no—
154C2×#10596833C2×D411yescharacteristic
155C2×#10596833C2×D411yescharacteristic
156C2×#10596833C2×D411yescharacteristic
157C2×#10596833C2×D411yescharacteristic
158C2×Dic146—21no—
159C2×Dic146C22×C411yescharacteristic
160C22×Dic73—21no—
161C22×Dic73—21no—
162C22×Dic73C2×D411yescharacteristic
163C22×Dic73#4111yescharacteristic
164#11924018—41no—
165#11924016—22no—
166C2×#10596831C22×C411yescharacteristic
167C2×#10596831C22×C411yescharacteristic
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73C22×C411yescharacteristic
172C4⋊Dic73C22×C411yescharacteristic
173C4⋊Dic73C22×C411yescharacteristic
174#11924007—21no—
175#11924007—21no—
176#11924007—21no—
177#11924007—21no—
178#11924007—21no—
179C22⋊Dic73—21no—
180C22⋊Dic73—21no—
181C22⋊Dic73—21no—
182C22⋊Dic73—21no—
183C4×Dic73—21no—
184C4×Dic73—21no—
185C4×Dic73—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73C22×C411yescharacteristic
189C4×Dic73C2×D411yescharacteristic
190C4×Dic73#4111yescharacteristic
191C4×Dic73C2411yescharacteristic
192#41×C73—22no—
193#41×C73—41no—
194C2×Q8×C73C22×C411yescharacteristic
195C2×Q8×C73C22×C411yescharacteristic
196C2×D4×C73—21no—
197C2×D4×C73C22×C411yescharacteristic
198C22×C4×C73—21no—
199C22×C4×C73C22×C411yescharacteristic
200Q16×C73—41no—
201SD16×C73—22no—
202C2×C8×C73C22×C411yescharacteristic
203C2×C8×C73C22×C411yescharacteristic
204C4⋊C4×C73—21no—
205C4⋊C4×C73C22×C411yescharacteristic
206C22⋊C4×C73—21no—
207C23×C73—21no—
208C23×C73C4×D411yescharacteristic
209Q8×C73—21no—
210Q8×C73—21no—
211Q8×C73—21no—
212Q8×C73—21no—
213D4×C73—21no—
214D4×C73—42no—
215D4×C73#13112yes—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—42no—
222C2×C4×C73C4×D411yescharacteristic
223C2×C4×C73C23×C411yescharacteristic
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73—22no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—41no—
232C2×Dic73—41no—
233C2×Dic73—42no—
234C2×Dic73C4×D411yescharacteristic
235C2×Dic73C4×D411yescharacteristic
236C2×Dic73C22×D411yescharacteristic
237C2×Dic73C2×#4111yescharacteristic
238Dic146—21no—
239Dic146—21no—
240Dic146—42no—
241#10596833—22no—
242#10596833—22no—
243#10596833—21no—
244#10596833—21no—
245#10596833—21no—
246#10596833—21no—
247#10596833—41no—
248#10596833—41no—
249#10596833—41no—
250#10596833—41no—
251C8×C73—21no—
252C8×C73—21no—
253#10596831—21no—
254#10596831—21no—
255C22×C73—21no—
256C22×C73—22no—
257C22×C73—21no—
258C22×C73—21no—
259C22×C73—41no—
260C22×C73—41no—
261C22×C73C2×C4×D411yescharacteristic
262C4×C73—21no—
263C4×C73—41no—
264C4×C73—41no—
265C4×C73—41no—
266C4×C73C2×#13111yescharacteristic
267C4×C73#56811yescharacteristic
268Dic73—21no—
269Dic73—21no—
270Dic73—41no—
271Dic73—41no—
272Dic73—41no—
273Dic73—41no—
274Dic73#44312yes—
275Dic73#57711yescharacteristic
276Dic73#57811yescharacteristic
277#18016—731nomaximal, Sylow
278C2×C73—41no—
279C2×C73—42no—
280C2×C73#293311yescharacteristic
281C2×C73#294711yescharacteristic
282C2×C73C2×#44311yescharacteristic
283#3107—731no—
284#2990—731no—
285#2986—731no—
286#1591—731no—
287#1591—731no—
288#1587—731no—
289#1587—731no—
290#1585—731no—
291#1585—731no—
292C8×SD16—731no—
293C8×SD16—731no—
294#1574—731no—
295#1493—731no—
296#1476—731no—
297#1450—731no—
298C73#1801611yescharacteristic, complemented, Sylow
299#533—731no—
300C4×#41—731no—
301#485—731no—
302#485—731no—
303#475—731no—
304#474—731no—
305#462—731no—
306#461—731no—
307#452—731no—
308#452#6440811yescharacteristic
309#446—731no—
310#446—731no—
311#446—731no—
312#446—731no—
313#446—1461no—
314C8×Q8—731no—
315C8×Q8—731no—
316C8×Q8—1461no—
317C4×Q16—731no—
318C4×Q16—731no—
319C4×SD16—731no—
320C4×SD16—731no—
321#435—1461no—
322#435—1461no—
323C8×D4—731no—
324C8×D4—731no—
325#433—1461no—
326#433—1461no—
327#432—1461no—
328#431—731no—
329#431—731no—
330#431—731no—
331#423—731no—
332C2×C4⋊C8—1461no—
333C2×C4×C8—731no—
334#335—731no—
335#335—731no—
336#334—731no—
337#334—731no—
338Q8⋊C8—731no—
339Q8⋊C8—731no—
340Q8⋊C8—731no—
341Q8⋊C8—731no—
342Q8⋊C8—731no—
343Q8⋊C8—731no—
344D4⋊C8—731no—
345D4⋊C8—731no—
346#322—731no—
347#322—731no—
348C82—731no—
349C82—731no—
350C2×#41C2×#6440811yescharacteristic
351C2×Q16—731no—
352C2×Q16C2×#6440811yescharacteristic
353C2×SD16—731no—
354C2×SD16C2×#6440811yescharacteristic
355C2×M16—1461no—
356C22×C8—1461no—
357C22×C8—1461no—
358C22×C8—1461no—
359C4⋊Q8—731no—
360C4⋊Q8—731no—
361#125—1461no—
362C22⋊Q8—731no—
363C22⋊Q8—1461no—
364C22⋊Q8C2×#6440811yescharacteristic
365C4×Q8—731no—
366C4×Q8—731no—
367C4×Q8—731no—
368C4×Q8—731no—
369C4×Q8—1461no—
370C4×D4—731no—
371C4×D4—731no—
372C4×D4—1461no—
373C4×D4—1461no—
374#117—731no—
375#117—1461no—
376#117—1461no—
377C2×C42—731no—
378C2×C42—1461no—
379#107—731no—
380#107—731no—
381#106—731no—
382#106—731no—
383C4⋊C8—731no—
384C4⋊C8—731no—
385C4⋊C8—731no—
386C4⋊C8—731no—
387C4⋊C8—731no—
388C4⋊C8—731no—
389C4⋊C8—731no—
390C4⋊C8—731no—
391C4⋊C8—1461no—
392C4⋊C8—1461no—
393C4⋊C8—1461no—
394C4⋊C8—1461no—
395C4⋊C8—1461no—
396C4⋊C8—1461no—
397C4⋊C8—1461no—
398#103—731no—
399#103—731no—
400#103—731no—
401#103—731no—
402#103—731no—
403#103C2×#6440811yescharacteristic
404#102—731no—
405#102C2×#6440811yescharacteristic
406C22⋊C8—731no—
407C22⋊C8—731no—
408C22⋊C8—731no—
409C22⋊C8—1461no—
410C22⋊C8—1461no—
411C22⋊C8—1461no—
412C22⋊C8C2×#6440811yescharacteristic
413#97—1461no—
414#97—1461no—
415#97—1461no—
416#97—1461no—
417C4×C8—731no—
418C4×C8—731no—
419C4×C8—731no—
420C4×C8—731no—
421C4×C8—731no—
422C4×C8—731no—
423C4×C8—1461no—
424C4×C8—1461no—
425C4×C8—1461no—
426C4×C8—1461no—
427#41—22no—
428#41—41no—
429C2×Q8—731no—
430C2×Q8—1461no—
431C2×Q8C22×#6440811yescharacteristic
432C2×Q8C22×#6440811yescharacteristic
433C2×D4—21no—
434C2×D4C22×#6440811yescharacteristic
435C22×C4—21no—
436C22×C4—731no—
437C22×C4—731no—
438C22×C4—1461no—
439C22×C4—1461no—
440C22×C4C22×#6440811yescharacteristic
441Q16—41no—
442Q16—2921no—
443SD16—22no—
444SD16—1462no—
445M16—5841no—
446C2×C8—5841no—
447C2×C8—5841no—
448C2×C8—731no—
449C2×C8—731no—
450C2×C8—731no—
451C2×C8—731no—
452C2×C8—731no—
453C2×C8—731no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1464no—
461C2×C8C22×#6440811yescharacteristic
462C2×C8C22×#6440811yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—731no—
465C4⋊C4—731no—
466C4⋊C4—731no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4C22×#6440811yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—1461no—
478C22⋊C4—1461no—
479C22⋊C4—1461no—
480C22⋊C4—1461no—
481C42—731no—
482C42—731no—
483C42—731no—
484C42—731no—
485C42—1461no—
486C42—1461no—
487C42—1461no—
488C42—1461no—
489C42—1461no—
490C23—21no—
491C23D4×#6440811yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—2922no—
497Q8—1461no—
498Q8—1461no—
499D4—21no—
500D4—42no—
501D4#1455667812yes—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—42no—
508C2×C4—2921no—
509C2×C4—2921no—
510C2×C4—2922no—
511C2×C4—731no—
512C2×C4—731no—
513C2×C4—731no—
514C2×C4—731no—
515C2×C4—1461no—
516C2×C4—1461no—
517C2×C4—1461no—
518C2×C4—1461no—
519C2×C4—1462no—
520C2×C4—1461no—
521C2×C4—1461no—
522C2×C4D4×#6440811yescharacteristic
523C2×C4C23×#6440811yescharacteristic
524C8—21no—
525C8—21no—
526C8—2921no—
527C8—2921no—
528C8—2921no—
529C8—2921no—
530C8—1461no—
531C8—1461no—
532C8—1461no—
533C8—1461no—
534C8—1462no—
535C8—1462no—
536C8—1461no—
537C8—1461no—
538C22—21no—
539C22—22no—
540C22—21no—
541C22—21no—
542C22—41no—
543C22—41no—
544C22C2×D4×#6440811yescharacteristic
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—2921no—
550C4—2921no—
551C4—2921no—
552C4—2921no—
553C4—731no—
554C4—731no—
555C4—732no—
556C4—1461no—
557C4—1461no—
558C4C2×#1455667811yescharacteristic
559C4#1798966511yescharacteristic
560C2—41no—
561C2—42no—
562C2#2262666311yescharacteristic
563C2#2262668911yescharacteristic
564C2D73⋊#44311yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#452×C73C4
socleC22×C73C2×C4×D4
prime radicalC2×#41×C73C2×C4
cosocleC4×Dic73C24
hypercenter#452#64408
hypocenterC73#18016
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. #22626840C2semidirect
3. #22625571C2semidirect
4. #22625569C2semidirect
5. #22625557C2semidirect
6. #22625556C2semidirect
7. #22625548C2semidirect
8. #22625546C2semidirect
9. #22625514C2semidirect
10. SD16×#10596833C2semidirect
11. #22625507C2semidirect
12. #22625279C2semidirect
14. #22625136C2semidirect
298. C73#18016semidirect

Maximal subgroups

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

subgroupindex
#226268572
#226268402
#226255712
#226255692
#226255572
#226255562
#226255482
#226255462
#226255142
SD16×#105968332
#226255072
#226252792
#226252222
#226251362
#226199952
#1801673