Finite Group Databasebeta

← #34019122

Subgroups — #34019122

Order 18688.

602 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626732C211yesmaximal, characteristic
2#22626726C211yesmaximal, characteristic
3#22625570C211yesmaximal, characteristic, complemented
4#22625569C211yesmaximal, characteristic, complemented
5#22625545C211yesmaximal, characteristic, complemented
6#22625543C211yesmaximal, characteristic, complemented
7#22625543C211yesmaximal, characteristic, complemented
8#22625542C211yesmaximal, characteristic, complemented
9#22625519C211yesmaximal, characteristic, complemented
10#22625518C211yesmaximal, characteristic, complemented
11#22625500C211yesmaximal, characteristic
12#22625279C211yesmaximal, characteristic
13C2×#17989437C211yesmaximal, characteristic
14#22625132C211yesmaximal, characteristic
15#22619967C211yesmaximal, characteristic
16#17989639—21no—
17#17989639—21no—
18#17989639—21no—
19#17989638C2211yescharacteristic
20#17989637—21no—
21#17989637C2211yescharacteristic
22#17989637C2211yescharacteristic
23#17989637C2211yescharacteristic
24#17989636C2211yescharacteristic
25#17989636C2211yescharacteristic
26#17989636C2211yescharacteristic
27Q8×#10596833C2211yescharacteristic
28#17989593—21no—
29#17989590—21no—
30#17989590C2211yescharacteristic
31C2×C4⋊#10596833C2211yescharacteristic
32#17989551—21no—
33C4×#11924037—21no—
34C4×#11924037C2211yescharacteristic
35C2×C4×#10596833C2211yescharacteristic
36#17989451C2211yescharacteristic
37#17989451C2211yescharacteristic
38#17989450C2211yescharacteristic
39#17989450C2211yescharacteristic
40#17989437C2211yescharacteristic
41#17989437C2211yescharacteristic
42#17989437C2211yescharacteristic
43#17989437C2211yescharacteristic
44#17989437C2211yescharacteristic
45#17989437C2211yescharacteristic
46Q8⋊#10596833C2211yescharacteristic
47Q8⋊#10596833C2211yescharacteristic
48#17989429C2211yescharacteristic
49#17989429C2211yescharacteristic
50C8×#10596833C2211yescharacteristic
51C8×#10596833C2211yescharacteristic
52#17989139C2211yescharacteristic
53C2×#14556556C2211yescharacteristic
54#17988922C411yescharacteristic
55#17988881C411yescharacteristic
56#17988881C411yescharacteristic
57#17988796C411yescharacteristic
58#17988796C411yescharacteristic
59#17988745C411yescharacteristic
60#17988742C411yescharacteristic
61#17988681C2211yescharacteristic
62#17988681C2211yescharacteristic
63#17988679C2211yescharacteristic
64#17988679C2211yescharacteristic
65#17988637C2211yescharacteristic
66#484×C73C411yescharacteristic
67C2×#11924037—21no—
68C2×#11924037D411yescharacteristic
69C2×#11924037D411yescharacteristic
70C22×#10596833—21no—
71C22×#10596833—21no—
72#14556658—21no—
73#14556658—21no—
74#14556658—21no—
75#14556658—21no—
76#14556658—21no—
77C4⋊#10596833—21no—
78C4⋊#10596833—21no—
79C4⋊#10596833—21no—
80C4⋊#10596833—21no—
81C4⋊#10596833—21no—
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84C4⋊#10596833C2311yescharacteristic
85C4⋊#10596833C2311yescharacteristic
86#14556646—21no—
87#14556646—21no—
88#14556646—21no—
89#14556646—21no—
90#14556646D411yescharacteristic
91#14556646D411yescharacteristic
92C4×#10596833—21no—
93C4×#10596833—21no—
94C4×#10596833—21no—
95C4×#10596833—21no—
96C4×#10596833—21no—
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C22×Dic146C2×C411yescharacteristic
102C2×Dic292C2×C411yescharacteristic
103C2×Dic292C2×C411yescharacteristic
104C2×#11924018C2×C411yescharacteristic
105C2×#11924018C2×C411yescharacteristic
106#14556590C2×C411yescharacteristic
107#14556589C2×C411yescharacteristic
108#14556581—21no—
109#14556570C2×C411yescharacteristic
110#14556568—21no—
111Q8×Dic73C2311yescharacteristic
112#14556556—21no—
113#14556556—21no—
114#14556556—21no—
115#14556556C2×C411yescharacteristic
116#14556556C2×C411yescharacteristic
117#14556556C2311yescharacteristic
118#14556556C2311yescharacteristic
119#14556556C2311yescharacteristic
120#14556547—21no—
121C2×#11924007—21no—
122C2×C4×Dic73—21no—
123C2×C4×Dic73C2311yescharacteristic
124#14556524C2×C411yescharacteristic
125#14556524C2×C411yescharacteristic
126#14556520C2×C411yescharacteristic
127#14556520C2×C411yescharacteristic
128#14556516C2×C411yescharacteristic
129#14556516C2×C411yescharacteristic
130#14556514C2×C411yescharacteristic
131#14556514C2×C411yescharacteristic
132#14556514C2×C411yescharacteristic
133#14556514C2×C411yescharacteristic
134C22⋊#10596831C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137C22⋊Q8×C73C2×C411yescharacteristic
138C2×C4⋊C4×C73C2×C411yescharacteristic
139#107×C73C2×C411yescharacteristic
140#107×C73C2×C411yescharacteristic
141#103×C73C2×C411yescharacteristic
142#103×C73C2×C411yescharacteristic
143C22⋊C8×C73C2×C411yescharacteristic
144#11924037—22no—
145#11924037—42no—
146#11924037Q1614yescomplemented
147C2×#10596833—21no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833—42no—
154C2×#10596833—42no—
155C2×#10596833C2×D411yescharacteristic
156C2×#10596833C2×D411yescharacteristic
157C2×#10596833C2×D411yescharacteristic
158C2×#10596833C2×D411yescharacteristic
159C2×Dic146—22no—
160C2×Dic146—21no—
161C2×Dic146—21no—
162C2×Dic146—41no—
163C2×Dic146C22×C411yescharacteristic
164C2×Dic146C22×C411yescharacteristic
165C2×Dic146C22×C411yescharacteristic
166C22×Dic73—21no—
167C22×Dic73C2×D411yescharacteristic
168C22×Dic73#4111yescharacteristic
169Dic292—22no—
170Dic292—41no—
171#11924018—22no—
172#11924018—41no—
173C2×#10596831C22×C411yescharacteristic
174C2×#10596831C22×C411yescharacteristic
175C4⋊Dic73—21no—
176C4⋊Dic73C22×C411yescharacteristic
177#11924007—21no—
178#11924007—21no—
179#11924007—21no—
180#11924007—21no—
181#11924007—21no—
182#11924007—21no—
183#11924007—21no—
184#11924007—21no—
185#11924007—21no—
186C22⋊Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73—21no—
191C4×Dic73—21no—
192C4×Dic73—21no—
193C4×Dic73C22×C411yescharacteristic
194C4×Dic73C2×D411yescharacteristic
195C4×Dic73#4111yescharacteristic
196C4×Dic73C2411yescharacteristic
197C2×Q8×C73C22×C411yescharacteristic
198C22×C4×C73—21no—
199C22×C4×C73C22×C411yescharacteristic
200C2×C8×C73C22×C411yescharacteristic
201C2×C8×C73C22×C411yescharacteristic
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73C22×C411yescharacteristic
205C4⋊C4×C73C22×C411yescharacteristic
206C4⋊C4×C73C22×C411yescharacteristic
207C22⋊C4×C73—21no—
208C23×C73C4×D411yescharacteristic
209Q8×C73—21no—
210Q8×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—81no—
218C2×C4×C73C4×D411yescharacteristic
219C2×C4×C73C23×C411yescharacteristic
220C2×Dic73—21no—
221C2×Dic73—21no—
222C2×Dic73—21no—
223C2×Dic73—21no—
224C2×Dic73—41no—
225C2×Dic73—42no—
226C2×Dic73C4×D411yescharacteristic
227C2×Dic73C4×D411yescharacteristic
228C2×Dic73C2×Q1612yes—
229C2×Dic73#13512yes—
230C2×Dic73C22×D411yescharacteristic
231C2×Dic73C2×#4111yescharacteristic
232Dic146—21no—
233Dic146—21no—
234Dic146—21no—
235Dic146—21no—
236Dic146—21no—
237Dic146—42no—
238Dic146—81no—
239Dic146#13112yes—
240#10596833—22no—
241#10596833—21no—
242#10596833—21no—
243#10596833—41no—
244#10596833—41no—
245#10596833—41no—
246#10596833—41no—
247#10596833—41no—
248#10596833C2×Q1612yes—
249#10596833C2×Q1612yes—
250C8×C73—21no—
251C8×C73—21no—
252#10596831—21no—
253#10596831—21no—
254C22×C73—21no—
255C22×C73—21no—
256C22×C73C4×Q1612yes—
257C22×C73C2×C4×D411yescharacteristic
258C4×C73—21no—
259C4×C73—41no—
260C4×C73—41no—
261C4×C73—41no—
262C4×C73—41no—
263C4×C73C2×#13111yescharacteristic
264C4×C73#56811yescharacteristic
265Dic73—22no—
266Dic73—21no—
267Dic73—21no—
268Dic73—41no—
269Dic73—41no—
270Dic73—41no—
271Dic73C4×Q1612yes—
272Dic73C22×Q1611yescharacteristic
273Dic73C2×#13511yescharacteristic
274#17998—731nomaximal, Sylow
275C2×C73—22no—
276C2×C73#293311yescharacteristic
277C2×C73C2×C4×Q1611yescharacteristic
278C2×C73#295611yescharacteristic
279#3104—731no—
280Q8×M16—731no—
281#2962—731no—
282#1591—731no—
283#1591—731no—
284#1584—731no—
285#1584—731no—
286#1584—731no—
287#1584—731no—
288C8×Q16—731no—
289C8×Q16—731no—
290#1571—731no—
291#1493—731no—
292C2×Q8⋊C8—731no—
293C8×M16—731no—
294C73#1799811yescharacteristic, complemented, Sylow
295#533—731no—
296C2×C4×Q8—731no—
297#484—731no—
298#484#6440811yescharacteristic
299#477—731no—
300#477—731no—
301#462—731no—
302#462—731no—
303#451—731no—
304#451—731no—
305#446—731no—
306#446—731no—
307#446—731no—
308#446—731no—
309#446—1461no—
310#446—1461no—
311#446—1461no—
312#446—1461no—
313C8×Q8—731no—
314C8×Q8—731no—
315C8×Q8—731no—
316C8×Q8—731no—
317C4×Q16—731no—
318C4×Q16—731no—
319C4×Q16—731no—
320C4×Q16—731no—
321#431—731no—
322#424—1461no—
323#423—731no—
324#423—1461no—
325C2×C4⋊C8—731no—
326#405—1461no—
327C4×M16—731no—
328C4×M16—1461no—
329C2×C4×C8—731no—
330#335—731no—
331#335—731no—
332#335—731no—
333#335—731no—
334Q8⋊C8—731no—
335Q8⋊C8—731no—
336Q8⋊C8—731no—
337Q8⋊C8—731no—
338Q8⋊C8—731no—
339Q8⋊C8—731no—
340Q8⋊C8—731no—
341Q8⋊C8—731no—
342#322—731no—
343#322—731no—
344C82—731no—
345C82—731no—
346C22×Q8—731no—
347C2×Q16—731no—
348C2×Q16—731no—
349C2×Q16—731no—
350C2×Q16—731no—
351C2×M16—731no—
352C2×M16—731no—
353C2×M16—1461no—
354C22×C8—1461no—
355C22×C8—1461no—
356C4⋊Q8—731no—
357C4⋊Q8—731no—
358#125—1461no—
359C22⋊Q8—731no—
360C22⋊Q8—1461no—
361C22⋊Q8C2×#6440811yescharacteristic
362C4×Q8—731no—
363C4×Q8—731no—
364C4×Q8—731no—
365C4×Q8—731no—
366C4×Q8—731no—
367C4×Q8—731no—
368C4×Q8—1461no—
369C4×Q8—1461no—
370C4×Q8—1461no—
371#117—1461no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4C2×#6440811yescharacteristic
374C2×C42—731no—
375C2×C42—1461no—
376#107—731no—
377#107—731no—
378#107C2×#6440811yescharacteristic
379#107C2×#6440811yescharacteristic
380C4⋊C8—731no—
381C4⋊C8—731no—
382C4⋊C8—731no—
383C4⋊C8—731no—
384C4⋊C8—731no—
385C4⋊C8—731no—
386C4⋊C8—1461no—
387C4⋊C8—1461no—
388C4⋊C8—1461no—
389C4⋊C8—1461no—
390C4⋊C8—1461no—
391C4⋊C8—1461no—
392C4⋊C8—1461no—
393C4⋊C8—1461no—
394C4⋊C8—1461no—
395C4⋊C8—1461no—
396#103—731no—
397#103—731no—
398#103—731no—
399#103—731no—
400#103—731no—
401#103—731no—
402#103C2×#6440811yescharacteristic
403#103C2×#6440811yescharacteristic
404C22⋊C8—731no—
405C22⋊C8C2×#6440811yescharacteristic
406#97—731no—
407#97—731no—
408#97—1461no—
409#97—1461no—
410#97—1461no—
411#97—1461no—
412C4×C8—731no—
413C4×C8—731no—
414C4×C8—731no—
415C4×C8—731no—
416C4×C8—731no—
417C4×C8—731no—
418C4×C8—1461no—
419C4×C8—1461no—
420C4×C8—1461no—
421C4×C8—1461no—
422C4×C8—1461no—
423C2×Q8—2921no—
424C2×Q8—731no—
425C2×Q8—731no—
426C2×Q8—731no—
427C2×Q8—1462no—
428C2×Q8—1461no—
429C2×Q8—1461no—
430C2×Q8C22×#6440811yescharacteristic
431C22×C4—21no—
432C22×C4—731no—
433C22×C4—731no—
434C22×C4—1461no—
435C22×C4C22×#6440811yescharacteristic
436Q16—2921no—
437Q16—2921no—
438Q16—1462no—
439Q16—1462no—
440M16—2922no—
441M16—734no—
442M16—1462no—
443C2×C8—2922no—
444C2×C8—2922no—
445C2×C8—731no—
446C2×C8—731no—
447C2×C8—731no—
448C2×C8—731no—
449C2×C8—731no—
450C2×C8—731no—
451C2×C8—1461no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8C22×#6440811yescharacteristic
458C2×C8C22×#6440811yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—731no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4C22×#6440811yescharacteristic
473C4⋊C4C22×#6440811yescharacteristic
474C4⋊C4C22×#6440811yescharacteristic
475C22⋊C4—21no—
476C22⋊C4—1461no—
477C42—731no—
478C42—731no—
479C42—731no—
480C42—731no—
481C42—1461no—
482C42—1461no—
483C42—1461no—
484C42—1461no—
485C42—1461no—
486C42—1461no—
487C23D4×#6440811yescharacteristic
488Q8—21no—
489Q8—21no—
490Q8—2922no—
491Q8—5841no—
492Q8—732no—
493Q8—1461no—
494Q8—1461no—
495Q8—1461no—
496Q8—1461no—
497Q8—1461no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—81no—
505C2×C4—2922no—
506C2×C4—2921no—
507C2×C4—731no—
508C2×C4—731no—
509C2×C4—732no—
510C2×C4—732no—
511C2×C4—731no—
512C2×C4—731no—
513C2×C4—1461no—
514C2×C4—1461no—
515C2×C4—1461no—
516C2×C4—1461no—
517C2×C4D4×#6440811yescharacteristic
518C2×C4C23×#6440811yescharacteristic
519C8—21no—
520C8—21no—
521C8—2921no—
522C8—2921no—
523C8—2921no—
524C8—2921no—
525C8—2921no—
526C8—732no—
527C8—732no—
528C8—1461no—
529C8—1461no—
530C8—1462no—
531C8—1461no—
532C8—1461no—
533C22—21no—
534C22—21no—
535C22Q16×#6440812yes—
536C22C2×D4×#6440811yescharacteristic
537C4—21no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—2921no—
543C4—2921no—
544C4—2921no—
545C4—731no—
546C4—731no—
547C4—732no—
548C4—1462no—
549C4—1461no—
550C4—1461no—
551C4D73⋊#13111yescharacteristic
552C4#1798966611yescharacteristic
553C2—22no—
554C2#2262666511yescharacteristic
555C2C2×Q16×#6440811yescharacteristic
556C2#2262670511yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#484×C73C4
socleC22×C73C2×C4×D4
prime radicalC23×C73C4×D4
cosocleC4×Dic73C24
hypercenter#484#64408
hypocenterC73#17998
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #22625570C2semidirect
4. #22625569C2semidirect
5. #22625545C2semidirect
6. #22625543C2semidirect
7. #22625543C2semidirect
8. #22625542C2semidirect
9. #22625519C2semidirect
10. #22625518C2semidirect
146. #11924037Q16semidirect
294. C73#17998semidirect

Maximal subgroups

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

subgroupindex
#226267322
#226267262
#226255702
#226255692
#226255452
#226255432
#226255422
#226255192
#226255182
#226255002
#226252792
C2×#179894372
#226251322
#226199672
#1799873