Finite Group Databasebeta

← #34018681

Subgroups — #34018681

Order 18688.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626847C211yesmaximal, characteristic
2#22626836C211yesmaximal, characteristic, complemented
3#22625557C211yesmaximal, characteristic, complemented
4#22625548C211yesmaximal, characteristic, complemented
5#22625545C211yesmaximal, characteristic, complemented
6#22625536C211yesmaximal, characteristic, complemented
7#22625525C211yesmaximal, characteristic, complemented
8SD16⋊#10596833C211yesmaximal, characteristic, complemented
9#22625518C211yesmaximal, characteristic, complemented
10SD16×#10596833C211yesmaximal, characteristic, complemented
11#22625505C211yesmaximal, characteristic, complemented
12#22625263C211yesmaximal, characteristic
13#22625240C211yesmaximal, characteristic, complemented
14#22625153C211yesmaximal, characteristic, complemented
15#22619999C211yesmaximal, characteristic
16#17989639—21no—
17#17989639—21no—
18#17989638C2211yescharacteristic
19#17989638C2211yescharacteristic
20#17989636C2211yescharacteristic
21#17989636C2211yescharacteristic
22Q8×#10596833C2211yescharacteristic
23Q8×#10596833C2211yescharacteristic
24#17989614—21no—
25#17989614—21no—
26D4×#10596833C2211yescharacteristic
27D4×#10596833C2211yescharacteristic
28#17989612—21no—
29#17989612—21no—
30#17989609C2211yescharacteristic
31#17989606C2211yescharacteristic
32#17989605C2211yescharacteristic
33#17989604—21no—
34C2×#14556658—21no—
35#17989588C2211yescharacteristic
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
50#17989428C2211yescharacteristic
51C8×#10596833C2211yescharacteristic
52#17989142C2211yescharacteristic
53C4×#11924027C2211yescharacteristic
54#17988926C411yescharacteristic
55#17988926C411yescharacteristic
56#17988873C411yescharacteristic
57#17988867C411yescharacteristic
58#17988794C411yescharacteristic
59#17988746C411yescharacteristic
60#17988746C411yescharacteristic
61#17988679C2211yescharacteristic
62#17988679C2211yescharacteristic
63SD16×Dic73C2211yescharacteristic
64SD16×Dic73C2211yescharacteristic
65#17988639C2211yescharacteristic
66#461×C73C411yescharacteristic
67C2×#11924036—21no—
68C22×#10596833—21no—
69C22×#10596833—21no—
70#14556658—21no—
71#14556658—21no—
72#14556658—21no—
73#14556658—21no—
74#14556658—21no—
75#14556658D411yescharacteristic
76#14556658D411yescharacteristic
77C4⋊#10596833—21no—
78C4⋊#10596833—21no—
79C4⋊#10596833—21no—
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84#14556648—21no—
85#14556648—21no—
86#14556648D411yescharacteristic
87#14556648D411yescharacteristic
88C22⋊#10596833—21no—
89C22⋊#10596833—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×#11924027C2×C411yescharacteristic
102C2×#11924018C2×C411yescharacteristic
103C2×#11924018C2×C411yescharacteristic
104#14556591C2×C411yescharacteristic
105#14556582—21no—
106#14556571—21no—
107#14556567C2×C411yescharacteristic
108#14556567C2×C411yescharacteristic
109Q8×Dic73C2311yescharacteristic
110Q8×Dic73C2311yescharacteristic
111C4×Dic146C2×C411yescharacteristic
112#14556556—21no—
113#14556556C2311yescharacteristic
114D4×Dic73C2311yescharacteristic
115C4×C22⋊D73—21no—
116#14556551—21no—
117#14556549—21no—
118#14556549C2×C411yescharacteristic
119#14556547—21no—
120C2×C4×Dic73—21no—
121C42×D73C2311yescharacteristic
122#14556524C2×C411yescharacteristic
123#14556524C2×C411yescharacteristic
124#14556523C2×C411yescharacteristic
125#14556523C2×C411yescharacteristic
126C4⋊#10596831C2×C411yescharacteristic
127#14556516C2×C411yescharacteristic
128#14556516C2×C411yescharacteristic
129#14556515C2×C411yescharacteristic
130#14556515C2×C411yescharacteristic
131#14556512C2×C411yescharacteristic
132#14556512C2×C411yescharacteristic
133#14556503C2×C411yescharacteristic
134#14556503C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137C2×SD16×C73C2×C411yescharacteristic
138C2×SD16×C73C2×C411yescharacteristic
139C4⋊Q8×C73C2×C411yescharacteristic
140C4×D4×C73C2×C411yescharacteristic
141C4⋊C8×C73C2×C411yescharacteristic
142#103×C73C2×C411yescharacteristic
143#103×C73C2×C411yescharacteristic
144#11924036—81no—
145C2×#10596833—21no—
146C2×#10596833—21no—
147C2×#10596833—21no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—81no—
152C2×#10596833—81no—
153C2×#10596833C2×D411yescharacteristic
154C2×#10596833C2×D411yescharacteristic
155C2×#10596833C2×D411yescharacteristic
156C2×#10596833C2×D411yescharacteristic
157#11924027—22no—
158#11924027—22no—
159C2×Dic146—21no—
160C2×Dic146C22×C411yescharacteristic
161C2×C22⋊D73—21no—
162C22×Dic73—21no—
163C22×Dic73—21no—
164C2×C4×D73C22×C411yescharacteristic
165C2×C4×D73C2×D411yescharacteristic
166C2×C4×D73#4111yescharacteristic
167#11924018—22no—
168#11924018—22no—
169C2×#10596831C22×C411yescharacteristic
170C2×#10596831C22×C411yescharacteristic
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73—21no—
174C4⋊Dic73C22×C411yescharacteristic
175C4⋊Dic73C22×C411yescharacteristic
176C4⋊Dic73C22×C411yescharacteristic
177#11924007—21no—
178#11924007—21no—
179#11924007—21no—
180#11924007—21no—
181C22⋊Dic73—21no—
182#11924005—21no—
183#11924005—21no—
184#11924005—21no—
185C4×Dic73—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73C2×D411yescharacteristic
191C4×Dic73#4111yescharacteristic
192C4×Dic73C2411yescharacteristic
193C2×Q8×C73C22×C411yescharacteristic
194C2×Q8×C73C22×C411yescharacteristic
195C2×D4×C73C22×C411yescharacteristic
196C22×C4×C73—21no—
197SD16×C73—22no—
198SD16×C73—22no—
199C2×C8×C73C22×C411yescharacteristic
200C2×C8×C73C22×C411yescharacteristic
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73C22×C411yescharacteristic
204C22⋊C4×C73—21no—
205C42×C73C22×C411yescharacteristic
206C23×C73—21no—
207C22×D73C4×D411yescharacteristic
208Q8×C73—21no—
209Q8×C73—21no—
210Q8×C73—21no—
211Q8×C73—21no—
212D4×C73—21no—
213D4×C73#13112yes—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—81no—
218C2×C4×C73C4×D411yescharacteristic
219C2×C4×C73C4×D411yescharacteristic
220C2×C4×C73C23×C411yescharacteristic
221C22⋊D73—24no—
222C2×Dic73—24no—
223C2×Dic73—21no—
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—41no—
228C2×Dic73—41no—
229C2×Dic73C4×D411yescharacteristic
230C2×Dic73C22×D411yescharacteristic
231C2×Dic73C2×#4111yescharacteristic
232C4×D73—21no—
233C4×D73—21no—
234C4×D73—21no—
235C4×D73—21no—
236C4×D73—41no—
237Dic146—21no—
238Dic146—81no—
239Dic146#13112yes—
240#10596833—21no—
241#10596833—21no—
242#10596833—21no—
243#10596833—21no—
244#10596833—41no—
245#10596833—41no—
246#10596833—41no—
247#10596833—41no—
248#10596833—41no—
249#10596833—41no—
250C8×C73—21no—
251C8×C73—21no—
252#10596831—21no—
253#10596831—21no—
254C22×C73—22no—
255C22×C73—41no—
256C22×C73—41no—
257C22×C73C2×C4×D411yescharacteristic
258C2×D73—21no—
259C2×D73—21no—
260C2×D73—21no—
261C4×C73—21no—
262C4×C73—21no—
263C4×C73—41no—
264C4×C73—41no—
265C4×C73—41no—
266C4×C73C2×#13111yescharacteristic
267C4×C73C2×#13111yescharacteristic
268Dic73—22no—
269Dic73—21no—
270Dic73—21no—
271Dic73—41no—
272Dic73—41no—
273Dic73—41no—
274Dic73#57711yescharacteristic
275Dic73#57811yescharacteristic
276#17921—731nomaximal, Sylow
277C2×C73—42no—
278C2×C73#292911yescharacteristic
279C2×C73#294711yescharacteristic
280C2×C73#295711yescharacteristic
281D73—41no—
282#3107—731no—
283#2988—731no—
284#2985—731no—
285#1587—731no—
286#1585—731no—
287#1584—731no—
288#1582—731no—
289#1579—731no—
290SD16⋊C8—731no—
291C8×Q16—731no—
292C8×SD16—731no—
293#1573—731no—
294#1488—731no—
295#1481—731no—
296#1454—731no—
297C73#1792111yescharacteristic, complemented, Sylow
298#533—731no—
299C4×#41—731no—
300#485—731no—
301#485—731no—
302#475—731no—
303#474—731no—
304#462—731no—
305#461#6440811yescharacteristic
306#452—731no—
307#452—731no—
308#446—731no—
309#446—731no—
310#446—1461no—
311#446—1461no—
312C8×Q8—731no—
313C8×Q8—731no—
314C8×Q8—731no—
315C8×Q8—731no—
316C4×Q16—731no—
317C4×Q16—731no—
318C4×SD16—731no—
319C4×SD16—731no—
320C8×D4—731no—
321C8×D4—731no—
322C8×D4—1461no—
323C8×D4—1461no—
324#433—1461no—
325#433—1461no—
326#432—731no—
327#431—731no—
328#431—731no—
329#431—731no—
330#431—1461no—
331C2×C4⋊C8—731no—
332C2×C4⋊C8—1461no—
333#335—731no—
334#335—731no—
335#334—731no—
336#334—731no—
337Q8⋊C8—731no—
338Q8⋊C8—731no—
339Q8⋊C8—731no—
340Q8⋊C8—731no—
341Q8⋊C8—731no—
342Q8⋊C8—731no—
343D4⋊C8—731no—
344D4⋊C8—731no—
345#322—731no—
346#322—731no—
347#322—731no—
348C82—731no—
349C2×#41—731no—
350C2×Q16—731no—
351C2×Q16—731no—
352C2×SD16C2×#6440811yescharacteristic
353C2×SD16C2×#6440811yescharacteristic
354C22×C8—1461no—
355C22×C8—1461no—
356C22×C8—1461no—
357C4⋊Q8—731no—
358C4⋊Q8C2×#6440811yescharacteristic
359#125—1461no—
360C22⋊Q8—731no—
361C22⋊Q8—731no—
362C22⋊Q8—1461no—
363C4×Q8—731no—
364C4×Q8—731no—
365C4×Q8—731no—
366C4×Q8—731no—
367C4×Q8—1461no—
368C4×D4—731no—
369C4×D4—1461no—
370C4×D4—1461no—
371C4×D4C2×#6440811yescharacteristic
372#117—731no—
373#117—1461no—
374#117—1461no—
375C2×C42—731no—
376C2×C42—1461no—
377#107—731no—
378#107—731no—
379#106—731no—
380#106—731no—
381C4⋊C8—731no—
382C4⋊C8—731no—
383C4⋊C8—731no—
384C4⋊C8—731no—
385C4⋊C8—731no—
386C4⋊C8—731no—
387C4⋊C8—731no—
388C4⋊C8—1461no—
389C4⋊C8—1461no—
390C4⋊C8—1461no—
391C4⋊C8—1461no—
392C4⋊C8—1461no—
393C4⋊C8—1461no—
394C4⋊C8—1461no—
395C4⋊C8—1461no—
396C4⋊C8C2×#6440811yescharacteristic
397#103—731no—
398#103—731no—
399#103—731no—
400#103—731no—
401#103C2×#6440811yescharacteristic
402#103C2×#6440811yescharacteristic
403#102—731no—
404#102—731no—
405C22⋊C8—731no—
406C22⋊C8—731no—
407C22⋊C8—731no—
408C22⋊C8—731no—
409C22⋊C8—1461no—
410C22⋊C8—1461no—
411C22⋊C8—1461no—
412C22⋊C8—1461no—
413#97—1461no—
414#97—1461no—
415#97—1461no—
416C4×C8—731no—
417C4×C8—731no—
418C4×C8—731no—
419C4×C8—731no—
420C4×C8—731no—
421C4×C8—731no—
422C4×C8—1461no—
423C4×C8—1461no—
424C4×C8—1461no—
425C4×C8—1461no—
426#41—1462no—
427#41—1462no—
428C2×Q8—731no—
429C2×Q8—1461no—
430C2×Q8C22×#6440811yescharacteristic
431C2×Q8C22×#6440811yescharacteristic
432C2×D4—1461no—
433C2×D4C22×#6440811yescharacteristic
434C22×C4—21no—
435C22×C4—731no—
436C22×C4—731no—
437C22×C4—731no—
438C22×C4—1461no—
439C22×C4—1461no—
440Q16—1462no—
441Q16—1462no—
442SD16—22no—
443SD16—22no—
444C2×C8—5841no—
445C2×C8—5841no—
446C2×C8—5841no—
447C2×C8—731no—
448C2×C8—731no—
449C2×C8—731no—
450C2×C8—731no—
451C2×C8—731no—
452C2×C8—731no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8C22×#6440811yescharacteristic
460C2×C8C22×#6440811yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—731no—
464C4⋊C4—731no—
465C4⋊C4—731no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4C22×#6440811yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—1461no—
476C22⋊C4—1461no—
477C22⋊C4—1461no—
478C22⋊C4—1461no—
479C42—731no—
480C42—731no—
481C42—731no—
482C42—1461no—
483C42—1461no—
484C42—1461no—
485C42—1461no—
486C42—1461no—
487C42C22×#6440811yescharacteristic
488C23—21no—
489C23—731no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—5841no—
495Q8—732no—
496Q8—1461no—
497D4—21no—
498D4—1464no—
499D4#1455667812yes—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—81no—
504C2×C4—2921no—
505C2×C4—2921no—
506C2×C4—2921no—
507C2×C4—731no—
508C2×C4—731no—
509C2×C4—731no—
510C2×C4—1461no—
511C2×C4—1461no—
512C2×C4—1461no—
513C2×C4—1461no—
514C2×C4—1461no—
515C2×C4—1464no—
516C2×C4—1461no—
517C2×C4—1461no—
518C2×C4—1461no—
519C2×C4D4×#6440811yescharacteristic
520C2×C4D4×#6440811yescharacteristic
521C2×C4C23×#6440811yescharacteristic
522C8—21no—
523C8—21no—
524C8—2921no—
525C8—2921no—
526C8—2921no—
527C8—2921no—
528C8—2921no—
529C8—2921no—
530C8—1461no—
531C8—1461no—
532C8—1461no—
533C8—1461no—
534C8—1461no—
535C8—1461no—
536C22—22no—
537C22—41no—
538C22—41no—
539C22—1461no—
540C22—1461no—
541C22—1461no—
542C22C2×D4×#6440811yescharacteristic
543C4—21no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—2921no—
549C4—2921no—
550C4—2921no—
551C4—731no—
552C4—731no—
553C4—1461no—
554C4—1462no—
555C4—1461no—
556C4C2×#1455667811yescharacteristic
557C4D73⋊#13111yescharacteristic
558C2—42no—
559C2—2921no—
560C2#2262664511yescharacteristic
561C2#2262668911yescharacteristic
562C2#2262670911yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#461×C73C4
socleC22×C73C2×C4×D4
prime radicalC2×#11924027C2×C4
cosocleC4×Dic73C24
hypercenter#461#64408
hypocenterC73#17921
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #22626836C2semidirect
3. #22625557C2semidirect
4. #22625548C2semidirect
5. #22625545C2semidirect
6. #22625536C2semidirect
7. #22625525C2semidirect
8. SD16⋊#10596833C2semidirect
9. #22625518C2semidirect
10. SD16×#10596833C2semidirect
11. #22625505C2semidirect
13. #22625240C2semidirect
14. #22625153C2semidirect
297. C73#17921semidirect

Maximal subgroups

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

subgroupindex
#226268472
#226268362
#226255572
#226255482
#226255452
#226255362
#226255252
SD16⋊#105968332
#226255182
SD16×#105968332
#226255052
#226252632
#226252402
#226251532
#226199992
#1792173