Finite Group Databasebeta

← #34019516

Subgroups — #34019516

Order 18688.

624 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626863C211yesmaximal, characteristic
2#22626834C211yesmaximal, characteristic, complemented
3#22625574C211yesmaximal, characteristic, complemented
4#22625567C211yesmaximal, characteristic, complemented
5#22625560C211yesmaximal, characteristic, complemented
6#22625556C211yesmaximal, characteristic, complemented
7#22625544C211yesmaximal, characteristic, complemented
8#22625540C211yesmaximal, characteristic, complemented
9#22625524C211yesmaximal, characteristic, complemented
10Q16×#10596833C211yesmaximal, characteristic, complemented
11#22625495C211yesmaximal, characteristic, complemented
12#22625262C211yesmaximal, characteristic
13#22625259C211yesmaximal, characteristic, complemented
14#22625150C211yesmaximal, characteristic, complemented
15#137×Dic73C211yesmaximal, characteristic
16#17989638—21no—
17#17989638—21no—
18#17989638C2211yescharacteristic
19#17989638C2211yescharacteristic
20Q8×#10596833C2211yescharacteristic
21Q8×#10596833C2211yescharacteristic
22Q8×#10596833C2211yescharacteristic
23Q8×#10596833C2211yescharacteristic
24#17989622—21no—
25#17989622—21no—
26#17989622C2211yescharacteristic
27#17989622C2211yescharacteristic
28#17989608—21no—
29#17989608C2211yescharacteristic
30#17989604—21no—
31#17989604—21no—
32#17989604C2211yescharacteristic
33#17989604C2211yescharacteristic
34#17989590—21no—
35#17989590C2211yescharacteristic
36#17989451C2211yescharacteristic
37#17989450C2211yescharacteristic
38#17989449C2211yescharacteristic
39#17989448C2211yescharacteristic
40Q8⋊#10596833C2211yescharacteristic
41Q8⋊#10596833C2211yescharacteristic
42Q8⋊#10596833C2211yescharacteristic
43Q8⋊#10596833C2211yescharacteristic
44Q8⋊#10596833C2211yescharacteristic
45Q8⋊#10596833C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic
47D4⋊#10596833C2211yescharacteristic
48#17989429C2211yescharacteristic
49#17989429C2211yescharacteristic
50#17989428C2211yescharacteristic
51C8×#10596833C2211yescharacteristic
52#41×Dic73C2211yescharacteristic
53C2×Q8×Dic73C2211yescharacteristic
54#17988694C411yescharacteristic
55#17988694C411yescharacteristic
56#17988688C411yescharacteristic
57#17988688C411yescharacteristic
58Q16×Dic73C2211yescharacteristic
59Q16×Dic73C2211yescharacteristic
60SD16×Dic73C2211yescharacteristic
61SD16×Dic73C2211yescharacteristic
62#17988629C411yescharacteristic
63#17988596C411yescharacteristic
64#17988588C411yescharacteristic
65M16×Dic73C2211yescharacteristic
66C2×#137×C73C411yescharacteristic
67C2×#11924037—21no—
68C2×#11924037—21no—
69C2×#11924037—21no—
70#14556658—21no—
71#14556658—21no—
72#14556658—21no—
73#14556658—21no—
74#14556658D411yescharacteristic
75#14556658D411yescharacteristic
76C4⋊#10596833—21no—
77C4⋊#10596833—21no—
78C4⋊#10596833—21no—
79C4⋊#10596833—21no—
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84C22⋊#10596833—21no—
85C22⋊#10596833—21no—
86C22⋊#10596833—21no—
87C22⋊#10596833—21no—
88C22⋊#10596833D411yescharacteristic
89C22⋊#10596833D411yescharacteristic
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×#11924012C2×C411yescharacteristic
102Q8×Dic73—21no—
103Q8×Dic73—21no—
104Q8×Dic73C2×C411yescharacteristic
105Q8×Dic73C2×C411yescharacteristic
106Q8×Dic73C2×C411yescharacteristic
107Q8×Dic73C2311yescharacteristic
108Q8×Dic73C2311yescharacteristic
109Q8×Dic73C2311yescharacteristic
110D4×Dic73—21no—
111D4×Dic73—21no—
112D4×Dic73C2×C411yescharacteristic
113D4×Dic73C2311yescharacteristic
114#14556550—21no—
115#14556550C2×C411yescharacteristic
116C2×C4⋊Dic73—21no—
117C2×C4⋊Dic73C2×C411yescharacteristic
118C2×C4×Dic73—21no—
119C2×C4×Dic73—21no—
120C2×C4×Dic73C2311yescharacteristic
121#14556525C2×C411yescharacteristic
122#14556525C2×C411yescharacteristic
123#14556523C2×C411yescharacteristic
124#14556523C2×C411yescharacteristic
125#14556515C2×C411yescharacteristic
126#14556515C2×C411yescharacteristic
127#14556515C2×C411yescharacteristic
128#14556515C2×C411yescharacteristic
129#14556515C2×C411yescharacteristic
130#14556515C2×C411yescharacteristic
131#14556512C2×C411yescharacteristic
132#14556512C2×C411yescharacteristic
133#14556501C2×C411yescharacteristic
134#14556501C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137C2×#41×C73C2×C411yescharacteristic
138C22×Q8×C73C2×C411yescharacteristic
139#137×C73—24no—
140C2×Q16×C73C2×C411yescharacteristic
141C2×Q16×C73C2×C411yescharacteristic
142C2×SD16×C73C2×C411yescharacteristic
143C2×SD16×C73C2×C411yescharacteristic
144C2×M16×C73C2×C411yescharacteristic
145#11924037—81no—
146#11924037—81no—
147#11924037—81no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833—21no—
154C2×#10596833C2×D411yescharacteristic
155C2×#10596833C2×D411yescharacteristic
156C2×#10596833C2×D411yescharacteristic
157C2×#10596833C2×D411yescharacteristic
158C22×Dic73—21no—
159C22×Dic73—21no—
160C22×Dic73—21no—
161C22×Dic73C2×D411yescharacteristic
162C22×Dic73#4111yescharacteristic
163#11924012—41no—
164C2×#10596831C22×C411yescharacteristic
165C2×#10596831C22×C411yescharacteristic
166C4⋊Dic73—21no—
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73C22×C411yescharacteristic
174C4⋊Dic73C22×C411yescharacteristic
175C4⋊Dic73C22×C411yescharacteristic
176C4⋊Dic73C22×C411yescharacteristic
177C22⋊Dic73—21no—
178C22⋊Dic73—21no—
179C22⋊Dic73—21no—
180C4×Dic73—21no—
181C4×Dic73—21no—
182C4×Dic73—21no—
183C4×Dic73—21no—
184C4×Dic73—21no—
185C4×Dic73—21no—
186C4×Dic73—21no—
187C4×Dic73C22×C411yescharacteristic
188C4×Dic73C2×D411yescharacteristic
189C4×Dic73#4111yescharacteristic
190C4×Dic73C2411yescharacteristic
191#41×C73—22no—
192#41×C73—41no—
193C2×Q8×C73—21no—
194C2×Q8×C73—22no—
195C2×Q8×C73—41no—
196C2×Q8×C73C22×C411yescharacteristic
197C2×Q8×C73C22×C411yescharacteristic
198C2×Q8×C73C22×C411yescharacteristic
199C2×D4×C73—21no—
200C2×D4×C73C22×C411yescharacteristic
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203C22×C4×C73C22×C411yescharacteristic
204Q16×C73—22no—
205Q16×C73M1614yescomplemented
206SD16×C73—22no—
207SD16×C73—22no—
208M16×C73—22no—
209C2×C8×C73C22×C411yescharacteristic
210C2×C8×C73C22×C411yescharacteristic
211C23×C73—21no—
212C23×C73C4×D411yescharacteristic
213Q8×C73—21no—
214Q8×C73—21no—
215Q8×C73—21no—
216Q8×C73—21no—
217Q8×C73—42no—
218Q8×C73C2×M1612yes—
219Q8×C73C2×M1612yes—
220D4×C73—21no—
221D4×C73—21no—
222D4×C73—42no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—42no—
229C2×C4×C73—42no—
230C2×C4×C73C4×D411yescharacteristic
231C2×C4×C73C23×C411yescharacteristic
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—41no—
239C2×Dic73—42no—
240C2×Dic73—42no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73C4×D411yescharacteristic
244C2×Dic73C4×D411yescharacteristic
245C2×Dic73C22×D411yescharacteristic
246C2×Dic73C2×#4111yescharacteristic
247#10596833—21no—
248#10596833—21no—
249#10596833—21no—
250#10596833—21no—
251#10596833—41no—
252#10596833—41no—
253#10596833—41no—
254#10596833—41no—
255#10596833—41no—
256#10596833—41no—
257C8×C73—21no—
258C8×C73C2×M1612yes—
259#10596831—21no—
260#10596831—21no—
261C22×C73—21no—
262C22×C73—21no—
263C22×C73—21no—
264C22×C73—41no—
265C22×C73—41no—
266C22×C73—41no—
267C22×C73C2×C4×D411yescharacteristic
268C4×C73—21no—
269C4×C73—22no—
270C4×C73—22no—
271C4×C73—41no—
272C4×C73C22×M1611yescharacteristic
273C4×C73#56811yescharacteristic
274Dic73—21no—
275Dic73—21no—
276Dic73—21no—
277Dic73—41no—
278Dic73—41no—
279Dic73—41no—
280Dic73—41no—
281Dic73#57711yescharacteristic
282Dic73#57811yescharacteristic
283#18053—731nomaximal, Sylow
284C2×C73—41no—
285C2×C73—81no—
286C2×C73D4×M1611yescharacteristic
287C2×C73#294711yescharacteristic
288C2×C73#295511yescharacteristic
289#2991—731no—
290#2984—731no—
291C4×#137—731no—
292#1592—731no—
293#1590—731no—
294#1588—731no—
295#1587—731no—
296#1584—731no—
297#1583—731no—
298#1579—731no—
299C8×Q16—731no—
300#1569—731no—
301#1488—731no—
302#1487—731no—
303#1453—731no—
304C73#1805311yescharacteristic, complemented, Sylow
305C2×#137#6440811yescharacteristic
306C4×#41—731no—
307C2×C4×Q8—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—
316#441—731no—
317#441—731no—
318#440—731no—
319#440—731no—
320C4×Q16—731no—
321C4×Q16—731no—
322C4×SD16—731no—
323C4×SD16—731no—
324#436—731no—
325#436—731no—
326#436—1461no—
327#436—1461no—
328#432—731no—
329#432—1461no—
330#431—731no—
331#431—731no—
332#431—1461no—
333#431—1461no—
334#428—731no—
335#423—731no—
336#423—1461no—
337#419—731no—
338#417—731no—
339C4×M16—731no—
340#335—731no—
341#335—731no—
342#334—731no—
343#334—731no—
344Q8⋊C8—731no—
345Q8⋊C8—731no—
346Q8⋊C8—731no—
347Q8⋊C8—731no—
348Q8⋊C8—731no—
349Q8⋊C8—731no—
350D4⋊C8—731no—
351D4⋊C8—731no—
352#322—731no—
353#322—731no—
354#322—731no—
355C82—731no—
356C2×#41C2×#6440811yescharacteristic
357C22×Q8C2×#6440811yescharacteristic
358#137—24no—
359C2×Q16C2×#6440811yescharacteristic
360C2×Q16C2×#6440811yescharacteristic
361C2×SD16C2×#6440811yescharacteristic
362C2×SD16C2×#6440811yescharacteristic
363C2×M16—731no—
364C2×M16—1461no—
365C2×M16—1461no—
366C2×M16—1461no—
367C2×M16C2×#6440811yescharacteristic
368C4×Q8—731no—
369C4×Q8—731no—
370C4×Q8—731no—
371C4×Q8—731no—
372C4×Q8—731no—
373C4×Q8—731no—
374C4×Q8—1461no—
375C4×Q8—1461no—
376C4×D4—731no—
377C4×D4—731no—
378C4×D4—1461no—
379C4×D4—1461no—
380#117—731no—
381#117—1461no—
382C2×C4⋊C4—731no—
383C2×C4⋊C4—1461no—
384C2×C42—731no—
385C2×C42—1461no—
386C2×C42—1461no—
387#107—731no—
388#107—731no—
389#106—731no—
390#106—731no—
391C4⋊C8—731no—
392C4⋊C8—731no—
393C4⋊C8—731no—
394C4⋊C8—731no—
395C4⋊C8—731no—
396C4⋊C8—731no—
397C4⋊C8—1461no—
398C4⋊C8—1461no—
399C4⋊C8—1461no—
400C4⋊C8—1461no—
401C4⋊C8—1461no—
402C4⋊C8—1461no—
403C4⋊C8—1461no—
404C4⋊C8—1461no—
405#103—731no—
406#103—731no—
407#103—731no—
408#103—731no—
409#103—731no—
410#103—731no—
411#102—731no—
412#102—731no—
413C22⋊C8—731no—
414C22⋊C8—731no—
415C22⋊C8—1461no—
416C22⋊C8—1461no—
417C22⋊C8—1461no—
418C22⋊C8—1461no—
419#97—731no—
420#97—731no—
421#97—1461no—
422#97—1461no—
423#97—1461no—
424C4×C8—731no—
425C4×C8—731no—
426C4×C8—731no—
427C4×C8—731no—
428C4×C8—731no—
429C4×C8—731no—
430C4×C8—1461no—
431C4×C8—1461no—
432C4×C8—1461no—
433C4×C8—1461no—
434#41—22no—
435#41—41no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—41no—
439C2×Q8C22×#6440811yescharacteristic
440C2×Q8C22×#6440811yescharacteristic
441C2×Q8C22×#6440811yescharacteristic
442C2×D4—21no—
443C2×D4C22×#6440811yescharacteristic
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—731no—
447C22×C4—731no—
448C22×C4—1461no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4C22×#6440811yescharacteristic
452Q16—22no—
453Q16#1192403714yescomplemented
454SD16—22no—
455SD16—22no—
456M16—22no—
457M16—2921no—
458M16—5841no—
459M16—5841no—
460M16—5841no—
461C2×C8—731no—
462C2×C8—731no—
463C2×C8—731no—
464C2×C8—731no—
465C2×C8—731no—
466C2×C8—731no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8C22×#6440811yescharacteristic
474C2×C8C22×#6440811yescharacteristic
475C4⋊C4—731no—
476C4⋊C4—731no—
477C4⋊C4—731no—
478C4⋊C4—731no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C22⋊C4—1461no—
487C22⋊C4—1461no—
488C22⋊C4—1461no—
489C42—731no—
490C42—731no—
491C42—731no—
492C42—731no—
493C42—1461no—
494C42—1461no—
495C42—1461no—
496C42—1461no—
497C42—1461no—
498C42—1461no—
499C42—1461no—
500C23—21no—
501C23D4×#6440811yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—21no—
505Q8—21no—
506Q8—42no—
507Q8C2×#1192403712yes—
508Q8C2×#1192403712yes—
509D4—21no—
510D4—21no—
511D4—42no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—2922no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2922no—
523C2×C4—2921no—
524C2×C4—731no—
525C2×C4—731no—
526C2×C4—731no—
527C2×C4—731no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4D4×#6440811yescharacteristic
535C2×C4C23×#6440811yescharacteristic
536C8—21no—
537C8—2921no—
538C8—2921no—
539C8—2921no—
540C8—2921no—
541C8—2921no—
542C8—2921no—
543C8—1461no—
544C8—1461no—
545C8—1461no—
546C8—1461no—
547C8—1461no—
548C8—1461no—
549C8C2×#1192403712yes—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22C2×D4×#6440811yescharacteristic
557C4—21no—
558C4—22no—
559C4—22no—
560C4—41no—
561C4—2921no—
562C4—2921no—
563C4—2921no—
564C4—2921no—
565C4—731no—
566C4—731no—
567C4—1461no—
568C4—1461no—
569C4—1461no—
570C4C22×#1192403711yescharacteristic
571C4#1798966511yescharacteristic
572C2—41no—
573C2—81no—
574C2D4×#1192403711yescharacteristic
575C2#2262668911yescharacteristic
576C2#2262670411yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
FittingC2×#137×C73C4
socleC22×C73C2×C4×D4
prime radicalC2×#41×C73C2×C4
cosocleC4×Dic73C24
hypercenterC2×#137#64408
hypocenterC73#18053
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. #22626834C2semidirect
3. #22625574C2semidirect
4. #22625567C2semidirect
5. #22625560C2semidirect
6. #22625556C2semidirect
7. #22625544C2semidirect
8. #22625540C2semidirect
9. #22625524C2semidirect
10. Q16×#10596833C2semidirect
11. #22625495C2semidirect
13. #22625259C2semidirect
14. #22625150C2semidirect
205. Q16×C73M16semidirect
304. C73#18053semidirect
453. Q16#11924037semidirect

Maximal subgroups

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

subgroupindex
#226268632
#226268342
#226255742
#226255672
#226255602
#226255562
#226255442
#226255402
#226255242
Q16×#105968332
#226254952
#226252622
#226252592
#226251502
#137×Dic732
#1805373