Finite Group Databasebeta

← #10782801

Subgroups — #10782801

Order 768.

580 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79358C211yesmaximal, characteristic
2#79296C211yesmaximal, characteristic
3#79272C211yesmaximal, characteristic, complemented
4#79029C211yesmaximal, characteristic
5C3×#1573C211yesmaximal, characteristic, complemented
6#70820C211yesmaximal, characteristic, complemented
7C2×#4482C211yesmaximal, characteristic, complemented
8#68595C211yesmaximal, characteristic, complemented
9#68588C211yesmaximal, characteristic, complemented
10#68585C211yesmaximal, characteristic, complemented
11#68578C211yesmaximal, characteristic, complemented
12#68552C211yesmaximal, characteristic, complemented
13#68549C211yesmaximal, characteristic, complemented
14#68542C211yesmaximal, characteristic, complemented
15#68539C211yesmaximal, characteristic, complemented
16#19590—31nomaximal, Sylow
17#5479C2211yescharacteristic
18#5470C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C3×#432C2211yescharacteristic
21C3×#431C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23#5116—21no—
24#5114—21no—
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26#4851C2211yescharacteristic
27#4850—21no—
28#4850—21no—
29#4849C2211yescharacteristic
30#4849C2211yescharacteristic
31#4849C2211yescharacteristic
32#4846C2211yescharacteristic
33#4845—21no—
34#4841C2211yescharacteristic
35#4841C2211yescharacteristic
36#4841C2211yescharacteristic
37#4840—21no—
38#4839—21no—
39#4829—21no—
40#4739—21no—
41#4738—21no—
42#4730—21no—
43#67⋊C8—21no—
44#4722—21no—
45#4722—21no—
46#4715—21no—
47#4703—21no—
48#4688C2211yescharacteristic
49C8×D12C2211yescharacteristic
50#4682C2211yescharacteristic
51#4682C2211yescharacteristic
52#4681C2211yescharacteristic
53#4681C2211yescharacteristic
54#4679C2211yescharacteristic
55C8×Dic6C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#335C2211yescharacteristic
58C3×#334C2211yescharacteristic
59C3×#334C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4482C2211yescharacteristic
64#4482C2211yescharacteristic
65#4482C2211yescharacteristic
66#4482C2211yescharacteristic
67#4482C2211yescharacteristic
68#3248—31no—
69#3230—31no—
70#3218—31no—
71#2994—31no—
72#1700—31no—
73#1699—31no—
74#1694—31no—
75#1693—31no—
76#1680—31no—
77#1679—31no—
78#1674—31no—
79#1673—31no—
80#1573D311yescharacteristic, complemented
81#1552—31no—
82C2×#332—31no—
83C22×Dic6—21no—
84C22×C3×C8—21no—
85C2×C3×C42C2311yescharacteristic
86#930D411yescharacteristic
87#929—21no—
88C4×#67—21no—
89#926—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—21no—
92#924—21no—
93#924—21no—
94#924D411yescharacteristic
95C22×C3⋊C8—21no—
96C2×Dic12—21no—
97C2×Dic12—21no—
98C2×Dic12—21no—
99C2×#204—21no—
100C2×#204—21no—
101C2×#204—21no—
102C2×ΓL(1,𝔽25)—21no—
103C2×D3×C8—21no—
104#876—21no—
105#875D411yescharacteristic
106C4×D12C2311yescharacteristic
107#872—21no—
108#870—21no—
109#870D411yescharacteristic
110#869—21no—
111#869—21no—
112#869—21no—
113#869C2311yescharacteristic
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116#869C2311yescharacteristic
117C4×Dic6—21no—
118C4×Dic6—21no—
119C4×Dic6C2311yescharacteristic
120C3×C4⋊C8—21no—
121C3×C4⋊C8D411yescharacteristic
122C3×C4⋊C8D411yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×C4⋊C8C2311yescharacteristic
125C3×C4⋊C8C2311yescharacteristic
126C3×C4⋊C8C2311yescharacteristic
127C3×C22⋊C8—21no—
128C3×C22⋊C8D411yescharacteristic
129C3×C22⋊C8D411yescharacteristic
130C3×#97—21no—
131C3×C4×C8C2311yescharacteristic
132C3×C4×C8C2311yescharacteristic
133#821—21no—
134(C2×D3)⋊C8—21no—
135(C2×D3)⋊C8—21no—
136#818—21no—
137#818—21no—
138#818—21no—
139#818—21no—
140#818—21no—
141#817—21no—
142#816—21no—
143#816—21no—
144#816—21no—
145#816—21no—
146#816—21no—
147#816—21no—
148#816—21no—
149#816—21no—
150#816—21no—
151#816—21no—
152#816—21no—
153#815—21no—
154Dic3⋊C8—21no—
155Dic3⋊C8—21no—
156C8×Dic3—21no—
157C4⋊(C3⋊C8)—21no—
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4⋊(C3⋊C8)C2311yescharacteristic
160#533—31no—
161C2×C4⋊Q8—31no—
162#529—31no—
163#500—31no—
164#500—31no—
165#497—61no—
166#491—61no—
167#487—31no—
168#487—31no—
169#484—61no—
170#484—61no—
171#479—61no—
172#477—61no—
173#477—61no—
174#475—31no—
175#474—31no—
176#467—61no—
177#465—61no—
178#464—61no—
179#462—31no—
180#462—31no—
181#462—31no—
182#461—31no—
183#461—31no—
184#461—31no—
185#451—61no—
186#448—61no—
187#446—31no—
188C8×Q8—31no—
189#436—31no—
190#435—61no—
191C8×D4—31no—
192C8×D4—61no—
193#433—61no—
194#432C2×D311yescharacteristic
195#431C2×D311yescharacteristic
196#424—61no—
197C2×C4⋊C8—31no—
198C2×C4⋊C8C2×D311yescharacteristic
199#335C2×D311yescharacteristic
200#335C2×D311yescharacteristic
201#334C2×D311yescharacteristic
202#334C2×D311yescharacteristic
203#332—31no—
204#332—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#332—31no—
210#332—31no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4#4111yescharacteristic
214C2×#67—21no—
215C22×Dic3—41no—
216C2×D12C2×D411yescharacteristic
217C2×C4×D3—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—41no—
223C2×Dic6—81no—
224C2×Dic6—81no—
225C2×Dic6C2×D411yescharacteristic
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—81no—
231C2×C3×C8C2×D411yescharacteristic
232C2×C3×C8C2×D411yescharacteristic
233C2×C3×C8C2×D411yescharacteristic
234C2×C3×C8C2×D411yescharacteristic
235C3×C42C2×D411yescharacteristic
236C3×C42C2×D411yescharacteristic
237C3×C42#4111yescharacteristic
238C3×C42C2411yescharacteristic
239#217—21no—
240#217—41no—
241(C2×D3)⋊C4—21no—
242(C2×D3)⋊C4—21no—
243(C2×D3)⋊C4—21no—
244(C2×D3)⋊C4—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—41no—
254C4⋊Dic3—41no—
255C4⋊Dic3C2×D411yescharacteristic
256C4⋊Dic3C2×D411yescharacteristic
257Dic3⋊C4—21no—
258Dic3⋊C4—21no—
259Dic3⋊C4—21no—
260Dic3⋊C4—21no—
261Dic3⋊C4—41no—
262Dic3⋊C4—41no—
263C4×Dic3—21no—
264C4×Dic3—41no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—81no—
268Dic12—81no—
269Dic12—81no—
270Dic12—81no—
271#204—81no—
272#204—81no—
273#204—81no—
274ΓL(1,𝔽25)—81no—
275D3×C8—81no—
276C22×Q8—61no—
277C2×Q16—61no—
278C2×Q16—61no—
279C2×Q16—61no—
280C2×SD16—61no—
281C2×SD16—61no—
282C2×SD16—61no—
283C2×M16—61no—
284C22×C8—21no—
285C22×C8—61no—
286C22×C8—61no—
287C4⋊Q8—31no—
288C4⋊Q8—31no—
289C4⋊Q8—31no—
290C4⋊Q8—31no—
291C4⋊Q8—61no—
292C4⋊Q8—61no—
293C4⋊Q8—61no—
294#126—61no—
295#125—31no—
296#125—61no—
297#124—31no—
298#123—61no—
299C22⋊Q8—31no—
300C22⋊Q8—61no—
301C22⋊Q8—61no—
302#121—31no—
303C4×Q8—31no—
304C4×Q8—61no—
305C4×Q8—61no—
306C4×D4—31no—
307C4×D4—61no—
308#117—61no—
309#117—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C42C22×D311yescharacteristic
313#107—61no—
314#107—61no—
315#107—61no—
316#107—61no—
317#107—61no—
318#106—61no—
319C4⋊C8—21no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—61no—
323C4⋊C8—61no—
324C4⋊C8—61no—
325C4⋊C8D1211yescharacteristic
326C4⋊C8D1211yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328C4⋊C8C22×D311yescharacteristic
329C4⋊C8C22×D311yescharacteristic
330C4⋊C8C22×D311yescharacteristic
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#103—61no—
338#103—61no—
339#103—61no—
340#103—61no—
341#103—61no—
342#102—61no—
343C22⋊C8—21no—
344C22⋊C8—61no—
345C22⋊C8—61no—
346C22⋊C8D1211yescharacteristic
347C22⋊C8D1211yescharacteristic
348#97—21no—
349#97—61no—
350C4×C8—61no—
351C4×C8C22×D311yescharacteristic
352C4×C8C22×D311yescharacteristic
353C23×C3#12111yescharacteristic
354C22×D3—21no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4#12111yescharacteristic
361C2×C3×C4#12111yescharacteristic
362C2×C3×C4#12111yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366#67—42no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—82no—
375D12—21no—
376D12—21no—
377C4×D3—41no—
378C4×D3—41no—
379Dic6—21no—
380Dic6—21no—
381Dic6—41no—
382Dic6—41no—
383Dic6—41no—
384Dic6—41no—
385Dic6—41no—
386Dic6—41no—
387Dic6—41no—
388Dic6—41no—
389Dic6—82no—
390C3×C8—21no—
391C3×C8—21no—
392C3×C8—21no—
393C3×C8—21no—
394C3×C8—41no—
395C3×C8—41no—
396C3×C8—41no—
397C3×C8—41no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C2×Q8—31no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—121no—
406C2×Q8—241no—
407C2×Q8—241no—
408C2×D4—31no—
409C2×D4—61no—
410C22×C4—61no—
411C22×C4—121no—
412C22×C4C2×D1211yescharacteristic
413C22×C4D3×D411yescharacteristic
414C22×C4#23911yescharacteristic
415Q16—241no—
416Q16—241no—
417Q16—241no—
418SD16—241no—
419SD16—241no—
420SD16—241no—
421M16—241no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—81no—
429C2×C8—241no—
430C2×C8—241no—
431C2×C8C2×D1211yescharacteristic
432C2×C8C2×D1211yescharacteristic
433C2×C8C2×D1211yescharacteristic
434C2×C8C2×D1211yescharacteristic
435C4⋊C4—31no—
436C4⋊C4—31no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—121no—
450C4⋊C4—121no—
451C4⋊C4—121no—
452C4⋊C4—121no—
453C22⋊C4—61no—
454C22⋊C4—61no—
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—61no—
458C22⋊C4—121no—
459C42—61no—
460C42—121no—
461C42C2×D1211yescharacteristic
462C42D3×D411yescharacteristic
463C42#23911yescharacteristic
464C42C23×D311yescharacteristic
465C22×C3—21no—
466C22×C3—21no—
467C22×C3#47312yes—
468C22×C3C2×#12111yescharacteristic
469C2×D3—41no—
470C2×D3—41no—
471C2×D3—41no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4#57611yescharacteristic
477C3×C4#57711yescharacteristic
478C3×C4#57711yescharacteristic
479C3×C4#57811yescharacteristic
480Dic3—41no—
481Dic3—41no—
482Dic3—41no—
483Dic3—81no—
484Dic3—81no—
485Dic3—81no—
486Dic3—81no—
487C23—61no—
488C23#89511yescharacteristic
489Q8—61no—
490Q8—61no—
491Q8—121no—
492Q8—121no—
493Q8—121no—
494Q8—121no—
495Q8—121no—
496Q8—121no—
497Q8—121no—
498Q8—121no—
499Q8—242no—
500D4—61no—
501D4—61no—
502D4—122no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—242no—
518C2×C4#89511yescharacteristic
519C2×C4#89511yescharacteristic
520C2×C4#89511yescharacteristic
521C2×C4C22×D1211yescharacteristic
522C2×C4C2×D3×D411yescharacteristic
523C2×C4C2×#23911yescharacteristic
524C8—21no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—41no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—121no—
533C8—121no—
534C2×C3—22no—
535C2×C3#304611yescharacteristic
536C2×C3#306211yescharacteristic
537C2×C3C2×#47311yescharacteristic
538D3—81no—
539C22—21no—
540C22—21no—
541C22—121no—
542C22—121no—
543C22—121no—
544C22#489612yes—
545C22C2×#89511yescharacteristic
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—121no—
551C4—121no—
552C4—121no—
553C4—241no—
554C4—241no—
555C4—241no—
556C4—241no—
557C4#575111yescharacteristic
558C4#575311yescharacteristic
559C4#577411yescharacteristic
560C4#577811yescharacteristic
561C3#1959011yescharacteristic, complemented, Sylow
562C2—22no—
563C2—241no—
564C2#7908911yescharacteristic
565C2#7921311yescharacteristic
566C2C2×#489611yescharacteristic
subgroupG/H
centerC22C2×#895
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1573C2
socleC22×C3C2×#121
prime radical#930D4
hypercenter#1573D3
hypocenterC3#19590
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1573

Complements

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

NH ≅ G/Ntype
3. #79272C2semidirect
5. C3×#1573C2semidirect
6. #70820C2semidirect
7. C2×#4482C2semidirect
8. #68595C2semidirect
9. #68588C2semidirect
10. #68585C2semidirect
11. #68578C2semidirect
12. #68552C2semidirect
13. #68549C2semidirect
14. #68542C2semidirect
15. #68539C2semidirect
80. #1573D3semidirect
561. C3#19590semidirect

Maximal subgroups

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

subgroupindex
#793582
#792962
#792722
#790292
C3×#15732
#708202
C2×#44822
#685952
#685882
#685852
#685782
#685522
#685492
#685422
#685392
#195903