Finite Group Databasebeta

← #10778223

Subgroups — #10778223

Order 768.

668 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79301C211yesmaximal, characteristic, complemented
2#79021C211yesmaximal, characteristic, complemented
3#78535C211yesmaximal, characteristic, complemented
4#78512C211yesmaximal, characteristic, complemented
5C3×#1565C211yesmaximal, characteristic, complemented
6#70775C211yesmaximal, characteristic, complemented
7#70433C211yesmaximal, characteristic, complemented
8#68590C211yesmaximal, characteristic, complemented
9#68586C211yesmaximal, characteristic, complemented
10#68584C211yesmaximal, characteristic, complemented
11#68583C211yesmaximal, characteristic, complemented
12#68548C211yesmaximal, characteristic, complemented
13#68547C211yesmaximal, characteristic, complemented
14#68544C211yesmaximal, characteristic, complemented
15#68540C211yesmaximal, characteristic, complemented
16#19184—31nomaximal, Sylow
17C2×#904—21no—
18#5477C2211yescharacteristic
19#5477C2211yescharacteristic
20C4×#235C2211yescharacteristic
21C3×#431C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C3×C4×M16C2211yescharacteristic
24#5135—21no—
25#5135—21no—
26#5128—21no—
27#5128—21no—
28#5113—21no—
29#4926C2211yescharacteristic
30#4850C2211yescharacteristic
31#4849C2211yescharacteristic
32#4848C2211yescharacteristic
33#4848C2211yescharacteristic
34#4845—21no—
35#4844—21no—
36#4843C2211yescharacteristic
37#4843C2211yescharacteristic
38#4842C2211yescharacteristic
39#4841C2211yescharacteristic
40#4737—21no—
41#4734—21no—
42#4711—21no—
43#4711—21no—
44#4707—21no—
45#4705—21no—
46#4695C2211yescharacteristic
47#4694C2211yescharacteristic
48#4694C2211yescharacteristic
49C4×D24—21no—
50C4×#204—21no—
51#4688C2211yescharacteristic
52#4688C2211yescharacteristic
53#4680C2211yescharacteristic
54C8×Dic6C2211yescharacteristic
55C8×Dic6C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#335C2211yescharacteristic
58C3×#334C2211yescharacteristic
59C3×#334C2211yescharacteristic
60#4485C2211yescharacteristic
61#4485C2211yescharacteristic
62#4485C2211yescharacteristic
63#4485C2211yescharacteristic
64#4482C2211yescharacteristic
65#4482C2211yescharacteristic
66#4481C2211yescharacteristic
67#4481C2211yescharacteristic
68#3184—31no—
69#3106—31no—
70#3035—31no—
71#2971—31no—
72#1701—31no—
73#1697—31no—
74#1691—31no—
75#1687—31no—
76#1682—31no—
77#1678—31no—
78#1672—31no—
79#1668—31no—
80#1565D311yescharacteristic, complemented
81#1553—31no—
82#1551—31no—
83C2×#235—21no—
84C2×#235—21no—
85C2×#235D411yescharacteristic
86C2×C3×M16D411yescharacteristic
87C2×C3×M16D411yescharacteristic
88C22×C3×C8—21no—
89C2×C3×C42C2311yescharacteristic
90#930—21no—
91#930—21no—
92#930—21no—
93#930—21no—
94C4×#67—21no—
95C4×#67—21no—
96#926D411yescharacteristic
97C2×Dic12—21no—
98#904—44no—
99C2×D24—21no—
100C2×D24—21no—
101C2×D24—21no—
102C2×D24—21no—
103C2×#204—21no—
104C2×#204—21no—
105C2×#204—21no—
106C2×ΓL(1,𝔽25)—21no—
107C2×ΓL(1,𝔽25)—21no—
108#875—21no—
109#875—21no—
110#874C2311yescharacteristic
111#874C2311yescharacteristic
112C4×D12—21no—
113C4×D12—21no—
114C4×D12—21no—
115C4×D12—21no—
116C4×D12D411yescharacteristic
117C4×D12C2311yescharacteristic
118#872—21no—
119C42×D3—21no—
120#869C2311yescharacteristic
121#869C2311yescharacteristic
122C4×Dic6D411yescharacteristic
123C4×Dic6C2311yescharacteristic
124C3×C4⋊C8—21no—
125C3×C4⋊C8C2311yescharacteristic
126C3×C4⋊C8C2311yescharacteristic
127C3×C4⋊C8C2311yescharacteristic
128C3×C4⋊C8C2311yescharacteristic
129C3×C22⋊C8—21no—
130C3×#97D411yescharacteristic
131C3×#97D411yescharacteristic
132C3×C4×C8—21no—
133C3×C4×C8C2311yescharacteristic
134C3×C4×C8C2311yescharacteristic
135#830—21no—
136#821—21no—
137#821—21no—
138#821—21no—
139#821—21no—
140#821—21no—
141#821—21no—
142#821—21no—
143#821—21no—
144#821—21no—
145(C2×D3)⋊C8—21no—
146(C2×D3)⋊C8—21no—
147#818—21no—
148#817—21no—
149#817—21no—
150#817—21no—
151#816—21no—
152#816—21no—
153#816—21no—
154Dic3⋊C8—21no—
155Dic3⋊C8—21no—
156C8×Dic3—21no—
157C8×Dic3—21no—
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4⋊(C3⋊C8)C2311yescharacteristic
160#803—21no—
161C2×#135—61no—
162#532—31no—
163#532—31no—
164C4×#41—31no—
165#498—31no—
166#493—31no—
167#486—31no—
168#486—31no—
169#482—61no—
170#477—31no—
171#476—31no—
172#469—61no—
173#469—61no—
174#464—61no—
175#463—61no—
176#462—31no—
177#461—31no—
178#460—31no—
179#460—31no—
180#459—31no—
181#459—31no—
182#449—61no—
183C8×Q8—31no—
184C8×Q8—31no—
185#442—61no—
186#442—61no—
187C4×SD16—61no—
188C4×D8—61no—
189#436—31no—
190#436—31no—
191#436—61no—
192#436—61no—
193#432—31no—
194#432—61no—
195#431C2×D311yescharacteristic
196C2×C4⋊C8C2×D311yescharacteristic
197#416—61no—
198C4×M16—61no—
199C4×M16C2×D311yescharacteristic
200#335C2×D311yescharacteristic
201#335C2×D311yescharacteristic
202#334C2×D311yescharacteristic
203#334C2×D311yescharacteristic
204#333—31no—
205#333—31no—
206#332—31no—
207#332—31no—
208#331—31no—
209#331—31no—
210#331—31no—
211#331—31no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4#4111yescharacteristic
215C2×#67—21no—
216C2×#67—41no—
217C2×#67—41no—
218#235—22no—
219#235—41no—
220#235—42no—
221#235—42no—
222#235—82no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12C2×D411yescharacteristic
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—41no—
234C2×C4×D3—41no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6C2×D411yescharacteristic
238C3×M16—22no—
239C3×M16—41no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—42no—
245C2×C3×C8C2×D411yescharacteristic
246C2×C3×C8C2×D411yescharacteristic
247C2×C3×C8C2×D411yescharacteristic
248C2×C3×C8C2×D411yescharacteristic
249C3×C42C2×D411yescharacteristic
250C3×C42C2×D411yescharacteristic
251C3×C42#4111yescharacteristic
252C3×C42C2411yescharacteristic
253#217—21no—
254(C2×D3)⋊C4—21no—
255(C2×D3)⋊C4—21no—
256(C2×D3)⋊C4—41no—
257(C2×D3)⋊C4—41no—
258(C2×D3)⋊C4—41no—
259(C2×D3)⋊C4—41no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3C2×D411yescharacteristic
265C4⋊Dic3C2×D411yescharacteristic
266Dic3⋊C4—21no—
267Dic3⋊C4—21no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273Dic12—42no—
274D24—42no—
275D24—42no—
276D24—42no—
277D24—42no—
278#204—42no—
279#204—42no—
280#204—42no—
281ΓL(1,𝔽25)—42no—
282ΓL(1,𝔽25)—42no—
283C2×#41—31no—
284C2×#41—61no—
285C2×#41—61no—
286#135—124no—
287C2×Q16—61no—
288C2×SD16—61no—
289C2×SD16—61no—
290C2×SD16—61no—
291C2×D8—61no—
292C2×D8—61no—
293C2×D8—61no—
294C2×D8—61no—
295C2×M16—61no—
296C2×M16—61no—
297C2×M16D1211yescharacteristic
298C2×M16D1211yescharacteristic
299C22×C8—21no—
300C4⋊Q8—31no—
301C4⋊Q8—31no—
302#127—31no—
303#127—31no—
304#124—61no—
305#124—61no—
306#121—61no—
307#121—61no—
308#121—61no—
309#121—61no—
310C4×Q8—31no—
311C4×Q8—31no—
312C4×D4—31no—
313C4×D4—31no—
314C4×D4—61no—
315C4×D4—61no—
316C4×D4—61no—
317C4×D4—61no—
318C4×D4—61no—
319C4×D4—61no—
320#117—31no—
321#117—61no—
322C2×C42—61no—
323C2×C42C22×D311yescharacteristic
324#107—61no—
325#106—61no—
326#106—61no—
327#106—61no—
328C4⋊C8—21no—
329C4⋊C8—31no—
330C4⋊C8—31no—
331C4⋊C8—61no—
332C4⋊C8—61no—
333C4⋊C8C22×D311yescharacteristic
334C4⋊C8C22×D311yescharacteristic
335C4⋊C8C22×D311yescharacteristic
336C4⋊C8C22×D311yescharacteristic
337#103—61no—
338#103—61no—
339#103—61no—
340#102—61no—
341#102—61no—
342#102—61no—
343#102—61no—
344#102—61no—
345#102—61no—
346#102—61no—
347#102—61no—
348#102—61no—
349C22⋊C8—21no—
350C22⋊C8—61no—
351C22⋊C8—61no—
352C22⋊C8—61no—
353#97—61no—
354#97D1211yescharacteristic
355#97D1211yescharacteristic
356C4×C8—21no—
357C4×C8—61no—
358C4×C8—61no—
359C4×C8C22×D311yescharacteristic
360C4×C8C22×D311yescharacteristic
361C23×C3#12111yescharacteristic
362C22×D3—21no—
363C22×D3—41no—
364C22×D3—41no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—41no—
370C2×C3×C4#12111yescharacteristic
371C2×C3×C4#12111yescharacteristic
372C2×C3×C4#12111yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376#67—42no—
377#67—82no—
378#67—82no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384D12—21no—
385D12—21no—
386D12—41no—
387D12—41no—
388D12—41no—
389D12—41no—
390D12—41no—
391D12—41no—
392D12—41no—
393D12—41no—
394D12—81no—
395D12—81no—
396C4×D3—42no—
397C4×D3—41no—
398C4×D3—41no—
399C4×D3—81no—
400C4×D3—82no—
401C4×D3—82no—
402Dic6—21no—
403Dic6—41no—
404Dic6—41no—
405Dic6—81no—
406Dic6AGL(1,ℤ/8)12yes—
407C3×C8—21no—
408C3×C8—21no—
409C3×C8—21no—
410C3×C8—41no—
411C3×C8—41no—
412C3×C8—41no—
413C3×C8—41no—
414C3×C8#13512yescomplemented
415C3⋊C8—41no—
416C3⋊C8—41no—
417#41—62no—
418#41—121no—
419#41—122no—
420#41—122no—
421#41—242no—
422C2×Q8—31no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×D4—31no—
426C2×D4—61no—
427C2×D4—61no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—121no—
434C2×D4—121no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—121no—
439C22×C4—121no—
440C22×C4C2×D1211yescharacteristic
441C22×C4D3×D411yescharacteristic
442C22×C4#23911yescharacteristic
443Q16—122no—
444SD16—122no—
445SD16—122no—
446SD16—122no—
447D8—122no—
448D8—122no—
449D8—122no—
450D8—122no—
451M16—22no—
452M16—41no—
453M16—122no—
454M16—122no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—42no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8C2×D1211yescharacteristic
463C2×C8C2×D1211yescharacteristic
464C2×C8C2×D1211yescharacteristic
465C2×C8C2×D1211yescharacteristic
466C4⋊C4—31no—
467C4⋊C4—31no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C22⋊C4—61no—
475C22⋊C4—61no—
476C22⋊C4—61no—
477C22⋊C4—121no—
478C22⋊C4—121no—
479C22⋊C4—121no—
480C22⋊C4—121no—
481C42—61no—
482C42—61no—
483C42—61no—
484C42C2×D1211yescharacteristic
485C42D3×D411yescharacteristic
486C42#23911yescharacteristic
487C42C23×D311yescharacteristic
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3C2×#12111yescharacteristic
492C2×D3—41no—
493C2×D3—41no—
494C2×D3—41no—
495C2×D3—81no—
496C2×D3—81no—
497C2×D3—81no—
498C2×D3—81no—
499C2×D3—81no—
500C2×D3—81no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4C2×#13511yescharacteristic
506C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
507C3×C4#57511yescharacteristic
508C3×C4#57511yescharacteristic
509Dic3—22no—
510Dic3—41no—
511Dic3—41no—
512Dic3—81no—
513Dic3—81no—
514C23—61no—
515C23—121no—
516C23—121no—
517C23#89511yescharacteristic
518Q8—32no—
519Q8—61no—
520Q8—121no—
521Q8—121no—
522Q8—241no—
523D4—61no—
524D4—61no—
525D4—122no—
526D4—121no—
527D4—121no—
528D4—121no—
529D4—121no—
530D4—121no—
531D4—121no—
532D4—121no—
533D4—121no—
534D4—241no—
535D4—241no—
536D4—242no—
537D4—242no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—122no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—241no—
552C2×C4—242no—
553C2×C4—242no—
554C2×C4#89511yescharacteristic
555C2×C4#89511yescharacteristic
556C2×C4#89511yescharacteristic
557C2×C4C22×D1211yescharacteristic
558C2×C4C2×D3×D411yescharacteristic
559C2×C4C2×#23911yescharacteristic
560C8—21no—
561C8—21no—
562C8—21no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—121no—
568C8—121no—
569C8#90412yescomplemented
570C2×C3—41no—
571C2×C3#303511yescharacteristic
572C2×C3#305511yescharacteristic
573C2×C3#306811yescharacteristic
574D3—81no—
575D3—161no—
576D3—161no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—121no—
581C22—121no—
582C22—121no—
583C22—241no—
584C22—241no—
585C22—241no—
586C22—241no—
587C22—241no—
588C22—241no—
589C22C2×#89511yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—62no—
595C4—121no—
596C4—121no—
597C4—241no—
598C4—241no—
599C4C2×#90411yescharacteristic
600C4#574811yescharacteristic
601C4D3×AGL(1,ℤ/8)11yescharacteristic
602C4#577711yescharacteristic
603C3#1918411yescharacteristic, complemented, Sylow
604C2—41no—
605C2—241no—
606C2—481no—
607C2—481no—
608C2#7902111yescharacteristic
609C2#7921111yescharacteristic
610C2#7926111yescharacteristic
subgroupG/H
centerC22C2×#895
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1565C2
socleC22×C3C2×#121
hypercenter#1565D3
hypocenterC3#19184
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
1. #79301C2semidirect
2. #79021C2semidirect
3. #78535C2semidirect
4. #78512C2semidirect
5. C3×#1565C2semidirect
6. #70775C2semidirect
7. #70433C2semidirect
8. #68590C2semidirect
9. #68586C2semidirect
10. #68584C2semidirect
11. #68583C2semidirect
12. #68548C2semidirect
13. #68547C2semidirect
14. #68544C2semidirect
15. #68540C2semidirect
80. #1565D3semidirect
414. C3×C8#135semidirect
569. C8#904semidirect
603. C3#19184semidirect

Maximal subgroups

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

subgroupindex
#793012
#790212
#785352
#785122
C3×#15652
#707752
#704332
#685902
#685862
#685842
#685832
#685482
#685472
#685442
#685402
#191843