Finite Group Databasebeta

← #10777931

Subgroups — #10777931

Order 768.

642 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83299C211yesmaximal, characteristic, complemented
2#83291C211yesmaximal, characteristic, complemented
3#79301C211yesmaximal, characteristic, complemented
4#79277C211yesmaximal, characteristic, complemented
5C3×#1543C211yesmaximal, characteristic, complemented
6#70773C211yesmaximal, characteristic, complemented
7#70745C211yesmaximal, characteristic, complemented
8#68514C211yesmaximal, characteristic, complemented
9#68513C211yesmaximal, characteristic, complemented
10#68502C211yesmaximal, characteristic, complemented
11#68501C211yesmaximal, characteristic, complemented
12#68414C211yesmaximal, characteristic, complemented
13#68413C211yesmaximal, characteristic, complemented
14#68400C211yesmaximal, characteristic, complemented
15#68399C211yesmaximal, characteristic, complemented
16#19160—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18#5477C2211yescharacteristic
19C4×#235C2211yescharacteristic
20C3×#431C2211yescharacteristic
21C3×#431C2211yescharacteristic
22#5091C2211yescharacteristic
23#5088C2211yescharacteristic
24#5084C2211yescharacteristic
25#5078C2211yescharacteristic
26#5075C2211yescharacteristic
27#5073C2211yescharacteristic
28#5072C2211yescharacteristic
29#5070C2211yescharacteristic
30#5038—21no—
31#5038—21no—
32#5037—21no—
33#5035—21no—
34C4×#214—21no—
35C4×#213—21no—
36C4×#213—21no—
37C4×#213—21no—
38#4925C2211yescharacteristic
39C2×C4×C3⋊C8C2211yescharacteristic
40#4850C2211yescharacteristic
41#4848C2211yescharacteristic
42#4843C2211yescharacteristic
43#4842C2211yescharacteristic
44#4838C2211yescharacteristic
45#4838C2211yescharacteristic
46#4831C2211yescharacteristic
47#4831C2211yescharacteristic
48#4737—21no—
49#4734—21no—
50#4730—21no—
51#4730—21no—
52C4×D24—21no—
53C4×#204—21no—
54C4×ΓL(1,𝔽25)—21no—
55#4685—21no—
56C3×#333C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#331C2211yescharacteristic
59C3×#331C2211yescharacteristic
60#4485C2211yescharacteristic
61#4485C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3169—31no—
69#3106—31no—
70#3106—31no—
71#2965—31no—
72#1698—31no—
73#1698—31no—
74#1697—31no—
75#1697—31no—
76#1696—31no—
77#1695—31no—
78#1688—31no—
79#1687—31no—
80#1564—31no—
81#1543—31no—
82#1543D311yescharacteristic, complemented
83C2×C3×#41—21no—
84C2×#235—21no—
85C2×#235D411yescharacteristic
86C3×C4⋊Q8C2311yescharacteristic
87C3×#127C2311yescharacteristic
88C3×#124—21no—
89C3×#121—21no—
90C3×#121—21no—
91C3×C4×D4—21no—
92C3×C4×D4—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#214—21no—
95C2×#213—21no—
96C2×#213—21no—
97C2×#213—21no—
98#930—21no—
99#930—21no—
100C4×#67—21no—
101C4×#67—21no—
102#926D411yescharacteristic
103C22×C3⋊C8D411yescharacteristic
104C22×C3⋊C8D411yescharacteristic
105C2×D24—21no—
106C2×#204—21no—
107C2×ΓL(1,𝔽25)—21no—
108C2×ΓL(1,𝔽25)—21no—
109#875—21no—
110#874C2311yescharacteristic
111C4×D12—21no—
112C4×D12—21no—
113C4×D12D411yescharacteristic
114C4×D12C2311yescharacteristic
115#872—21no—
116C42×D3—21no—
117#869C2311yescharacteristic
118C4×Dic6D411yescharacteristic
119C4×Dic6C2311yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×C22⋊C8—21no—
125C3×C22⋊C8—21no—
126C3×C4×C8—21no—
127C3×C4×C8—21no—
128#832—21no—
129#832—21no—
130#832—21no—
131#832—21no—
132#830—21no—
133#821—21no—
134#821—21no—
135#821—21no—
136(C2×D3)⋊C8—21no—
137(C2×D3)⋊C8—21no—
138#818—21no—
139#817—21no—
140#816—21no—
141#815—21no—
142#815—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8—21no—
145#810—21no—
146#809—21no—
147#809—21no—
148#809—21no—
149#808—21no—
150#807—21no—
151#807—21no—
152#807—21no—
153C4⋊(C3⋊C8)C2311yescharacteristic
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4×C3⋊C8—21no—
156C4×C3⋊C8D411yescharacteristic
157C4×C3⋊C8D411yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#532—31no—
161#532C2×D311yescharacteristic
162C4×#41—31no—
163#500—31no—
164#493—31no—
165#486—31no—
166#486—31no—
167#482—61no—
168#482—61no—
169#477—31no—
170#477—31no—
171#476—31no—
172#476—31no—
173#466—61no—
174#466—61no—
175#460—31no—
176#460—31no—
177#459—31no—
178#459—31no—
179#449—61no—
180#449—61no—
181#446—31no—
182#446—31no—
183C4×SD16—61no—
184C4×SD16—61no—
185C4×D8—61no—
186C4×D8—61no—
187C4×D8—61no—
188C4×D8—61no—
189#436—31no—
190#436—31no—
191#435—61no—
192#435—61no—
193#431—31no—
194#431—61no—
195#431C2×D311yescharacteristic
196#431C2×D311yescharacteristic
197C4×M16—61no—
198C2×C4×C8—31no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#333C2×D311yescharacteristic
206#333C2×D311yescharacteristic
207#331—31no—
208#331—31no—
209#331C2×D311yescharacteristic
210#331C2×D311yescharacteristic
211C3×#41—82no—
212C2×C3×Q8—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—41no—
217C22×C3×C4—41no—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4#4111yescharacteristic
221C2×#67—21no—
222C2×#67—41no—
223#235—41no—
224#235—41no—
225#235—82no—
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×Dic6—21no—
235C2×Dic6C2×D411yescharacteristic
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C22⋊C4—41no—
243C3×C22⋊C4—41no—
244C3×C42C2×D411yescharacteristic
245C3×C42C2×D411yescharacteristic
246C3×C42#4111yescharacteristic
247C3×C42C2411yescharacteristic
248#217—21no—
249#214—42no—
250#213—42no—
251#213—42no—
252#213—42no—
253(C2×D3)⋊C4—21no—
254(C2×D3)⋊C4—21no—
255(C2×D3)⋊C4—41no—
256(C2×D3)⋊C4—41no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3C2×D411yescharacteristic
260C4⋊Dic3C2×D411yescharacteristic
261Dic3⋊C4—21no—
262Dic3⋊C4—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C2×C3⋊C8—22no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—41no—
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274D24—42no—
275#204—42no—
276ΓL(1,𝔽25)—42no—
277ΓL(1,𝔽25)—42no—
278C2×#41—21no—
279C2×#41—31no—
280C2×#41—61no—
281C2×SD16—61no—
282C2×SD16—61no—
283C2×D8—61no—
284C2×D8—61no—
285C2×D8—61no—
286C2×D8—61no—
287C2×M16—61no—
288C2×M16—61no—
289C22×C8—31no—
290C22×C8—31no—
291C4⋊Q8—31no—
292C4⋊Q8C22×D311yescharacteristic
293#127—31no—
294#127C22×D311yescharacteristic
295#124—21no—
296#124—61no—
297#121—21no—
298#121—21no—
299#121—61no—
300#121—61no—
301C4×Q8—31no—
302C4×Q8—31no—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—31no—
306C4×D4—31no—
307C4×D4—61no—
308C4×D4—61no—
309C4×D4—61no—
310C4×D4—61no—
311#117—31no—
312#117—61no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—61no—
316#107—61no—
317#107—61no—
318#107—61no—
319#106—61no—
320#106—61no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—61no—
324C4⋊C8—61no—
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328C4⋊C8C22×D311yescharacteristic
329#103—61no—
330#103—61no—
331#102—61no—
332#102—61no—
333#102—61no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339#102—61no—
340#102—61no—
341C22⋊C8—21no—
342C22⋊C8—21no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345C22⋊C8—61no—
346#97—61no—
347#97—61no—
348C4×C8—21no—
349C4×C8—21no—
350C4×C8—31no—
351C4×C8—31no—
352C4×C8—31no—
353C4×C8—31no—
354C4×C8—61no—
355C23×C3—41no—
356C23×C3#12111yescharacteristic
357C22×D3—21no—
358C22×D3—41no—
359C3×Q8—81no—
360C3×D4—41no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—41no—
364C3×D4—81no—
365C3×D4—82no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—22no—
371C2×C3×C4—41no—
372C2×C3×C4—82no—
373C2×C3×C4#12111yescharacteristic
374C2×C3×C4#12111yescharacteristic
375C2×C3×C4#12111yescharacteristic
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379#67—42no—
380#67—82no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—41no—
385D12—21no—
386D12—21no—
387D12—41no—
388D12—41no—
389D12—41no—
390D12—41no—
391D12—81no—
392C4×D3—42no—
393C4×D3—41no—
394C4×D3—41no—
395C4×D3—42no—
396C4×D3—82no—
397Dic6—21no—
398Dic6—21no—
399Dic6—81no—
400C3×C8—41no—
401C3×C8—41no—
402C3×C8—41no—
403C3×C8—41no—
404C3⋊C8—21no—
405C3⋊C8—21no—
406C3⋊C8—21no—
407C3⋊C8—41no—
408C3⋊C8—41no—
409C3⋊C8#13512yes—
410#41—82no—
411#41—121no—
412#41—121no—
413#41—242no—
414C2×Q8—21no—
415C2×Q8—31no—
416C2×Q8—61no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—31no—
421C2×D4—41no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—121no—
427C22×C4—41no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—121no—
432C22×C4#23511yescharacteristic
433C22×C4D3×D411yescharacteristic
434C22×C4D3×D411yescharacteristic
435SD16—122no—
436SD16—122no—
437D8—122no—
438D8—122no—
439D8—122no—
440D8—122no—
441M16—122no—
442M16—122no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—31no—
448C2×C8—31no—
449C2×C8—31no—
450C2×C8—31no—
451C2×C8—62no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—121no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—31no—
458C4⋊C4—31no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—61no—
466C22⋊C4—61no—
467C22⋊C4—61no—
468C22⋊C4—121no—
469C22⋊C4—121no—
470C42—61no—
471C42—61no—
472C42—61no—
473C42#23511yescharacteristic
474C42D3×D411yescharacteristic
475C42D3×D411yescharacteristic
476C42C23×D311yescharacteristic
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3C2×#12111yescharacteristic
484C2×D3—41no—
485C2×D3—41no—
486C2×D3—41no—
487C2×D3—81no—
488C2×D3—81no—
489C2×D3—81no—
490C3×C4—21no—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—81no—
494C3×C4#47212yes—
495C3×C4C2×#13511yescharacteristic
496C3×C4#57511yescharacteristic
497C3×C4#57511yescharacteristic
498C3×C4#57511yescharacteristic
499Dic3—41no—
500Dic3—41no—
501Dic3—41no—
502Dic3—81no—
503C23—41no—
504C23—61no—
505C23—121no—
506C23#88411yescharacteristic
507Q8—61no—
508Q8—61no—
509Q8—81no—
510Q8—241no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—61no—
516D4—61no—
517D4—81no—
518D4—82no—
519D4—122no—
520D4—121no—
521D4—121no—
522D4—121no—
523D4—121no—
524D4—241no—
525D4—242no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—82no—
536C2×C4—122no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—122no—
540C2×C4—121no—
541C2×C4—242no—
542C2×C4#88411yescharacteristic
543C2×C4#88411yescharacteristic
544C2×C4#88411yescharacteristic
545C2×C4C2×#23511yescharacteristic
546C2×C4C2×D3×D411yescharacteristic
547C2×C4C2×D3×D411yescharacteristic
548C8—32no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—41no—
553C8—61no—
554C8—61no—
555C8—61no—
556C8—121no—
557C8—121no—
558C2×C3—41no—
559C2×C3—161no—
560C2×C3#303511yescharacteristic
561C2×C3#305411yescharacteristic
562C2×C3C2×#47211yescharacteristic
563D3—81no—
564D3—161no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—121no—
572C22—121no—
573C22—121no—
574C22—241no—
575C22—241no—
576C22—241no—
577C22C2×#88411yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—81no—
582C4—121no—
583C4—121no—
584C4—121no—
585C4—241no—
586C4#474912yes—
587C4D3×#13511yescharacteristic
588C4#576811yescharacteristic
589C4#576811yescharacteristic
590C4#576811yescharacteristic
591C3#1916011yescharacteristic, complemented, Sylow
592C2—41no—
593C2—161no—
594C2—241no—
595C2—481no—
596C2C2×#474911yescharacteristic
597C2#7883911yescharacteristic
598C2#7886111yescharacteristic
subgroupG/H
centerC2×C4#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1543C2
socleC22×C3C2×#121
hypercenter#1543D3
hypocenterC3#19160
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1543

Complements

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

NH ≅ G/Ntype
1. #83299C2semidirect
2. #83291C2semidirect
3. #79301C2semidirect
4. #79277C2semidirect
5. C3×#1543C2semidirect
6. #70773C2semidirect
7. #70745C2semidirect
8. #68514C2semidirect
9. #68513C2semidirect
10. #68502C2semidirect
11. #68501C2semidirect
12. #68414C2semidirect
13. #68413C2semidirect
14. #68400C2semidirect
15. #68399C2semidirect
82. #1543D3semidirect
591. C3#19160semidirect

Maximal subgroups

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

subgroupindex
#832992
#832912
#793012
#792772
C3×#15432
#707732
#707452
#685142
#685132
#685022
#685012
#684142
#684132
#684002
#683992
#191603