Finite Group Databasebeta

← #10778227

Subgroups — #10778227

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#83299C211yesmaximal, characteristic, complemented
2#83295C211yesmaximal, characteristic, complemented
3#79329C211yesmaximal, characteristic, complemented
4#79021C211yesmaximal, characteristic, complemented
5C3×#1551C211yesmaximal, characteristic, complemented
6#70817C211yesmaximal, characteristic, complemented
7#70420C211yesmaximal, characteristic, complemented
8#68510C211yesmaximal, characteristic, complemented
9#68508C211yesmaximal, characteristic, complemented
10#68502C211yesmaximal, characteristic, complemented
11#68500C211yesmaximal, characteristic, complemented
12#68411C211yesmaximal, characteristic, complemented
13#68409C211yesmaximal, characteristic, complemented
14#68401C211yesmaximal, characteristic, complemented
15#68399C211yesmaximal, characteristic, complemented
16#19184—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18C2×#904—21no—
19#5477C2211yescharacteristic
20C4×#235C2211yescharacteristic
21C3×#432C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23#5113—21no—
24#5091C2211yescharacteristic
25#5088C2211yescharacteristic
26#5084C2211yescharacteristic
27#5080C2211yescharacteristic
28#5075C2211yescharacteristic
29#5073C2211yescharacteristic
30#5072C2211yescharacteristic
31#5070C2211yescharacteristic
32#5039—21no—
33#5039—21no—
34#5037—21no—
35#5035—21no—
36C4×#214—21no—
37#5014—21no—
38#5014—21no—
39C4×#213—21no—
40#4925C2211yescharacteristic
41C4×#208C2211yescharacteristic
42#4849C2211yescharacteristic
43#4848C2211yescharacteristic
44#4845—21no—
45#4844—21no—
46#4843C2211yescharacteristic
47#4841C2211yescharacteristic
48#4838C2211yescharacteristic
49#4838C2211yescharacteristic
50#4830C2211yescharacteristic
51#4830C2211yescharacteristic
52#4729—21no—
53#4729—21no—
54#4707—21no—
55#4705—21no—
56C3×#333C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#331C2211yescharacteristic
59C3×#331C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4485C2211yescharacteristic
63#4485C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
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#1565—31no—
81#1553—31no—
82#1551D311yescharacteristic, complemented
83C2×C3×#41—21no—
84C2×#235—21no—
85C2×#235D411yescharacteristic
86C22×C3×C8—21no—
87C3×C4⋊Q8C2311yescharacteristic
88C3×#127C2311yescharacteristic
89C3×#124—21no—
90C3×#121—21no—
91C3×#121—21no—
92C3×C4×D4—21no—
93C3×C4×D4—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#214—21no—
96C2×#213—21no—
97C2×#213—21no—
98C2×#213—21no—
99#930—21no—
100#930—21no—
101C4×#67—21no—
102C4×#67—21no—
103#926D411yescharacteristic
104C2×#208D411yescharacteristic
105C2×#208D411yescharacteristic
106C2×Dic12—21no—
107#904—44no—
108C2×D24—21no—
109C2×#204—21no—
110C2×#204—21no—
111C2×ΓL(1,𝔽25)—21no—
112C2×ΓL(1,𝔽25)—21no—
113#875—21no—
114#874C2311yescharacteristic
115C4×D12—21no—
116C4×D12—21no—
117C4×D12D411yescharacteristic
118C4×D12C2311yescharacteristic
119#872—21no—
120C42×D3—21no—
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×#97—21no—
131#832—21no—
132#832—21no—
133#832—21no—
134#832—21no—
135#830—21no—
136#821—21no—
137#821—21no—
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8—21no—
140#816—21no—
141#816—21no—
142Dic3⋊C8—21no—
143Dic3⋊C8—21no—
144C8×Dic3—21no—
145C8×Dic3—21no—
146#810—21no—
147#809—21no—
148#809—21no—
149#809—21no—
150#808—21no—
151#808—21no—
152#808—21no—
153#807—21no—
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156#803D411yescharacteristic
157#803D411yescharacteristic
158C4×C3⋊C8—21no—
159C4×C3⋊C8C2311yescharacteristic
160C4×C3⋊C8C2311yescharacteristic
161C2×#135—61no—
162#532—31no—
163#532C2×D311yescharacteristic
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—61no—
194#432C2×D311yescharacteristic
195#431—31no—
196C2×C4⋊C8C2×D311yescharacteristic
197#416—61no—
198C4×M16—31no—
199C4×M16—61no—
200#335—31no—
201#335—31no—
202#334—31no—
203#334—31no—
204#333C2×D311yescharacteristic
205#333C2×D311yescharacteristic
206#332—31no—
207#332—31no—
208#331—31no—
209#331—31no—
210#331C2×D311yescharacteristic
211#331C2×D311yescharacteristic
212C3×#41—82no—
213C2×C3×Q8—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—41no—
218C22×C3×C4—41no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4#4111yescharacteristic
222C2×#67—21no—
223C2×#67—41no—
224#235—22no—
225#235—41no—
226#235—42no—
227#235—42no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12C2×D411yescharacteristic
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—41no—
236C2×Dic6—21no—
237C2×Dic6C2×D411yescharacteristic
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—42no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C22⋊C4—41no—
246C3×C22⋊C4—41no—
247C3×C42C2×D411yescharacteristic
248C3×C42C2×D411yescharacteristic
249C3×C42#4111yescharacteristic
250C3×C42C2411yescharacteristic
251#217—21no—
252#214—42no—
253#213—42no—
254#213—42no—
255#213—42no—
256(C2×D3)⋊C4—21no—
257(C2×D3)⋊C4—21no—
258(C2×D3)⋊C4—41no—
259(C2×D3)⋊C4—41no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3C2×D411yescharacteristic
263C4⋊Dic3C2×D411yescharacteristic
264Dic3⋊C4—21no—
265Dic3⋊C4—21no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269#208—22no—
270#208—41no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8C2×D411yescharacteristic
274C2×C3⋊C8C2×D411yescharacteristic
275C2×C3⋊C8C2×D411yescharacteristic
276C2×C3⋊C8C2×D411yescharacteristic
277Dic12—42no—
278D24—42no—
279#204—42no—
280#204—42no—
281ΓL(1,𝔽25)—42no—
282ΓL(1,𝔽25)—42no—
283C2×#41—21no—
284C2×#41—31no—
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—31no—
296C2×M16—31no—
297C2×M16—61no—
298C2×M16—61no—
299C22×C8—21no—
300C4⋊Q8—31no—
301C4⋊Q8C22×D311yescharacteristic
302#127—31no—
303#127C22×D311yescharacteristic
304#124—21no—
305#124—61no—
306#121—21no—
307#121—21no—
308#121—61no—
309#121—61no—
310C4×Q8—31no—
311C4×Q8—31no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—31no—
315C4×D4—31no—
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—21no—
354#97—31no—
355#97—31no—
356C4×C8—31no—
357C4×C8—31no—
358C4×C8—61no—
359C4×C8—61no—
360C4×C8—61no—
361C23×C3—41no—
362C23×C3#12111yescharacteristic
363C22×D3—21no—
364C22×D3—41no—
365C3×Q8—81no—
366C3×D4—41no—
367C3×D4—41no—
368C3×D4—41no—
369C3×D4—41no—
370C3×D4—81no—
371C3×D4—82no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—82no—
379C2×C3×C4#12111yescharacteristic
380C2×C3×C4#12111yescharacteristic
381C2×C3×C4#12111yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385#67—42no—
386#67—82no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—41no—
391D12—21no—
392D12—21no—
393D12—41no—
394D12—41no—
395D12—41no—
396D12—41no—
397D12—81no—
398C4×D3—42no—
399C4×D3—41no—
400C4×D3—41no—
401C4×D3—81no—
402C4×D3—82no—
403Dic6—21no—
404Dic6—41no—
405Dic6—41no—
406Dic6AGL(1,ℤ/8)12yes—
407C3×C8—41no—
408C3×C8—41no—
409C3×C8—41no—
410C3×C8—41no—
411C3⋊C8—21no—
412C3⋊C8—21no—
413C3⋊C8—21no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C3⋊C8#13512yescomplemented
417#41—62no—
418#41—82no—
419#41—121no—
420#41—122no—
421#41—122no—
422C2×Q8—21no—
423C2×Q8—31no—
424C2×Q8—61no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—31no—
429C2×D4—41no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—121no—
435C22×C4—41no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—121no—
440C22×C4#23511yescharacteristic
441C22×C4D3×D411yescharacteristic
442C22×C4D3×D411yescharacteristic
443Q16—122no—
444SD16—122no—
445SD16—122no—
446SD16—122no—
447D8—122no—
448D8—122no—
449D8—122no—
450D8—122no—
451M16—62no—
452M16—122no—
453M16—122no—
454M16—121no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—31no—
460C2×C8—31no—
461C2×C8—31no—
462C2×C8—31no—
463C2×C8—42no—
464C2×C8—61no—
465C2×C8—61no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—31no—
469C4⋊C4—31no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—61no—
477C22⋊C4—61no—
478C22⋊C4—61no—
479C22⋊C4—121no—
480C22⋊C4—121no—
481C42—61no—
482C42—61no—
483C42—61no—
484C42#23511yescharacteristic
485C42D3×D411yescharacteristic
486C42D3×D411yescharacteristic
487C42C23×D311yescharacteristic
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3C2×#12111yescharacteristic
495C2×D3—41no—
496C2×D3—41no—
497C2×D3—41no—
498C2×D3—81no—
499C2×D3—81no—
500C2×D3—81no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—81no—
506C3×C4C2×#13511yescharacteristic
507C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
508C3×C4#57511yescharacteristic
509C3×C4#57511yescharacteristic
510Dic3—22no—
511Dic3—41no—
512Dic3—41no—
513Dic3—81no—
514C23—41no—
515C23—61no—
516C23—121no—
517C23#88411yescharacteristic
518Q8—32no—
519Q8—61no—
520Q8—81no—
521Q8—121no—
522Q8—121no—
523D4—41no—
524D4—41no—
525D4—41no—
526D4—41no—
527D4—61no—
528D4—61no—
529D4—81no—
530D4—82no—
531D4—122no—
532D4—121no—
533D4—121no—
534D4—121no—
535D4—121no—
536D4—241no—
537D4—242no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—82no—
548C2×C4—122no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—241no—
553C2×C4—242no—
554C2×C4#88411yescharacteristic
555C2×C4#88411yescharacteristic
556C2×C4#88411yescharacteristic
557C2×C4C2×#23511yescharacteristic
558C2×C4C2×D3×D411yescharacteristic
559C2×C4C2×D3×D411yescharacteristic
560C8—32no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—41no—
565C8—61no—
566C8—61no—
567C8—61no—
568C8—121no—
569C8—121no—
570C2×C3—41no—
571C2×C3—161no—
572C2×C3#303511yescharacteristic
573C2×C3#305511yescharacteristic
574C2×C3#306811yescharacteristic
575D3—81no—
576D3—161no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—121no—
584C22—121no—
585C22—121no—
586C22—241no—
587C22—241no—
588C22—241no—
589C22C2×#88411yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—62no—
595C4—81no—
596C4—121no—
597C4—121no—
598C4—241no—
599C4D3×#13511yescharacteristic
600C4#576811yescharacteristic
601C4D3×AGL(1,ℤ/8)11yescharacteristic
602C4#577711yescharacteristic
603C3#1918411yescharacteristic, complemented, Sylow
604C2—41no—
605C2—161no—
606C2—241no—
607C2—481no—
608C2#7870311yescharacteristic
609C2#7886111yescharacteristic
610C2#7888811yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1551C2
socleC22×C3C2×#121
hypercenter#1551D3
hypocenterC3#19184
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1551

Complements

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

NH ≅ G/Ntype
1. #83299C2semidirect
2. #83295C2semidirect
3. #79329C2semidirect
4. #79021C2semidirect
5. C3×#1551C2semidirect
6. #70817C2semidirect
7. #70420C2semidirect
8. #68510C2semidirect
9. #68508C2semidirect
10. #68502C2semidirect
11. #68500C2semidirect
12. #68411C2semidirect
13. #68409C2semidirect
14. #68401C2semidirect
15. #68399C2semidirect
82. #1551D3semidirect
416. C3⋊C8#135semidirect
603. C3#19184semidirect

Maximal subgroups

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

subgroupindex
#832992
#832952
#793292
#790212
C3×#15512
#708172
#704202
#685102
#685082
#685022
#685002
#684112
#684092
#684012
#683992
#191843