Finite Group Databasebeta

← #10868307

Subgroups — #10868307

Order 768.

672 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73510C211yesmaximal, characteristic
2#73468C211yesmaximal, characteristic
3C3×#1924C211yesmaximal, characteristic
4#70488C211yesmaximal, characteristic, complemented
5#70455C211yesmaximal, characteristic, complemented
6#69937C211yesmaximal, characteristic
7#69936C211yesmaximal, characteristic
8#69929C211yesmaximal, characteristic
9#69926C211yesmaximal, characteristic
10#69819C211yesmaximal, characteristic, complemented
11#69813C211yesmaximal, characteristic, complemented
12#68683C211yesmaximal, characteristic, complemented
13#68638C211yesmaximal, characteristic, complemented
14#68624C211yesmaximal, characteristic, complemented
15#68623C211yesmaximal, characteristic, complemented
16#27061—31nomaximal, Sylow
17C2×#926C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C2×C4×Dic6C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C22⋊C8C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24C3×C4×C22⋊C4C2211yescharacteristic
25#5212C2211yescharacteristic
26#5212C2211yescharacteristic
27C2×#818C2211yescharacteristic
28C2×#818C2211yescharacteristic
29C2×#817C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35#4952—21no—
36#4945—21no—
37#4945—21no—
38#4944—21no—
39#4936C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42#4929C2211yescharacteristic
43#4928C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46C3⋊#401—21no—
47#4652—21no—
48#4652—21no—
49#4649—21no—
50C3×#336C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2494—31no—
61#2362—31no—
62#2074—31no—
63#2066—31no—
64#2036—31no—
65#2035—31no—
66#1949—31no—
67#1943—31no—
68#1924D311yescharacteristic
69#1921—31no—
70#1920—31no—
71#1863—31no—
72#1854—31no—
73#1825—31no—
74#1763—31no—
75C23×C3×C4C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C22⋊C4D411yescharacteristic
82C2×C3×C22⋊C4D411yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×#217—21no—
86C2×#217—21no—
87C2×#217—21no—
88C2×#217—21no—
89#926—22no—
90#926—41no—
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98#924—22no—
99#924—22no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C4×Dic6—22no—
110C4×Dic6—41no—
111C3×C4⋊C8—22no—
112C3×C22⋊C8—22no—
113C3×C22⋊C8—41no—
114C3×#97—41no—
115C3×#95D411yescharacteristic
116C3×#95D411yescharacteristic
117C3⋊#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128#818—22no—
129#818—22no—
130#817—22no—
131#817—41no—
132#816—22no—
133#816—22no—
134#816—22no—
135#816—41no—
136C2×C22⋊Q8—31no—
137C2×C4×Q8—31no—
138C2×#117—31no—
139C2×#107—31no—
140C2×#107—31no—
141C2×#106—31no—
142C2×#106—31no—
143C2×C4⋊C8C2×D311yescharacteristic
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×#103—31no—
148C2×C22⋊C8C2×D311yescharacteristic
149C2×C22⋊C8C2×D311yescharacteristic
150C2×#97C2×D311yescharacteristic
151#401—61no—
152#397—61no—
153#391—31no—
154#389—31no—
155#389—31no—
156#389—31no—
157#388—61no—
158#388—61no—
159#388—61no—
160#387—61no—
161#387—61no—
162#382—31no—
163#380—31no—
164#380—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—61no—
168C4×C22⋊C4C2×D311yescharacteristic
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#336C2×D311yescharacteristic
178#336C2×D311yescharacteristic
179C24×C3C2×D411yescharacteristic
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C2×Dic6—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6—21no—
205C2×Dic6#4112yes—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8#4112yes—
218C3×C22⋊C4—41no—
219C3×C22⋊C4—41no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—21no—
224#217—24no—
225#217—42no—
226#217—42no—
227#217—81no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3#4112yes—
241C4⋊Dic3#4112yes—
242Dic3⋊C4—24no—
243Dic3⋊C4—42no—
244Dic3⋊C4—42no—
245Dic3⋊C4—42no—
246Dic3⋊C4—81no—
247C4×Dic3—42no—
248C4×Dic3—42no—
249C4×Dic3—42no—
250C4×Dic3—81no—
251C22×Q8—31no—
252C23×C4C22×D311yescharacteristic
253C22×C8C22×D311yescharacteristic
254C22×C8C22×D311yescharacteristic
255C22×C8C22×D311yescharacteristic
256C22×C8C22×D311yescharacteristic
257C22⋊Q8—62no—
258C22⋊Q8—62no—
259C4×Q8—62no—
260C4×Q8—121no—
261#117—62no—
262#117—121no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4D1211yescharacteristic
280C2×C22⋊C4D1211yescharacteristic
281C2×C42—61no—
282C2×C42—61no—
283C2×C42—61no—
284C2×C42—61no—
285C2×C42C22×D311yescharacteristic
286C2×C42C22×D311yescharacteristic
287#107—62no—
288#107—62no—
289#106—62no—
290#106—121no—
291C4⋊C8—22no—
292#103—62no—
293#103—62no—
294#103—62no—
295#103—121no—
296C22⋊C8—22no—
297C22⋊C8—41no—
298#97—41no—
299#95—61no—
300#95—61no—
301#95—61no—
302#95—61no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95D1211yescharacteristic
311#95D1211yescharacteristic
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3C22×D411yescharacteristic
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4H(5,𝔽2)11yescharacteristic
344C2×C3×C42−1+411yescharacteristic
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369Dic6—22no—
370Dic6—22no—
371Dic6—41no—
372Dic6—41no—
373C3×C8—22no—
374C3×C8—41no—
375C3×C8—41no—
376C3×C8—41no—
377C24C2×D1211yescharacteristic
378C2×Q8—32no—
379C2×Q8—61no—
380C2×Q8—61no—
381C2×Q8—61no—
382C2×Q8—61no—
383C2×Q8—61no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4C2×D1211yescharacteristic
399C22×C4C2×D1211yescharacteristic
400C22×C4C2×D1211yescharacteristic
401C22×C4C2×D1211yescharacteristic
402C22×C4C2×D1211yescharacteristic
403C22×C4C23×D311yescharacteristic
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8#23712yes—
416C4⋊C4—32no—
417C4⋊C4—32no—
418C4⋊C4—64no—
419C4⋊C4—61no—
420C4⋊C4—61no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—122no—
432C4⋊C4—122no—
433C4⋊C4—122no—
434C4⋊C4—241no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—64no—
438C22⋊C4—122no—
439C22⋊C4—122no—
440C22⋊C4—241no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—122no—
446C42—122no—
447C42—122no—
448C42—241no—
449C22×C3—41no—
450C22×C3—41no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3#52511yescharacteristic
457C22×C3#54611yescharacteristic
458C22×C3#54711yescharacteristic
459C22×C3#57511yescharacteristic
460C22×C3#57511yescharacteristic
461C22×C3#57711yescharacteristic
462C22×C3#57811yescharacteristic
463C3×C4—41no—
464C3×C4—41no—
465C3×C4—41no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4#52911yescharacteristic
469C3×C4#53911yescharacteristic
470C3×C4#55311yescharacteristic
471C3×C4#55611yescharacteristic
472Dic3—42no—
473Dic3—42no—
474Dic3—42no—
475Dic3—81no—
476Dic3—81no—
477Dic3—81no—
478Dic3—81no—
479Dic3—81no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23C22×D1211yescharacteristic
488Q8—62no—
489Q8—62no—
490Q8—121no—
491Q8—121no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4C2×#23711yescharacteristic
536C2×C4#100411yescharacteristic
537C2×C4D3×#4111yescharacteristic
538C2×C4D3×#4111yescharacteristic
539C2×C4D3×#4111yescharacteristic
540C2×C4#101011yescharacteristic
541C8—22no—
542C8—41no—
543C8—41no—
544C8—41no—
545C2×C3—81no—
546C2×C3#246611yescharacteristic
547C2×C3#311111yescharacteristic
548C2×C3#312111yescharacteristic
549C2×C3#330711yescharacteristic
550C2×C3#332711yescharacteristic
551C2×C3#334911yescharacteristic
552C2×C3#335011yescharacteristic
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22#552411yescharacteristic
561C22#555411yescharacteristic
562C22#555511yescharacteristic
563C22#574811yescharacteristic
564C22#574811yescharacteristic
565C22#575111yescharacteristic
566C22#575311yescharacteristic
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—122no—
573C4—122no—
574C4—122no—
575C4—241no—
576C4—241no—
577C4—241no—
578C4—241no—
579C4—241no—
580C4#560111yescharacteristic
581C4#566011yescharacteristic
582C4#568011yescharacteristic
583C4#570211yescharacteristic
584C3#2706111yescharacteristic, complemented, Sylow
585C2—81no—
586C2#7551611yescharacteristic
587C2#7934511yescharacteristic
588C2#7939411yescharacteristic
589C2#7947511yescharacteristic
590C2#7947611yescharacteristic
591C2#7950011yescharacteristic
592C2#7956211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1924C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1924D3
hypocenterC3#27061
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1924

Complements

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

NH ≅ G/Ntype
4. #70488C2semidirect
5. #70455C2semidirect
10. #69819C2semidirect
11. #69813C2semidirect
12. #68683C2semidirect
13. #68638C2semidirect
14. #68624C2semidirect
15. #68623C2semidirect
584. C3#27061semidirect

Maximal subgroups

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

subgroupindex
#735102
#734682
C3×#19242
#704882
#704552
#699372
#699362
#699292
#699262
#698192
#698132
#686832
#686382
#686242
#686232
#270613