Finite Group Databasebeta

← #10868306

Subgroups — #10868306

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#82243C211yesmaximal, characteristic
2#73919C211yesmaximal, characteristic
3C3×#1854C211yesmaximal, characteristic
4#71575C211yesmaximal, characteristic
5#70642C211yesmaximal, characteristic, complemented
6#70610C211yesmaximal, characteristic, complemented
7#70589C211yesmaximal, characteristic, complemented
8#70561C211yesmaximal, characteristic, complemented
9#69937C211yesmaximal, characteristic
10#69915C211yesmaximal, characteristic
11#69526C211yesmaximal, characteristic
12#68915C211yesmaximal, characteristic, complemented
13#68798C211yesmaximal, characteristic, complemented
14#68789C211yesmaximal, characteristic, complemented
15#68783C211yesmaximal, characteristic, complemented
16#27061—31nomaximal, Sylow
17C2×C3×#117C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C2×#887C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C3×#380C2211yescharacteristic
24#5212C2211yescharacteristic
25C4×#217C2211yescharacteristic
26C2×#830C2211yescharacteristic
27C2×#818C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×#815C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32#4984C2211yescharacteristic
33#4977C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36#4973C2211yescharacteristic
37#4972C2211yescharacteristic
38#4972C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#808C2211yescharacteristic
42#4953—21no—
43#4945—21no—
44#4944—21no—
45#4944—21no—
46#4933—21no—
47C4×Dic3⋊C4—21no—
48C3⋊#401—21no—
49#4655—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
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#1924—31no—
69#1921—31no—
70#1920—31no—
71#1863—31no—
72#1854D311yescharacteristic
73#1825—31no—
74#1763—31no—
75C23×C3×C4C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×#117—22no—
80C3×#117—41no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42—21no—
88C2×#217—21no—
89C2×#217—21no—
90C2×#217D411yescharacteristic
91C2×#217D411yescharacteristic
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96#924—22no—
97#924—22no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3C2311yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#887—22no—
110#887—41no—
111C3×#107—22no—
112C3×#106—41no—
113C3×C22⋊C8—22no—
114C3×#95—21no—
115C3⋊#95—21no—
116C3⋊#95—21no—
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⋊#95D411yescharacteristic
126C3⋊#95D411yescharacteristic
127#830—41no—
128#818—22no—
129#816—22no—
130#816—22no—
131#815—41no—
132Dic3⋊C8—22no—
133#810—22no—
134#810—41no—
135#808—22no—
136C2×C22⋊Q8—31no—
137C2×C4×Q8—31no—
138C2×#117C2×D311yescharacteristic
139C2×#107—31no—
140C2×#107C2×D311yescharacteristic
141C2×#106—31no—
142C2×#106C2×D311yescharacteristic
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×#103—31no—
148C2×C22⋊C8—31no—
149C2×C22⋊C8C2×D311yescharacteristic
150C2×#97—31no—
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#380C2×D311yescharacteristic
165C4×C4⋊C4—31no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—61no—
168C4×C22⋊C4—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336—31no—
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×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
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#4112yes—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4#4112yes—
220C3×C22⋊C4—42no—
221C3×C22⋊C4—81no—
222C3×C42—42no—
223#217—24no—
224#217—41no—
225#217—41no—
226#217—42no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—81no—
233C4⋊Dic3#4112yes—
234Dic3⋊C4—24no—
235Dic3⋊C4—42no—
236Dic3⋊C4—42no—
237Dic3⋊C4—42no—
238C4×Dic3—21no—
239C4×Dic3—21no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C4×Dic3—42no—
243C4×Dic3—42no—
244C4×Dic3—81no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C22×Q8—31no—
252C23×C4C22×D311yescharacteristic
253C22×C8—31no—
254C22×C8—31no—
255C22×C8C22×D311yescharacteristic
256C22×C8C22×D311yescharacteristic
257C22⋊Q8—62no—
258C22⋊Q8—62no—
259C4×Q8—62no—
260C4×Q8—121no—
261#117—22no—
262#117—41no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4C22×D311yescharacteristic
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—31no—
278C2×C22⋊C4—31no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—61no—
281C2×C42—21no—
282C2×C42—31no—
283C2×C42—31no—
284C2×C42—61no—
285C2×C42—61no—
286C2×C42—61no—
287#107—22no—
288#107—62no—
289#106—41no—
290#106—62no—
291C4⋊C8—62no—
292#103—62no—
293#103—62no—
294#103—62no—
295#103—121no—
296C22⋊C8—22no—
297C22⋊C8—121no—
298#97—121no—
299#95—21no—
300#95—31no—
301#95—31no—
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#95—61no—
311#95—61no—
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—22no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
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—21no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
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—
377C24D3×D411yescharacteristic
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—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4D3×D411yescharacteristic
403C22×C4C23×D311yescharacteristic
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8#23512yes—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—32no—
424C4⋊C4—64no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—122no—
431C4⋊C4—122no—
432C4⋊C4—122no—
433C4⋊C4—241no—
434C4⋊C4#23712yes—
435C22⋊C4—42no—
436C22⋊C4—64no—
437C22⋊C4—81no—
438C22⋊C4—121no—
439C22⋊C4—121no—
440C22⋊C4—122no—
441C42—42no—
442C42—61no—
443C42—61no—
444C42—61no—
445C42—61no—
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—42no—
465C3×C4—81no—
466C3×C4—81no—
467C3×C4—81no—
468C3×C4#52911yescharacteristic
469C3×C4#53911yescharacteristic
470C3×C4#55311yescharacteristic
471C3×C4#55611yescharacteristic
472Dic3—41no—
473Dic3—41no—
474Dic3—42no—
475Dic3—42no—
476Dic3—41no—
477Dic3—41no—
478Dic3—81no—
479Dic3—81no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23C2×D3×D411yescharacteristic
488Q8—62no—
489Q8—62no—
490Q8—121no—
491Q8—121no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
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×#23511yescharacteristic
536C2×C4C2×#23711yescharacteristic
537C2×C4#100411yescharacteristic
538C2×C4#100711yescharacteristic
539C2×C4D3×#4111yescharacteristic
540C2×C4D3×#4111yescharacteristic
541C8—22no—
542C8—41no—
543C8—121no—
544C8—121no—
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#550711yescharacteristic
561C22#559111yescharacteristic
562C22#565611yescharacteristic
563C22#576811yescharacteristic
564C22#577111yescharacteristic
565C22#577511yescharacteristic
566C22#577711yescharacteristic
567C4—41no—
568C4—42no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—122no—
573C4—121no—
574C4—121no—
575C4—121no—
576C4—122no—
577C4—121no—
578C4—241no—
579C4—241no—
580C4#554011yescharacteristic
581C4#555711yescharacteristic
582C4#563811yescharacteristic
583C4#564111yescharacteristic
584C3#2706111yescharacteristic, complemented, Sylow
585C2—81no—
586C2#7443511yescharacteristic
587C2#7919411yescharacteristic
588C2#7923011yescharacteristic
589C2#8010711yescharacteristic
590C2#8025911yescharacteristic
591C2#8135311yescharacteristic
592C2#8149411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1854C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1854D3
hypocenterC3#27061
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
5. #70642C2semidirect
6. #70610C2semidirect
7. #70589C2semidirect
8. #70561C2semidirect
12. #68915C2semidirect
13. #68798C2semidirect
14. #68789C2semidirect
15. #68783C2semidirect
584. C3#27061semidirect

Maximal subgroups

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

subgroupindex
#822432
#739192
C3×#18542
#715752
#706422
#706102
#705892
#705612
#699372
#699152
#695262
#689152
#687982
#687892
#687832
#270613