Finite Group Databasebeta

← #10812967

Subgroups — #10812967

Order 768.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4861C211yesmaximal, characteristic
2#75979C211yesmaximal, characteristic
3#73767C211yesmaximal, characteristic
4C3×#1920C211yesmaximal, characteristic, complemented
5#70557C211yesmaximal, characteristic, complemented
6#69618C211yesmaximal, characteristic, complemented
7C4×#807C211yesmaximal, characteristic, complemented
8#68873C211yesmaximal, characteristic
9#68872C211yesmaximal, characteristic
10#68870C211yesmaximal, characteristic
11#68861C211yesmaximal, characteristic, complemented
12#68820C211yesmaximal, characteristic, complemented
13#68786C211yesmaximal, characteristic, complemented
14#68649C211yesmaximal, characteristic
15#68621C211yesmaximal, characteristic, complemented
16#22624—31nomaximal, Sylow
17C22×ΓL(1,𝔽25)C2211yescharacteristic
18C2×#892C2211yescharacteristic
19C2×D3×C4⋊C4C2211yescharacteristic
20C2×#872—21no—
21C2×C3×#106C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×C4×C4⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C8C2211yescharacteristic
25C2×#818C2211yescharacteristic
26C2×#815C2211yescharacteristic
27#4990—21no—
28#4987C2211yescharacteristic
29#4987C2211yescharacteristic
30#4984C2211yescharacteristic
31#4977—21no—
32Dic3×C4⋊C4C2211yescharacteristic
33Dic3×C4⋊C4C2211yescharacteristic
34#4972C2211yescharacteristic
35C2×#808C2211yescharacteristic
36C2×#808C2211yescharacteristic
37C2×#807C2211yescharacteristic
38C2×#807C2211yescharacteristic
39C4×(C2×D3)⋊C4C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41#4933—21no—
42#4928C2211yescharacteristic
43C2×C4×C3⋊C8C2211yescharacteristic
44#4861C418yescomplemented
45#4670—21no—
46#4664—21no—
47#4652—21no—
48#4648—21no—
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61C2×#428—31no—
62#2346—31no—
63#2344—31no—
64#1944—31no—
65#1941—31no—
66#1921—31no—
67#1920D311yescharacteristic, complemented
68#1919—31no—
69#1918—31no—
70#1825—31no—
71#1824—31no—
72#1809—31no—
73#1776—31no—
74C4×#107—31no—
75#1753—31no—
76C22×C4×D3—21no—
77C22×C4×D3C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4D411yescharacteristic
91C2×(C2×D3)⋊C4D411yescharacteristic
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C2×ΓL(1,𝔽25)—22no—
108C2×ΓL(1,𝔽25)—22no—
109C2×ΓL(1,𝔽25)C2×C414yes—
110#892—22no—
111#892C2×C414yes—
112D3×C4⋊C4—22no—
113D3×C4⋊C4C2×C414yes—
114#872—44no—
115C3×#106C2×C414yes—
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⋊#95—21no—
126C3⋊#95D411yescharacteristic
127C3⋊#95D411yescharacteristic
128(C2×D3)⋊C8—22no—
129#818C2×C414yes—
130#815—22no—
131#808—22no—
132#808C2×C414yes—
133#807—22no—
134#807C2×C414yes—
135C4×C3⋊C8—22no—
136C22×M16—31no—
137C2×#117—31no—
138C2×#117—61no—
139C22×C4⋊C4—31no—
140#428—38no—
141C2×#107—31no—
142C2×#107—31no—
143C2×#107—31no—
144C2×#106—31no—
145C2×#106—31no—
146C2×#106C2×D311yescharacteristic
147C2×C22⋊C8—31no—
148C2×#97—31no—
149C2×C4×C8—31no—
150#389—31no—
151#389—31no—
152#389—31no—
153#389C2×D311yescharacteristic
154#388—61no—
155#388—61no—
156#387—61no—
157#387—61no—
158#382—61no—
159#381—61no—
160#380—31no—
161#380—31no—
162C4×C4⋊C4—31no—
163C4×C4⋊C4—31no—
164C4×C4⋊C4—31no—
165C4×C4⋊C4C2×D311yescharacteristic
166C4×C22⋊C4—31no—
167#376—61no—
168#340—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#340C2×D311yescharacteristic
178#336—31no—
179#336C2×D311yescharacteristic
180C23×D3C2×D411yescharacteristic
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C22×Dic3C2×D411yescharacteristic
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3—21no—
198C2×C4×D3—21no—
199C2×C4×D3—24no—
200C2×C4×D3—42no—
201C2×C4×D3—42no—
202C2×C4×D3C22×C412yes—
203C2×C4×D3#4112yes—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8C22×C412yes—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—81no—
217C3×C4⋊C4C22×C412yes—
218C3×C4⋊C4C22×C412yes—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—42no—
222(C2×D3)⋊C4—24no—
223(C2×D3)⋊C4—22no—
224(C2×D3)⋊C4—41no—
225(C2×D3)⋊C4—42no—
226(C2×D3)⋊C4—81no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—81no—
234C4⋊Dic3C22×C412yes—
235C4⋊Dic3C22×C412yes—
236Dic3⋊C4—24no—
237Dic3⋊C4—42no—
238Dic3⋊C4—42no—
239C4×Dic3—21no—
240C4×Dic3—21no—
241C4×Dic3—24no—
242C4×Dic3—42no—
243C4×Dic3—42no—
244C4×Dic3—42no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8C22×C412yes—
249C2×C3⋊C8#4112yes—
250C2×C3⋊C8#4112yes—
251ΓL(1,𝔽25)—28no—
252C23×C4—31no—
253C23×C4—61no—
254C2×M16—34no—
255C2×M16—62no—
256C2×M16—62no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8C22×D311yescharacteristic
260C22×C8C22×D311yescharacteristic
261#117—34no—
262#117—62no—
263#117—124no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—34no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4C22×D311yescharacteristic
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C22⋊C4—31no—
280C2×C22⋊C4—31no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C42—21no—
285C2×C42—31no—
286C2×C42—61no—
287C2×C42—61no—
288C2×C42—61no—
289C2×C42—61no—
290C2×C42C22×D311yescharacteristic
291#107—34no—
292#107—34no—
293#107—62no—
294#106—34no—
295#106—62no—
296#106C4×D314yes—
297C22⋊C8—62no—
298#97—62no—
299C4×C8—62no—
300#95—21no—
301#95—21no—
302#95—31no—
303#95—31no—
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×C3C22×D411yescharacteristic
313C22×D3—21no—
314C22×D3—21no—
315C22×D3—21no—
316C22×D3—21no—
317C22×D3—21no—
318C22×D3—21no—
319C22×D3C4×D412yes—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
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×C4AGL(1,ℤ/8)12yes—
338C2×C3×C4C23×C411yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C42−1+411yescharacteristic
343C2×C3×C42−1+411yescharacteristic
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
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×Dic3C4×D412yes—
365C4×D3—22no—
366C4×D3—22no—
367C4×D3—22no—
368C4×D3—22no—
369C4×D3—48no—
370C3×C8—22no—
371C3×C8—41no—
372C3⋊C8—41no—
373C3⋊C8#11714yescomplemented
374C24—31no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—31no—
380C22×C4—32no—
381C22×C4—32no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—64no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—122no—
394C22×C4—122no—
395C22×C4D3×D411yescharacteristic
396C22×C4D3×D411yescharacteristic
397C22×C4C23×D311yescharacteristic
398M16—68no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—32no—
405C2×C8—32no—
406C2×C8—32no—
407C2×C8—61no—
408C2×C8—61no—
409C2×C8—61no—
410C2×C8C2×C4×D312yes—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—32no—
418C4⋊C4—32no—
419C4⋊C4—64no—
420C4⋊C4—61no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—81no—
427C4⋊C4—122no—
428C4⋊C4—122no—
429C4⋊C4—241no—
430C4⋊C4C2×C4×D312yes—
431C4⋊C4C2×C4×D312yes—
432C22⋊C4—64no—
433C22⋊C4—62no—
434C22⋊C4—121no—
435C22⋊C4—122no—
436C22⋊C4—241no—
437C42—21no—
438C42—21no—
439C42—42no—
440C42—64no—
441C42—61no—
442C42—61no—
443C42—122no—
444C42—122no—
445C42—122no—
446C22×C3C2×C4×D411yescharacteristic
447C22×C3#54711yescharacteristic
448C22×C3#54711yescharacteristic
449C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
450C22×C3#57511yescharacteristic
451C22×C3#57711yescharacteristic
452C22×C3#57811yescharacteristic
453C2×D3—22no—
454C2×D3—22no—
455C2×D3—22no—
456C2×D3—41no—
457C2×D3—41no—
458C2×D3—41no—
459C2×D3—41no—
460C3×C4—22no—
461C3×C4—41no—
462C3×C4—42no—
463C3×C4—42no—
464C3×C4—81no—
465C3×C4—81no—
466C3×C4C2×#11711yescharacteristic
467C3×C4#51911yescharacteristic
468C3×C4#55211yescharacteristic
469C3×C4#55211yescharacteristic
470Dic3—22no—
471Dic3—41no—
472Dic3—41no—
473Dic3—42no—
474Dic3—42no—
475Dic3—81no—
476Dic3—81no—
477C23—32no—
478C23—61no—
479C23—61no—
480C23—61no—
481C23—61no—
482C23—61no—
483C23—61no—
484C23C2×D3×D411yescharacteristic
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—32no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—128no—
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#91412yes—
529C2×C4C22×C4×D311yescharacteristic
530C2×C4#100711yescharacteristic
531C2×C4D3×#4111yescharacteristic
532C2×C4D3×#4111yescharacteristic
533C2×C4D3×#4111yescharacteristic
534C2×C4#101011yescharacteristic
535C8—22no—
536C8—34no—
537C8—41no—
538C8—121no—
539C2×C3#234411yescharacteristic
540C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
541C2×C3#294711yescharacteristic
542C2×C3#334311yescharacteristic
543C2×C3#334811yescharacteristic
544C2×C3#335211yescharacteristic
545C2×C3#335411yescharacteristic
546D3—42no—
547C22—62no—
548C22—62no—
549C22—62no—
550C22—121no—
551C22—121no—
552C22—121no—
553C22—121no—
554C22C4×D3×D411yescharacteristic
555C22#563111yescharacteristic
556C22#565511yescharacteristic
557C22C2×#91411yescharacteristic
558C22#576811yescharacteristic
559C22#577511yescharacteristic
560C22#577911yescharacteristic
561C4—22no—
562C4—42no—
563C4—42no—
564C4—41no—
565C4—62no—
566C4—81no—
567C4—81no—
568C4—122no—
569C4—122no—
570C4—121no—
571C4—121no—
572C4—241no—
573C4—241no—
574C4D3×#11711yescharacteristic
575C4#557211yescharacteristic
576C4#569311yescharacteristic
577C4#570811yescharacteristic
578C3#2262411yescharacteristic, complemented, Sylow
579C2—122no—
580C2#7496111yescharacteristic
581C2C4×#91411yescharacteristic
582C2#7977611yescharacteristic
583C2#8108511yescharacteristic
584C2#8126011yescharacteristic
585C2#8136611yescharacteristic
586C2#8142411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1920C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
cosocleC22×C3×C4C24
hypercenter#1920D3
hypocenterC3#22624
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
4. C3×#1920C2semidirect
5. #70557C2semidirect
6. #69618C2semidirect
7. C4×#807C2semidirect
11. #68861C2semidirect
12. #68820C2semidirect
13. #68786C2semidirect
15. #68621C2semidirect
44. #4861C4semidirect
67. #1920D3semidirect
373. C3⋊C8#117semidirect
578. C3#22624semidirect

Maximal subgroups

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

subgroupindex
C2×#48612
#759792
#737672
C3×#19202
#705572
#696182
C4×#8072
#688732
#688722
#688702
#688612
#688202
#687862
#686492
#686212
#226243