Finite Group Databasebeta

← #10876077

Subgroups — #10876077

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73770C211yesmaximal, characteristic
2#73696C211yesmaximal, characteristic, complemented
3C3×#1917C211yesmaximal, characteristic, complemented
4#70488C211yesmaximal, characteristic, complemented
5#70480C211yesmaximal, characteristic
6#69817C211yesmaximal, characteristic, complemented
7#69812C211yesmaximal, characteristic
8#68672C211yesmaximal, characteristic
9#68672C211yesmaximal, characteristic
10#68672C211yesmaximal, characteristic
11#68671C211yesmaximal, characteristic
12#68644C211yesmaximal, characteristic, complemented
13#68644C211yesmaximal, characteristic, complemented
14#68638C211yesmaximal, characteristic, complemented
15#68622C211yesmaximal, characteristic, complemented
16#27579—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×C4×D12C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#97C2211yescharacteristic
23C3×C4×C4⋊C4C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#817C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×#816C2211yescharacteristic
31#4989—21no—
32#4989—21no—
33#4985—21no—
34#4978—21no—
35#4936C2211yescharacteristic
36#4936C2211yescharacteristic
37C4×C4⋊Dic3C2211yescharacteristic
38#4929C2211yescharacteristic
39#4929C2211yescharacteristic
40#4929C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4667—21no—
45D3×#95—21no—
46C3⋊#401—21no—
47#4651—21no—
48C3×#336C2211yescharacteristic
49C3×#336C2211yescharacteristic
50C3×#336C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2514—31no—
61#2349—31no—
62#2101—31no—
63#2101—31no—
64#2074—31no—
65#2066—31no—
66#1935—31no—
67#1919—31no—
68#1917D311yescharacteristic, complemented
69#1877—31no—
70#1877—31no—
71#1877—31no—
72#1864—31no—
73#1804—31no—
74#1762—31no—
75C22×D12C2311yescharacteristic
76C22×C4×D3—21no—
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3—21no—
102C2×#204—22no—
103C2×#204—22no—
104C2×#204—22no—
105C4×D12—22no—
106C4×D12—22no—
107C4×D12—22no—
108C4×D12—22no—
109C4×D12—41no—
110C4×D12—41no—
111C3×C4⋊C8—41no—
112C3×#97—41no—
113C3×#95D411yescharacteristic
114C3×#95D411yescharacteristic
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⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129#821—22no—
130#821—22no—
131#821—22no—
132#817—22no—
133#816—22no—
134#816—41no—
135#816—41no—
136C22×SD16—31no—
137C2×C4×D4—31no—
138C2×C4×D4—31no—
139C2×C4×D4—31no—
140C2×#106—31no—
141C2×C4⋊C8C2×D311yescharacteristic
142C2×#103—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#97C2×D311yescharacteristic
149#401—61no—
150#400—61no—
151#391—31no—
152#391—31no—
153#391—31no—
154#389—31no—
155#389—31no—
156#389—31no—
157#387—61no—
158#385—61no—
159#385—61no—
160#383—61no—
161#382—31no—
162#382—31no—
163#380—61no—
164C4×C4⋊C4—31no—
165C4×C4⋊C4C2×D311yescharacteristic
166C2×#95—61no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#336C2×D311yescharacteristic
176#336C2×D311yescharacteristic
177#336C2×D311yescharacteristic
178#336C2×D311yescharacteristic
179C23×D3—21no—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12#4112yes—
198C2×D12#4112yes—
199C2×D12#4112yes—
200C2×C4×D3—24no—
201C2×C4×D3—42no—
202C2×C4×D3—42no—
203C2×C4×D3—42no—
204C2×C4×D3—42no—
205C2×C4×D3—81no—
206C2×C4×D3—81no—
207C2×C4×D3—81no—
208C2×C4×D3—81no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C3×C4⋊C4—41no—
228C3×C4⋊C4—41no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—42no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3#4112yes—
250Dic3⋊C4—42no—
251Dic3⋊C4—81no—
252Dic3⋊C4—81no—
253C4×Dic3—42no—
254#204—44no—
255C22×Q8—31no—
256C22×D4—31no—
257C23×C4—61no—
258C23×C4—61no—
259C23×C4—61no—
260C2×SD16—62no—
261C2×SD16—62no—
262C2×SD16—62no—
263C22×C8C22×D311yescharacteristic
264C22×C8C22×D311yescharacteristic
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267C4×D4—62no—
268C4×D4—62no—
269C4×D4—62no—
270C4×D4—62no—
271C4×D4—121no—
272C4×D4—121no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4D1211yescharacteristic
283C2×C4⋊C4D1211yescharacteristic
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C42—61no—
288C2×C42C22×D311yescharacteristic
289C2×C42C22×D311yescharacteristic
290C2×C42C22×D311yescharacteristic
291#106—62no—
292C4⋊C8—41no—
293#103—62no—
294#103—121no—
295#103—121no—
296#102—62no—
297#102—62no—
298#102—62no—
299#97—41no—
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#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95D1211yescharacteristic
315#95D1211yescharacteristic
316C23×C3C22×D411yescharacteristic
317C22×D3—22no—
318C22×D3—22no—
319C22×D3—22no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3—41no—
323C22×D3—41no—
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—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4H(5,𝔽2)11yescharacteristic
347C2×C3×C42−1+411yescharacteristic
348C2×Dic3—22no—
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—
369D12—22no—
370D12—22no—
371D12—22no—
372D12#12614yes—
373C4×D3—84no—
374C4×D3—84no—
375C4×D3—84no—
376Dic6—41no—
377Dic6—41no—
378Dic6—41no—
379Dic6—41no—
380C3×C8—41no—
381C3×C8—41no—
382C3×C8—41no—
383C3×C8—41no—
384C24—61no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×Q8—61no—
389C2×Q8—61no—
390C2×Q8—61no—
391C2×D4—32no—
392C2×D4—32no—
393C2×D4—32no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—61no—
397C22×C4—64no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—122no—
406C22×C4—122no—
407C22×C4—122no—
408C22×C4—122no—
409C22×C4—241no—
410C22×C4—241no—
411C22×C4—241no—
412C22×C4—241no—
413C22×C4C2×D1211yescharacteristic
414C22×C4C2×D1211yescharacteristic
415C22×C4C2×D1211yescharacteristic
416C22×C4C2×D1211yescharacteristic
417C22×C4C2×D1211yescharacteristic
418C22×C4C2×D1211yescharacteristic
419C22×C4C23×D311yescharacteristic
420SD16—124no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C4⋊C4—32no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—122no—
448C4⋊C4—241no—
449C4⋊C4—241no—
450C22⋊C4—64no—
451C22⋊C4—122no—
452C22⋊C4—122no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—122no—
460C22×C3#52511yescharacteristic
461C22×C3#54611yescharacteristic
462C22×C3#54811yescharacteristic
463C22×C3#57511yescharacteristic
464C22×C3#57511yescharacteristic
465C22×C3#57711yescharacteristic
466C22×C3#57711yescharacteristic
467C2×D3—24no—
468C2×D3—42no—
469C2×D3—42no—
470C2×D3—42no—
471C2×D3—42no—
472C2×D3—42no—
473C2×D3—42no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4C2×#12611yescharacteristic
481C3×C4#53911yescharacteristic
482C3×C4#55211yescharacteristic
483C3×C4#55311yescharacteristic
484Dic3—42no—
485Dic3—81no—
486Dic3—81no—
487Dic3—81no—
488Dic3—81no—
489Dic3—81no—
490Dic3—81no—
491C23—62no—
492C23—62no—
493C23—62no—
494C23—121no—
495C23—121no—
496C23—121no—
497C23—121no—
498C23C22×D1211yescharacteristic
499Q8—121no—
500Q8—121no—
501Q8—121no—
502Q8—121no—
503D4—34no—
504D4—62no—
505D4—62no—
506D4—62no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—62no—
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×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—244no—
547C2×C4—244no—
548C2×C4—244no—
549C2×C4#100411yescharacteristic
550C2×C4#100711yescharacteristic
551C2×C4D3×#4111yescharacteristic
552C2×C4D3×#4111yescharacteristic
553C2×C4D3×#4111yescharacteristic
554C2×C4D3×#4111yescharacteristic
555C8—41no—
556C8—41no—
557C8—41no—
558C8—41no—
559C2×C3#248111yescharacteristic
560C2×C3#311111yescharacteristic
561C2×C3#312311yescharacteristic
562C2×C3#331111yescharacteristic
563C2×C3#332311yescharacteristic
564C2×C3#337611yescharacteristic
565C2×C3#337611yescharacteristic
566D3—44no—
567C22—64no—
568C22—122no—
569C22—122no—
570C22—122no—
571C22—122no—
572C22—122no—
573C22—122no—
574C22#552411yescharacteristic
575C22#555411yescharacteristic
576C22#557611yescharacteristic
577C22#574811yescharacteristic
578C22#574811yescharacteristic
579C22#575111yescharacteristic
580C22#575111yescharacteristic
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—122no—
588C4—241no—
589C4—241no—
590C4—241no—
591C4—241no—
592C4—241no—
593C4—241no—
594C4#563811yescharacteristic
595C4#566111yescharacteristic
596C4#569311yescharacteristic
597C4D3×#12611yescharacteristic
598C3#2757911yescharacteristic, complemented, Sylow
599C2—124no—
600C2#7502011yescharacteristic
601C2#7934511yescharacteristic
602C2#7938711yescharacteristic
603C2#7950211yescharacteristic
604C2#7955411yescharacteristic
605C2#7997411yescharacteristic
606C2#7997411yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1917C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1917D3
hypocenterC3#27579
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1917

Complements

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

NH ≅ G/Ntype
2. #73696C2semidirect
3. C3×#1917C2semidirect
4. #70488C2semidirect
6. #69817C2semidirect
12. #68644C2semidirect
13. #68644C2semidirect
14. #68638C2semidirect
15. #68622C2semidirect
68. #1917D3semidirect
598. C3#27579semidirect

Maximal subgroups

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

subgroupindex
#737702
#736962
C3×#19172
#704882
#704802
#698172
#698122
#686722
#686712
#686442
#686382
#686222
#275793