Finite Group Databasebeta

← #10876047

Subgroups — #10876047

Order 768.

696 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73694C211yesmaximal, characteristic
2#73637C211yesmaximal, characteristic, complemented
3C3×#1917C211yesmaximal, characteristic, complemented
4#70486C211yesmaximal, characteristic, complemented
5#70483C211yesmaximal, characteristic
6#69818C211yesmaximal, characteristic
7#69813C211yesmaximal, characteristic, complemented
8#68686C211yesmaximal, characteristic
9#68686C211yesmaximal, characteristic
10#68686C211yesmaximal, characteristic
11#68684C211yesmaximal, characteristic, complemented
12#68644C211yesmaximal, characteristic, complemented
13#68644C211yesmaximal, characteristic, complemented
14#68637C211yesmaximal, characteristic
15#68623C211yesmaximal, characteristic, complemented
16#27577—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×C4×Dic6C2211yescharacteristic
19C2×C4×Dic6C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
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#4992—21no—
32#4990—21no—
33#4976—21no—
34#4976—21no—
35#4939C2211yescharacteristic
36#4939C2211yescharacteristic
37#4939C2211yescharacteristic
38#4936C2211yescharacteristic
39#4936C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4675—21no—
45#4669—21no—
46#4650—21no—
47#4649—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#2513—31no—
61#2351—31no—
62#2101—31no—
63#2101—31no—
64#2073—31no—
65#2065—31no—
66#1936—31no—
67#1920—31no—
68#1917D311yescharacteristic, complemented
69#1880—31no—
70#1880—31no—
71#1880—31no—
72#1865—31no—
73#1805—31no—
74#1763—31no—
75C22×D12C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4D411yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
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×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×#204—22no—
109C2×#204—22no—
110C2×#204—22no—
111C4×Dic6—22no—
112C4×Dic6—22no—
113C4×Dic6—22no—
114C4×Dic6—22no—
115C4×Dic6—41no—
116C4×Dic6—41no—
117C3×C4⋊C8—41no—
118C3×#97—41no—
119C3×#95D411yescharacteristic
120C3×#95D411yescharacteristic
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—41no—
131#821—41no—
132#817—22no—
133#816—22no—
134#816—22no—
135#816—22no—
136C22×SD16—31no—
137C2×C4×Q8—31no—
138C2×C4×Q8—31no—
139C2×C4×Q8—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#397—61no—
150#397—61no—
151#390—31no—
152#390—31no—
153#390—31no—
154#389—31no—
155#389—31no—
156#389—31no—
157#389—61no—
158#389—61no—
159#388—61no—
160#388—61no—
161#382—31no—
162#382—31no—
163#382—61no—
164C4×C4⋊C4—31no—
165C4×C4⋊C4—61no—
166C4×C4⋊C4C2×D311yescharacteristic
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—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×Dic6—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6#4112yes—
204C2×Dic6#4112yes—
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—21no—
218C3×C4⋊C4—41no—
219C3×C4⋊C4—41no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—21no—
226(C2×D3)⋊C4—42no—
227(C2×D3)⋊C4—42no—
228(C2×D3)⋊C4—81no—
229(C2×D3)⋊C4—81no—
230(C2×D3)⋊C4—81no—
231(C2×D3)⋊C4—81no—
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—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3#4112yes—
244Dic3⋊C4—24no—
245Dic3⋊C4—42no—
246Dic3⋊C4—42no—
247Dic3⋊C4—42no—
248Dic3⋊C4—42no—
249Dic3⋊C4—81no—
250C4×Dic3—24no—
251C4×Dic3—42no—
252C4×Dic3—42no—
253C4×Dic3—81no—
254#204—44no—
255C22×Q8—31no—
256C22×D4—31no—
257C2×SD16—62no—
258C2×SD16—62no—
259C2×SD16—62no—
260C22×C8C22×D311yescharacteristic
261C22×C8C22×D311yescharacteristic
262C22×C8C22×D311yescharacteristic
263C22×C8C22×D311yescharacteristic
264C4×Q8—62no—
265C4×Q8—62no—
266C4×Q8—62no—
267C4×Q8—62no—
268C4×Q8—121no—
269C4×Q8—121no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
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×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C42—61no—
291C2×C42—61no—
292C2×C42—61no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295C2×C42C22×D311yescharacteristic
296C2×C42C22×D311yescharacteristic
297#106—62no—
298C4⋊C8—41no—
299#103—62no—
300#103—62no—
301#103—62no—
302#102—62no—
303#102—121no—
304#102—121no—
305#97—41no—
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—41no—
318C22×D3—41no—
319C22×D3—41no—
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—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
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—41no—
370D12—41no—
371D12—41no—
372D12—41no—
373Dic6—22no—
374Dic6—22no—
375Dic6—22no—
376Dic6#12614yes—
377C3×C8—41no—
378C3×C8—41no—
379C3×C8—41no—
380C3×C8—41no—
381C24—61no—
382C2×Q8—32no—
383C2×Q8—32no—
384C2×Q8—32no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—61no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4C2×D1211yescharacteristic
402C22×C4C2×D1211yescharacteristic
403C22×C4C2×D1211yescharacteristic
404C22×C4C2×D1211yescharacteristic
405C22×C4C2×D1211yescharacteristic
406C22×C4C2×D1211yescharacteristic
407C22×C4C23×D311yescharacteristic
408SD16—124no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C4⋊C4—32no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—64no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—122no—
437C4⋊C4—122no—
438C4⋊C4—122no—
439C4⋊C4—122no—
440C4⋊C4—241no—
441C22⋊C4—122no—
442C22⋊C4—122no—
443C22⋊C4—241no—
444C22⋊C4—241no—
445C22⋊C4—241no—
446C22⋊C4—241no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—64no—
454C42—122no—
455C42—122no—
456C42—241no—
457C22×C3#52511yescharacteristic
458C22×C3#54611yescharacteristic
459C22×C3#54811yescharacteristic
460C22×C3#57511yescharacteristic
461C22×C3#57511yescharacteristic
462C22×C3#57711yescharacteristic
463C22×C3#57711yescharacteristic
464C2×D3—81no—
465C2×D3—81no—
466C2×D3—81no—
467C2×D3—81no—
468C2×D3—81no—
469C2×D3—81no—
470C2×D3—81no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4C2×#12611yescharacteristic
478C3×C4#53911yescharacteristic
479C3×C4#55211yescharacteristic
480C3×C4#55311yescharacteristic
481Dic3—24no—
482Dic3—42no—
483Dic3—81no—
484Dic3—81no—
485Dic3—81no—
486Dic3—81no—
487Dic3—81no—
488C23—121no—
489C23—121no—
490C23—121no—
491C23—121no—
492C23—121no—
493C23—121no—
494C23—121no—
495C23C22×D1211yescharacteristic
496Q8—34no—
497Q8—62no—
498Q8—62no—
499Q8—62no—
500D4—121no—
501D4—121no—
502D4—121no—
503D4—121no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
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—62no—
523C2×C4—62no—
524C2×C4—62no—
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#100411yescharacteristic
544C2×C4#100711yescharacteristic
545C2×C4D3×#4111yescharacteristic
546C2×C4D3×#4111yescharacteristic
547C2×C4D3×#4111yescharacteristic
548C2×C4D3×#4111yescharacteristic
549C8—41no—
550C8—41no—
551C8—41no—
552C8—41no—
553C2×C3#248111yescharacteristic
554C2×C3#311111yescharacteristic
555C2×C3#312611yescharacteristic
556C2×C3#332111yescharacteristic
557C2×C3#332611yescharacteristic
558C2×C3#337111yescharacteristic
559C2×C3#337111yescharacteristic
560D3—161no—
561C22—241no—
562C22—241no—
563C22—241no—
564C22—241no—
565C22—241no—
566C22—241no—
567C22—241no—
568C22#552411yescharacteristic
569C22#555411yescharacteristic
570C22#557611yescharacteristic
571C22#574811yescharacteristic
572C22#574811yescharacteristic
573C22#575111yescharacteristic
574C22#575111yescharacteristic
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—64no—
582C4—122no—
583C4—241no—
584C4—241no—
585C4—241no—
586C4—241no—
587C4—241no—
588C4#563811yescharacteristic
589C4#566111yescharacteristic
590C4#569311yescharacteristic
591C4D3×#12611yescharacteristic
592C3#2757711yescharacteristic, complemented, Sylow
593C2—481no—
594C2#7502011yescharacteristic
595C2#7934511yescharacteristic
596C2#7939511yescharacteristic
597C2#7951611yescharacteristic
598C2#7956111yescharacteristic
599C2#7998211yescharacteristic
600C2#7998211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1917C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1917D3
hypocenterC3#27577
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. #73637C2semidirect
3. C3×#1917C2semidirect
4. #70486C2semidirect
7. #69813C2semidirect
11. #68684C2semidirect
12. #68644C2semidirect
13. #68644C2semidirect
15. #68623C2semidirect
68. #1917D3semidirect
592. C3#27577semidirect

Maximal subgroups

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

subgroupindex
#736942
#736372
C3×#19172
#704862
#704832
#698182
#698132
#686862
#686842
#686442
#686372
#686232
#275773