Finite Group Databasebeta

← #10876066

Subgroups — #10876066

Order 768.

724 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#82461C211yesmaximal, characteristic, complemented
2#76338C211yesmaximal, characteristic
3C3×#1880C211yesmaximal, characteristic
4#72048C211yesmaximal, characteristic
5#72048C211yesmaximal, characteristic
6#71631C211yesmaximal, characteristic, complemented
7#70236C211yesmaximal, characteristic, complemented
8#70221C211yesmaximal, characteristic, complemented
9#70180C211yesmaximal, characteristic
10#70166C211yesmaximal, characteristic
11#69645C211yesmaximal, characteristic
12#68846C211yesmaximal, characteristic, complemented
13#68810C211yesmaximal, characteristic, complemented
14#68656C211yesmaximal, characteristic, complemented
15#68624C211yesmaximal, characteristic, complemented
16#27578—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×#879—21no—
22C2×C3×#103C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C3×#390C2211yescharacteristic
25C2×#835C2211yescharacteristic
26C2×#835C2211yescharacteristic
27#5221C2211yescharacteristic
28#5221C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33#4984C2211yescharacteristic
34#4977—21no—
35#4977C2211yescharacteristic
36#4975C2211yescharacteristic
37#4972—21no—
38#4972C2211yescharacteristic
39#4952—21no—
40#4948—21no—
41#4944—21no—
42C4×C4⋊Dic3C2211yescharacteristic
43C4×Dic3⋊C4—21no—
44C4×Dic3⋊C4C2211yescharacteristic
45#4928—21no—
46#4928C2211yescharacteristic
47C2×#803C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#336C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2512—31no—
61#2351—31no—
62#2101—31no—
63#2099—31no—
64#2073—31no—
65#2067—31no—
66#1936—31no—
67#1921—31no—
68#1917—31no—
69#1880—31no—
70#1880—31no—
71#1880D311yescharacteristic
72#1863—31no—
73#1803—31no—
74#1763—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C2×C3×SD16—22no—
78C2×C3×SD16—22no—
79C2×C3×SD16—22no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4×Q8—22no—
83C3×C4×Q8—41no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95Q8×Dic3—22no—
96Q8×Dic3—22no—
97Q8×Dic3—22no—
98Q8×Dic3—41no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4D411yescharacteristic
111C2×Dic3⋊C4D411yescharacteristic
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#879—44no—
120C3×#103—22no—
121C3×#102—41no—
122#835—22no—
123#835—22no—
124#832—22no—
125#832—41no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134#818—22no—
135Dic3⋊C8—41no—
136#803—41no—
137C22×SD16C2×D311yescharacteristic
138C2×#126—61no—
139C2×C4×Q8—31no—
140C2×C4×Q8—31no—
141C2×C4×Q8C2×D311yescharacteristic
142C2×#107—31no—
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103C2×D311yescharacteristic
147C2×#102—31no—
148C2×#102—31no—
149C2×#102C2×D311yescharacteristic
150C2×#97—31no—
151#397—61no—
152#396—61no—
153#390—31no—
154#390—31no—
155#390C2×D311yescharacteristic
156#389—31no—
157#389—31no—
158#389—31no—
159#389—61no—
160#389—61no—
161#388—61no—
162#384—31no—
163#382—31no—
164#382—61no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—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#336—31no—
177#336—31no—
178#336—31no—
179#336C2×D311yescharacteristic
180C24×C3—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8#4112yes—
185C2×C3×Q8#4112yes—
186C2×C3×Q8#4112yes—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C3×SD16—44no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—42no—
219C3×C22⋊C4—42no—
220C3×C22⋊C4—81no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—42no—
224#217—42no—
225#217—42no—
226#217—42no—
227#217—81no—
228C4⋊Dic3—24no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—42no—
237C4⋊Dic3—42no—
238C4⋊Dic3#4112yes—
239Dic3⋊C4—22no—
240Dic3⋊C4—41no—
241Dic3⋊C4—42no—
242Dic3⋊C4—42no—
243Dic3⋊C4—42no—
244Dic3⋊C4—81no—
245C4×Dic3—24no—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—42no—
251C4×Dic3—42no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C22×Q8C22×D311yescharacteristic
259C22×D4C22×D311yescharacteristic
260C2×SD16—22no—
261C2×SD16—22no—
262C2×SD16—22no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267#126—124no—
268C4×Q8—22no—
269C4×Q8—41no—
270C4×Q8—62no—
271C4×Q8—62no—
272C4×Q8—62no—
273C4×Q8—121no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C42—21no—
297C2×C42—31no—
298C2×C42—31no—
299C2×C42—61no—
300C2×C42—61no—
301C2×C42—61no—
302C2×C42C22×D311yescharacteristic
303#107—62no—
304C4⋊C8—121no—
305#103—22no—
306#103—62no—
307#103—62no—
308#102—41no—
309#102—62no—
310#102—121no—
311#97—121no—
312#95—31no—
313#95—31no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3C22×D411yescharacteristic
328C3×Q8—22no—
329C3×Q8—22no—
330C3×Q8—22no—
331C3×Q8#12614yescomplemented
332C3×D4—41no—
333C3×D4—41no—
334C3×D4—41no—
335C3×D4—41no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4H(5,𝔽2)11yescharacteristic
356C2×C3×C42−1+411yescharacteristic
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—22no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3AGL(1,ℤ/8)12yes—
381C3×C8—41no—
382C3×C8—41no—
383C3⋊C8—41no—
384C3⋊C8—41no—
385C24—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8#23512yes—
390C2×Q8#23712yes—
391C2×Q8#23712yes—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—31no—
402C22×C4—31no—
403C22×C4—31no—
404C22×C4—31no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4D3×D411yescharacteristic
410C22×C4D3×D411yescharacteristic
411C22×C4C23×D311yescharacteristic
412SD16—44no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—32no—
430C4⋊C4—42no—
431C4⋊C4—62no—
432C4⋊C4—64no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—121no—
441C4⋊C4—122no—
442C4⋊C4—122no—
443C4⋊C4—122no—
444C4⋊C4—122no—
445C4⋊C4—122no—
446C4⋊C4—241no—
447C22⋊C4—42no—
448C22⋊C4—81no—
449C22⋊C4—122no—
450C22⋊C4—122no—
451C22⋊C4—122no—
452C22⋊C4—241no—
453C42—21no—
454C42—21no—
455C42—42no—
456C42—64no—
457C42—61no—
458C42—61no—
459C42—61no—
460C42—61no—
461C42—122no—
462C42—122no—
463C22×C3—81no—
464C22×C3—81no—
465C22×C3—81no—
466C22×C3—81no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3#52511yescharacteristic
471C22×C3#54611yescharacteristic
472C22×C3#54811yescharacteristic
473C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
474C22×C3#57511yescharacteristic
475C22×C3#57711yescharacteristic
476C22×C3#57811yescharacteristic
477C3×C4—24no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—81no—
481C3×C4—81no—
482C3×C4C2×#12611yescharacteristic
483C3×C4#53911yescharacteristic
484C3×C4#55211yescharacteristic
485C3×C4#55311yescharacteristic
486Dic3—22no—
487Dic3—41no—
488Dic3—42no—
489Dic3—41no—
490Dic3—41no—
491Dic3—81no—
492Dic3—81no—
493Dic3—81no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23C2×D3×D411yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—22no—
505Q8#87914yescomplemented
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—32no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—62no—
536C2×C4—61no—
537C2×C4—61no—
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—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4C2×#23511yescharacteristic
550C2×C4C2×#23711yescharacteristic
551C2×C4C2×#23711yescharacteristic
552C2×C4#100411yescharacteristic
553C2×C4D3×#4111yescharacteristic
554C2×C4#101011yescharacteristic
555C8—41no—
556C8—41no—
557C8—121no—
558C8—121no—
559C2×C3—161no—
560C2×C3#248111yescharacteristic
561C2×C3#311511yescharacteristic
562C2×C3#312111yescharacteristic
563C2×C3#330811yescharacteristic
564C2×C3#332611yescharacteristic
565C2×C3#336811yescharacteristic
566C2×C3#337711yescharacteristic
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22#554711yescharacteristic
575C22#559111yescharacteristic
576C22#559811yescharacteristic
577C22#576211yescharacteristic
578C22#577111yescharacteristic
579C22D3×AGL(1,ℤ/8)11yescharacteristic
580C22#577711yescharacteristic
581C4—24no—
582C4—41no—
583C4—41no—
584C4—62no—
585C4—81no—
586C4—81no—
587C4—121no—
588C4—121no—
589C4—121no—
590C4—122no—
591C4—241no—
592C4—241no—
593C4—241no—
594C4C2×#87911yescharacteristic
595C4#556411yescharacteristic
596C4#560111yescharacteristic
597C4#565011yescharacteristic
598C3#2757811yescharacteristic, complemented, Sylow
599C2—161no—
600C2#7473511yescharacteristic
601C2#7965811yescharacteristic
602C2#7981111yescharacteristic
603C2#8005511yescharacteristic
604C2#8010911yescharacteristic
605C2#8037211yescharacteristic
606C2#8058711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1880C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1880D3
hypocenterC3#27578
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
1. #82461C2semidirect
6. #71631C2semidirect
7. #70236C2semidirect
8. #70221C2semidirect
12. #68846C2semidirect
13. #68810C2semidirect
14. #68656C2semidirect
15. #68624C2semidirect
331. C3×Q8#126semidirect
505. Q8#879semidirect
598. C3#27578semidirect

Maximal subgroups

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

subgroupindex
#824612
#763382
C3×#18802
#720482
#716312
#702362
#702212
#701802
#701662
#696452
#688462
#688102
#686562
#686242
#275783