Finite Group Databasebeta

← #10876057

Subgroups — #10876057

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#76047C211yesmaximal, characteristic
2#74157C211yesmaximal, characteristic, complemented
3C3×#2101C211yesmaximal, characteristic, complemented
4#69631C211yesmaximal, characteristic, complemented
5#69557C211yesmaximal, characteristic
6#68909C211yesmaximal, characteristic
7#68909C211yesmaximal, characteristic
8#68904C211yesmaximal, characteristic, complemented
9#68901C211yesmaximal, characteristic
10#68856C211yesmaximal, characteristic
11#68810C211yesmaximal, characteristic, complemented
12#68790C211yesmaximal, characteristic, complemented
13#68783C211yesmaximal, characteristic, complemented
14#68686C211yesmaximal, characteristic
15#68612C211yesmaximal, characteristic, complemented
16#27578—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×#898—21no—
19C2×#887C2211yescharacteristic
20C2×#887C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#382C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#816C2211yescharacteristic
26C2×#815C2211yescharacteristic
27C2×Dic3⋊C8C2211yescharacteristic
28#4994—21no—
29#4988C2211yescharacteristic
30#4988C2211yescharacteristic
31#4975C2211yescharacteristic
32Dic3×C4⋊C4C2211yescharacteristic
33#4972C2211yescharacteristic
34#4972C2211yescharacteristic
35C2×#810C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#807C2211yescharacteristic
40#4939C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42#4675—21no—
43#4669—21no—
44#4650—21no—
45#4649—21no—
46#4647—21no—
47#4647—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2512—31no—
61#2351—31no—
62#2101D311yescharacteristic, complemented
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#1880—31no—
72#1863—31no—
73#1803—31no—
74#1763—31no—
75C22×D12C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×(C2×D3)⋊C4—21no—
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×C4⋊Dic3D411yescharacteristic
92C2×C4⋊Dic3D411yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
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×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C2×#204—22no—
111C2×#204—22no—
112C2×#204—22no—
113#898—44no—
114#887—22no—
115#887—22no—
116#887—41no—
117#887—41no—
118C4×Dic6—22no—
119C4×Dic6—22no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95D411yescharacteristic
127C3⋊#95D411yescharacteristic
128#821—22no—
129#816—22no—
130#815—41no—
131Dic3⋊C8—41no—
132#810—22no—
133#810—22no—
134#809—41no—
135#809—41no—
136#807—22no—
137C22×SD16—31no—
138C2×#126—61no—
139C2×C4×Q8—31no—
140C2×C4×Q8—31no—
141C2×C4×Q8—31no—
142C2×#107—31no—
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#97—31no—
151#397—61no—
152#396—61no—
153#390—31no—
154#390—31no—
155#390—31no—
156#389—31no—
157#389—31no—
158#389—61no—
159#389—61no—
160#389C2×D311yescharacteristic
161#388—61no—
162#384—31no—
163#382—61no—
164#382C2×D311yescharacteristic
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#340C2×D311yescharacteristic
173#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#336—31no—
177#336—31no—
178#336—31no—
179#336C2×D311yescharacteristic
180C23×D3—21no—
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×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C22×Dic3C2×D411yescharacteristic
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6#4112yes—
205C2×Dic6#4112yes—
206C2×Dic6#4112yes—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—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—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—42no—
222C3×C42—21no—
223C3×C42—21no—
224(C2×D3)⋊C4—42no—
225(C2×D3)⋊C4—42no—
226(C2×D3)⋊C4—42no—
227(C2×D3)⋊C4—42no—
228(C2×D3)⋊C4—81no—
229(C2×D3)⋊C4—81no—
230C4⋊Dic3—21no—
231C4⋊Dic3—22no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—41no—
235C4⋊Dic3#4112yes—
236Dic3⋊C4—24no—
237Dic3⋊C4—42no—
238Dic3⋊C4—42no—
239Dic3⋊C4—42no—
240Dic3⋊C4—42no—
241Dic3⋊C4—42no—
242Dic3⋊C4—81no—
243C4×Dic3—24no—
244C4×Dic3—21no—
245C4×Dic3—21no—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—42no—
249C4×Dic3—42no—
250C4×Dic3—42no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257#204—44no—
258C22×Q8—31no—
259C22×D4—31no—
260C2×SD16—62no—
261C2×SD16—62no—
262C2×SD16—62no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267#126—124no—
268C4×Q8—62no—
269C4×Q8—62no—
270C4×Q8—62no—
271C4×Q8—62no—
272C4×Q8—121no—
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—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C42—31no—
297C2×C42—31no—
298C2×C42—61no—
299C2×C42—61no—
300C2×C42—61no—
301C2×C42—61no—
302C2×C42C22×D311yescharacteristic
303#107—62no—
304C4⋊C8—121no—
305#103—62no—
306#103—62no—
307#103—62no—
308#102—62no—
309#102—121no—
310#102—121no—
311#97—121no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—31no—
316#95—31no—
317#95—61no—
318#95—61no—
319#95—61no—
320C23×C3C22×D411yescharacteristic
321C22×D3—41no—
322C22×D3—41no—
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—41no—
326C22×D3—41no—
327C22×D3—41no—
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—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4H(5,𝔽2)11yescharacteristic
351C2×C3×C42−1+411yescharacteristic
352C2×Dic3—22no—
353C2×Dic3—21no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—21no—
357C2×Dic3—22no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3AGL(1,ℤ/8)12yes—
373D12—41no—
374D12—41no—
375D12—41no—
376D12—41no—
377Dic6—22no—
378Dic6—22no—
379Dic6—22no—
380Dic6#12614yescomplemented
381C3×C8—41no—
382C3×C8—41no—
383C3⋊C8—41no—
384C3⋊C8—41no—
385C24—61no—
386C2×Q8—32no—
387C2×Q8—32no—
388C2×Q8—32no—
389C2×Q8—61no—
390C2×Q8—61no—
391C2×Q8—61no—
392C2×D4—61no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—31no—
403C22×C4—31no—
404C22×C4—31no—
405C22×C4—31no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4D3×D411yescharacteristic
410C22×C4D3×D411yescharacteristic
411C22×C4C23×D311yescharacteristic
412SD16—124no—
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—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—32no—
434C4⋊C4—42no—
435C4⋊C4—62no—
436C4⋊C4—64no—
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—122no—
448C22⋊C4—122no—
449C22⋊C4—122no—
450C22⋊C4—122no—
451C22⋊C4—241no—
452C22⋊C4—241no—
453C42—21no—
454C42—21no—
455C42—64no—
456C42—61no—
457C42—61no—
458C42—61no—
459C42—61no—
460C42—122no—
461C42—122no—
462C42—122no—
463C22×C3#52511yescharacteristic
464C22×C3#54611yescharacteristic
465C22×C3#54811yescharacteristic
466C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
467C22×C3#57511yescharacteristic
468C22×C3#57711yescharacteristic
469C22×C3#57811yescharacteristic
470C2×D3—81no—
471C2×D3—81no—
472C2×D3—81no—
473C2×D3—81no—
474C2×D3—81no—
475C2×D3—81no—
476C2×D3—81no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—81no—
480C3×C4—81no—
481C3×C4—81no—
482C3×C4—81no—
483C3×C4C2×#12611yescharacteristic
484C3×C4#53911yescharacteristic
485C3×C4#55211yescharacteristic
486C3×C4#55311yescharacteristic
487Dic3—24no—
488Dic3—22no—
489Dic3—42no—
490Dic3—41no—
491Dic3—41no—
492Dic3—41no—
493Dic3—81no—
494C23—121no—
495C23—121no—
496C23—121no—
497C23—121no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23C2×D3×D411yescharacteristic
502Q8—34no—
503Q8—62no—
504Q8—62no—
505Q8—62no—
506D4—121no—
507D4—121no—
508D4—121no—
509D4—121no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—32no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—61no—
532C2×C4—62no—
533C2×C4—61no—
534C2×C4—62no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4#100711yescharacteristic
550C2×C4D3×#4111yescharacteristic
551C2×C4D3×#4111yescharacteristic
552C2×C4D3×#4111yescharacteristic
553C2×C4D3×#4111yescharacteristic
554C2×C4#100911yescharacteristic
555C8—41no—
556C8—41no—
557C8—121no—
558C8—121no—
559C2×C3#248111yescharacteristic
560C2×C3#311511yescharacteristic
561C2×C3#312111yescharacteristic
562C2×C3#330811yescharacteristic
563C2×C3#332611yescharacteristic
564C2×C3#336811yescharacteristic
565C2×C3#337711yescharacteristic
566D3—161no—
567C22—241no—
568C22—241no—
569C22—241no—
570C22—241no—
571C22—241no—
572C22—241no—
573C22—241no—
574C22#563311yescharacteristic
575C22#565611yescharacteristic
576C22#567611yescharacteristic
577C22#576811yescharacteristic
578C22#577011yescharacteristic
579C22D3×AGL(1,ℤ/8)11yescharacteristic
580C22#577911yescharacteristic
581C4—41no—
582C4—41no—
583C4—62no—
584C4—64no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—121no—
590C4—122no—
591C4—121no—
592C4—121no—
593C4—241no—
594C4#569211yescharacteristic
595C4#569311yescharacteristic
596C4D3×#12611yescharacteristic
597C4#570911yescharacteristic
598C3#2757811yescharacteristic, complemented, Sylow
599C2—481no—
600C2#7814411yescharacteristic
601C2#8115311yescharacteristic
602C2#8126711yescharacteristic
603C2#8135111yescharacteristic
604C2#8149711yescharacteristic
605C2#8165811yescharacteristic
606C2#8169711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2101C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2101D3
hypocenterC3#27578
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2101

Complements

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

NH ≅ G/Ntype
2. #74157C2semidirect
3. C3×#2101C2semidirect
4. #69631C2semidirect
8. #68904C2semidirect
11. #68810C2semidirect
12. #68790C2semidirect
13. #68783C2semidirect
15. #68612C2semidirect
62. #2101D3semidirect
380. Dic6#126semidirect
598. C3#27578semidirect

Maximal subgroups

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

subgroupindex
#760472
#741572
C3×#21012
#696312
#695572
#689092
#689042
#689012
#688562
#688102
#687902
#687832
#686862
#686122
#275783