Finite Group Databasebeta

← #10858311

Subgroups — #10858311

Order 768.

932 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83236C211yesmaximal, characteristic, complemented
2C2×#5084C211yesmaximal, characteristic, complemented
3C2×#4872C211yesmaximal, characteristic
4#76203C211yesmaximal, characteristic, complemented
5C3×#2057C211yesmaximal, characteristic, complemented
6#70640C211yesmaximal, characteristic, complemented
7#70638C211yesmaximal, characteristic, complemented
8#70620C211yesmaximal, characteristic, complemented
9#69629C211yesmaximal, characteristic, complemented
10#68897C211yesmaximal, characteristic
11#68895C211yesmaximal, characteristic, complemented
12#68895C211yesmaximal, characteristic, complemented
13#68858C211yesmaximal, characteristic, complemented
14#68673C211yesmaximal, characteristic
15#68611C211yesmaximal, characteristic, complemented
16#26380—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#946—21no—
19C22×D24C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#107C2211yescharacteristic
26C3×#389C2211yescharacteristic
27#5231—21no—
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×C8×Dic3C2211yescharacteristic
31#5084—24no—
32#4994—21no—
33#4989—21no—
34#4989—21no—
35#4988C2211yescharacteristic
36#4987—21no—
37#4984C2211yescharacteristic
38#4984C2211yescharacteristic
39#4977C2211yescharacteristic
40#4972C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#808C2211yescharacteristic
48#4939C2211yescharacteristic
49C4×Dic3⋊C4C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51#4872—24no—
52#4668—21no—
53D3×#95—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#476—31no—
63C2×C4×D8—31no—
64#2452—31no—
65#2349—31no—
66#2092—31no—
67#2085—31no—
68#2065—31no—
69#2057D311yescharacteristic, complemented
70#1941—31no—
71#1916—31no—
72#1879—31no—
73#1879—31no—
74#1864—31no—
75#1864—31no—
76#1862—31no—
77C22×C3×Q8—21no—
78C22×D12C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C4×D3—21no—
83C22×C4×D3—21no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4⋊Q8—22no—
87C3×C4⋊Q8—41no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95#946—44no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4D411yescharacteristic
110C2×Dic3⋊C4D411yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C2×D24—22no—
116C2×D24—22no—
117C2×D24Q814yescomplemented
118#893—22no—
119#893—22no—
120#893—22no—
121#893—22no—
122#893—41no—
123#893—41no—
124C4×D12—22no—
125C4×D12—41no—
126C3×#107—22no—
127C3×#107—41no—
128C3×#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95D411yescharacteristic
135C3⋊#95D411yescharacteristic
136#821—22no—
137#821—41no—
138C8×Dic3—22no—
139#809—22no—
140#809—22no—
141#809—22no—
142#809D414yes—
143#808—22no—
144#808—22no—
145#808—22no—
146C4⋊(C3⋊C8)—22no—
147C22×D8—31no—
148C2×C4⋊Q8C2×D311yescharacteristic
149C2×C22⋊Q8—61no—
150C2×C4×D4—31no—
151C2×C4×D4—31no—
152C2×C4×D4—31no—
153C2×C4×D4—31no—
154#476—64no—
155C4×D8—64no—
156C2×#107C2×D311yescharacteristic
157C2×#107C2×D311yescharacteristic
158C2×#106—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C4×C8—31no—
169#396—61no—
170#393—61no—
171#390—31no—
172#390—31no—
173#389—31no—
174#389—31no—
175#389—31no—
176#389C2×D311yescharacteristic
177#385—61no—
178#385—61no—
179#385—61no—
180#382—31no—
181#380—61no—
182C4×C4⋊C4—31no—
183C2×#95—61no—
184#340—31no—
185#340—31no—
186#340C2×D311yescharacteristic
187#340C2×D311yescharacteristic
188#336—31no—
189#336—31no—
190#336—31no—
191#336C2×D311yescharacteristic
192C23×D3—21no—
193C23×D3—21no—
194C2×C3×Q8—42no—
195C2×C3×Q8—42no—
196C2×C3×Q8—81no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12C2×D412yes—
219C2×D12C2×Q812yes—
220C2×D12C2×Q812yes—
221C2×D12#4112yes—
222C2×C4×D3—24no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×C4×D3—42no—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—42no—
229C2×C4×D3—81no—
230C2×C4×D3—81no—
231C2×C4×D3—81no—
232C2×C4×D3—81no—
233C2×C4×D3—81no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8C2×Q812yes—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4—81no—
248C3×C4⋊C4C2×D412yes—
249C3×C4⋊C4#4112yes—
250C3×C42—21no—
251C3×C42—21no—
252(C2×D3)⋊C4—24no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—81no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—42no—
265C4⋊Dic3—81no—
266Dic3⋊C4—22no—
267Dic3⋊C4—22no—
268Dic3⋊C4—42no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8C2×D412yes—
279D24—28no—
280C22×Q8—21no—
281C22×D4—31no—
282C22×D4—31no—
283C23×C4—61no—
284C23×C4—61no—
285C23×C4—61no—
286C23×C4—61no—
287C2×D8—34no—
288C2×D8—62no—
289C2×D8—62no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294C4⋊Q8—22no—
295C4⋊Q8—41no—
296C22⋊Q8—124no—
297C4×D4—62no—
298C4×D4—62no—
299C4×D4—62no—
300C4×D4—62no—
301C4×D4—62no—
302C4×D4—121no—
303C4×D4—121no—
304C4×D4—121no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42C22×D311yescharacteristic
329#107—22no—
330#107—41no—
331#106—62no—
332#106—62no—
333#106—62no—
334C4⋊C8—62no—
335#102—34no—
336#102—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#102—121no—
341C4×C8—62no—
342#95—21no—
343#95—31no—
344#95—31no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3C22×D411yescharacteristic
351C22×D3—22no—
352C22×D3—22no—
353C22×D3—22no—
354C22×D3—22no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C3×Q8—84no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4#13512yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×Q811yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4H(5,𝔽2)11yescharacteristic
389C2×C3×C42−1+411yescharacteristic
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3AGL(1,ℤ/8)12yes—
408D12—22no—
409D12—22no—
410D12—22no—
411D12—22no—
412D12—22no—
413D12—41no—
414D12—41no—
415D12C22⋊Q814yescomplemented
416C4×D3—84no—
417C4×D3—84no—
418C4×D3—84no—
419C4×D3—84no—
420C3×C8—22no—
421C3×C8—41no—
422C3⋊C8—22no—
423C3⋊C8—41no—
424C24—61no—
425C24—61no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—81no—
429C2×D4—32no—
430C2×D4—32no—
431C2×D4—32no—
432C2×D4—32no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—64no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—122no—
453C22×C4—122no—
454C22×C4—122no—
455C22×C4—122no—
456C22×C4—122no—
457C22×C4—122no—
458C22×C4—241no—
459C22×C4—241no—
460C22×C4—241no—
461C22×C4—241no—
462C22×C4—241no—
463C22×C4D3×D411yescharacteristic
464C22×C4D3×D411yescharacteristic
465C22×C4C23×D311yescharacteristic
466D8—68no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—32no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8D3×Q812yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—62no—
487C4⋊C4—62no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—81no—
493C4⋊C4—122no—
494C4⋊C4—122no—
495C4⋊C4—241no—
496C4⋊C4D3×D412yes—
497C4⋊C4#23912yes—
498C22⋊C4—64no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—241no—
506C42—21no—
507C42—21no—
508C42—61no—
509C42—61no—
510C42—61no—
511C42—61no—
512C22×C3#52511yescharacteristic
513C22×C3D4211yescharacteristic
514C22×C3D4×Q811yescharacteristic
515C22×C3C2×#13511yescharacteristic
516C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
517C22×C3#57611yescharacteristic
518C22×C3#57711yescharacteristic
519C2×D3—24no—
520C2×D3—42no—
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—42no—
524C2×D3—42no—
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C2×D3—81no—
533C3×C4—22no—
534C3×C4—42no—
535C3×C4—42no—
536C3×C4—41no—
537C3×C4—81no—
538C3×C4—81no—
539C3×C4C2×C22⋊Q811yescharacteristic
540C3×C4#53911yescharacteristic
541C3×C4#54711yescharacteristic
542C3×C4#55411yescharacteristic
543Dic3—22no—
544Dic3—41no—
545Dic3—41no—
546Dic3—41no—
547Dic3—81no—
548Dic3—81no—
549C23—62no—
550C23—62no—
551C23—62no—
552C23—62no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23—121no—
563C23C2×D3×D411yescharacteristic
564Q8—84no—
565D4—34no—
566D4—62no—
567D4—62no—
568D4—62no—
569D4—62no—
570D4—62no—
571D4—121no—
572D4—121no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—32no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—244no—
609C2×C4—244no—
610C2×C4—244no—
611C2×C4—244no—
612C2×C4#91912yes—
613C2×C4C2×D3×D411yescharacteristic
614C2×C4#100411yescharacteristic
615C2×C4C2×D3×Q811yescharacteristic
616C2×C4C2×#23911yescharacteristic
617C2×C4#100711yescharacteristic
618C2×C4D3×#4111yescharacteristic
619C8—22no—
620C8—41no—
621C8—62no—
622C8—121no—
623C2×C3#244911yescharacteristic
624C2×C3#310611yescharacteristic
625C2×C3#311811yescharacteristic
626C2×C3#326411yescharacteristic
627C2×C3#328211yescharacteristic
628C2×C3#338011yescharacteristic
629C2×C3#338411yescharacteristic
630D3—44no—
631D3—82no—
632C22—64no—
633C22—122no—
634C22—122no—
635C22—122no—
636C22—122no—
637C22—122no—
638C22—122no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22#559011yescharacteristic
647C22#562711yescharacteristic
648C22#572711yescharacteristic
649C22#575711yescharacteristic
650C22C2×#91911yescharacteristic
651C22D3×AGL(1,ℤ/8)11yescharacteristic
652C22#577811yescharacteristic
653C4—22no—
654C4—42no—
655C4—41no—
656C4—42no—
657C4—62no—
658C4—81no—
659C4—81no—
660C4—121no—
661C4—121no—
662C4—121no—
663C4—241no—
664C4—241no—
665C4D3×C22⋊Q811yescharacteristic
666C4#563811yescharacteristic
667C4#572611yescharacteristic
668C4#573011yescharacteristic
669C3#2638011yescharacteristic, complemented, Sylow
670C2—124no—
671C2—242no—
672C2#7725211yescharacteristic
673C2#8001211yescharacteristic
674C2#8020711yescharacteristic
675C2#8095211yescharacteristic
676C2#8109311yescharacteristic
677C2#8195311yescharacteristic
678C2#8196411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2057C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#2057D3
hypocenterC3#26380
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2057

Complements

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

NH ≅ G/Ntype
1. #83236C2semidirect
2. C2×#5084C2semidirect
4. #76203C2semidirect
5. C3×#2057C2semidirect
6. #70640C2semidirect
7. #70638C2semidirect
8. #70620C2semidirect
9. #69629C2semidirect
11. #68895C2semidirect
12. #68895C2semidirect
13. #68858C2semidirect
15. #68611C2semidirect
69. #2057D3semidirect
117. C2×D24Q8semidirect
415. D12C22⋊Q8semidirect
669. C3#26380semidirect

Maximal subgroups

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

subgroupindex
#832362
C2×#50842
C2×#48722
#762032
C3×#20572
#706402
#706382
#706202
#696292
#688972
#688952
#688582
#686732
#686112
#263803