Finite Group Databasebeta

← #10876002

Subgroups — #10876002

Order 768.

800 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73770C211yesmaximal, characteristic, complemented
2#73692C211yesmaximal, characteristic, complemented
3C3×#1917C211yesmaximal, characteristic, complemented
4#70483C211yesmaximal, characteristic, complemented
5#70479C211yesmaximal, characteristic, complemented
6#69818C211yesmaximal, characteristic, complemented
7#69812C211yesmaximal, characteristic, complemented
8#68673C211yesmaximal, characteristic
9#68673C211yesmaximal, characteristic
10#68673C211yesmaximal, characteristic
11#68671C211yesmaximal, characteristic, complemented
12#68644C211yesmaximal, characteristic, complemented
13#68642C211yesmaximal, characteristic, complemented
14#68637C211yesmaximal, characteristic, complemented
15#68622C211yesmaximal, characteristic, complemented
16#27574—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C2×#898—21no—
19C2×C4×D12C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24C3×C4×C4⋊C4C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#817C2211yescharacteristic
32#4994—21no—
33#4989—21no—
34#4989—21no—
35#4939C2211yescharacteristic
36#4939C2211yescharacteristic
37#4939C2211yescharacteristic
38#4936C2211yescharacteristic
39#4935C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4674—21no—
45#4670—21no—
46#4667—21no—
47D3×#95—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#2512—31no—
61#2349—31no—
62#2101—31no—
63#2099—31no—
64#2073—31no—
65#2065—31no—
66#1936—31no—
67#1919—31no—
68#1917D311yescharacteristic, complemented
69#1879—31no—
70#1879—31no—
71#1879—31no—
72#1864—31no—
73#1802—31no—
74#1762—31no—
75C22×D12C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3—21no—
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×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
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×C4×Dic3—21no—
110C2×D24—22no—
111C2×D24—22no—
112C2×D24—22no—
113#898—44no—
114C4×D12—22no—
115C4×D12—22no—
116C4×D12—22no—
117C4×D12—22no—
118C4×D12—41no—
119C4×D12—41no—
120C3×C4⋊C8—41no—
121C3×#97—41no—
122C3×#95D411yescharacteristic
123C3×#95D411yescharacteristic
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130#821—22no—
131#821—22no—
132#821—22no—
133#821—22no—
134#821—41no—
135#821—41no—
136#817—22no—
137C22×D8—31no—
138C2×#126—61no—
139C2×C4×D4—31no—
140C2×C4×D4—31no—
141C2×C4×D4—31no—
142C2×#106—31no—
143C2×C4⋊C8C2×D311yescharacteristic
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#97C2×D311yescharacteristic
151#397—61no—
152#396—61no—
153#390—31no—
154#390—31no—
155#390—31no—
156#389—31no—
157#389—31no—
158#389—31no—
159#388—61no—
160#385—61no—
161#385—61no—
162#384—31no—
163#382—31no—
164#380—61no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4C2×D311yescharacteristic
167C2×#95—61no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#336C2×D311yescharacteristic
177#336C2×D311yescharacteristic
178#336C2×D311yescharacteristic
179#336C2×D311yescharacteristic
180C23×D3—21no—
181C23×D3—21no—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2411yescharacteristic
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12#4112yes—
205C2×D12#4112yes—
206C2×D12#4112yes—
207C2×C4×D3—24no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—42no—
212C2×C4×D3—81no—
213C2×C4×D3—81no—
214C2×C4×D3—81no—
215C2×C4×D3—81no—
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—
227C2×C3×C8—21no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—41no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—42no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—42no—
241(C2×D3)⋊C4—42no—
242(C2×D3)⋊C4—42no—
243(C2×D3)⋊C4—81no—
244(C2×D3)⋊C4—81no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—42no—
257C4⋊Dic3—81no—
258C4⋊Dic3#4112yes—
259Dic3⋊C4—42no—
260Dic3⋊C4—81no—
261Dic3⋊C4—81no—
262C4×Dic3—42no—
263D24—44no—
264C22×D4—31no—
265C22×D4—31no—
266C23×C4—61no—
267C23×C4—61no—
268C23×C4—61no—
269C2×D8—62no—
270C2×D8—62no—
271C2×D8—62no—
272C22×C8C22×D311yescharacteristic
273C22×C8C22×D311yescharacteristic
274C22×C8C22×D311yescharacteristic
275C22×C8C22×D311yescharacteristic
276#126—124no—
277C4×D4—62no—
278C4×D4—62no—
279C4×D4—62no—
280C4×D4—62no—
281C4×D4—121no—
282C4×D4—121no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4D1211yescharacteristic
295C2×C4⋊C4D1211yescharacteristic
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C42—61no—
306C2×C42C22×D311yescharacteristic
307C2×C42C22×D311yescharacteristic
308C2×C42C22×D311yescharacteristic
309#106—62no—
310C4⋊C8—41no—
311#102—62no—
312#102—62no—
313#102—62no—
314#102—62no—
315#102—121no—
316#102—121no—
317#97—41no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95D1211yescharacteristic
325#95D1211yescharacteristic
326C23×C3C22×D411yescharacteristic
327C22×D3—22no—
328C22×D3—22no—
329C22×D3—22no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—41no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4AGL(1,ℤ/8)12yes—
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4H(5,𝔽2)11yescharacteristic
364C2×C3×C42−1+411yescharacteristic
365C2×Dic3—22no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
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×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383D12—22no—
384D12—22no—
385D12—22no—
386D12—41no—
387D12—41no—
388D12—41no—
389D12—41no—
390D12#12614yescomplemented
391C4×D3—84no—
392C4×D3—84no—
393C4×D3—84no—
394C3×C8—41no—
395C3×C8—41no—
396C3×C8—41no—
397C3×C8—41no—
398C24—61no—
399C24—61no—
400C2×D4—32no—
401C2×D4—32no—
402C2×D4—32no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—61no—
412C22×C4—64no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—122no—
420C22×C4—122no—
421C22×C4—122no—
422C22×C4—122no—
423C22×C4—241no—
424C22×C4—241no—
425C22×C4—241no—
426C22×C4—241no—
427C22×C4C2×D1211yescharacteristic
428C22×C4C2×D1211yescharacteristic
429C22×C4C2×D1211yescharacteristic
430C22×C4C2×D1211yescharacteristic
431C22×C4C2×D1211yescharacteristic
432C22×C4C2×D1211yescharacteristic
433C22×C4C23×D311yescharacteristic
434D8—124no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C4⋊C4—22no—
448C4⋊C4—32no—
449C4⋊C4—41no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—241no—
464C4⋊C4—241no—
465C4⋊C4—241no—
466C22⋊C4—64no—
467C22⋊C4—122no—
468C22⋊C4—122no—
469C22⋊C4—122no—
470C22⋊C4—122no—
471C22⋊C4—122no—
472C22⋊C4—122no—
473C22⋊C4—241no—
474C22⋊C4—241no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—122no—
482C22×C3#52511yescharacteristic
483C22×C3#54611yescharacteristic
484C22×C3#54811yescharacteristic
485C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
486C22×C3#57511yescharacteristic
487C22×C3#57611yescharacteristic
488C22×C3#57711yescharacteristic
489C2×D3—24no—
490C2×D3—42no—
491C2×D3—42no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—42no—
495C2×D3—42no—
496C2×D3—81no—
497C2×D3—81no—
498C2×D3—81no—
499C2×D3—81no—
500C2×D3—81no—
501C2×D3—81no—
502C2×D3—81no—
503C3×C4—22no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—41no—
509C3×C4C2×#12611yescharacteristic
510C3×C4#53911yescharacteristic
511C3×C4#55211yescharacteristic
512C3×C4#55311yescharacteristic
513Dic3—42no—
514Dic3—81no—
515Dic3—81no—
516Dic3—81no—
517Dic3—81no—
518Dic3—81no—
519C23—62no—
520C23—62no—
521C23—62no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23C22×D1211yescharacteristic
534D4—34no—
535D4—62no—
536D4—62no—
537D4—62no—
538D4—121no—
539D4—121no—
540D4—121no—
541D4—121no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—62no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—244no—
578C2×C4—244no—
579C2×C4—244no—
580C2×C4#90812yes—
581C2×C4#100411yescharacteristic
582C2×C4#100711yescharacteristic
583C2×C4D3×#4111yescharacteristic
584C2×C4D3×#4111yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4D3×#4111yescharacteristic
587C8—41no—
588C8—41no—
589C8—41no—
590C8—41no—
591C2×C3#248111yescharacteristic
592C2×C3#311511yescharacteristic
593C2×C3#311811yescharacteristic
594C2×C3#330611yescharacteristic
595C2×C3#332211yescharacteristic
596C2×C3#336611yescharacteristic
597C2×C3#337611yescharacteristic
598D3—44no—
599D3—161no—
600C22—64no—
601C22—122no—
602C22—122no—
603C22—122no—
604C22—122no—
605C22—122no—
606C22—122no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22#552411yescharacteristic
615C22#555411yescharacteristic
616C22#557611yescharacteristic
617C22C2×#90811yescharacteristic
618C22#574811yescharacteristic
619C22#575111yescharacteristic
620C22#575211yescharacteristic
621C4—22no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—122no—
628C4—241no—
629C4—241no—
630C4—241no—
631C4—241no—
632C4—241no—
633C4#563811yescharacteristic
634C4#566111yescharacteristic
635C4#569311yescharacteristic
636C4D3×#12611yescharacteristic
637C3#2757411yescharacteristic, complemented, Sylow
638C2—124no—
639C2—481no—
640C2#7502011yescharacteristic
641C2#7934711yescharacteristic
642C2#7938311yescharacteristic
643C2#7949911yescharacteristic
644C2#7955311yescharacteristic
645C2#7996911yescharacteristic
646C2#7997411yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1917C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1917D3
hypocenterC3#27574
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
1. #73770C2semidirect
2. #73692C2semidirect
3. C3×#1917C2semidirect
4. #70483C2semidirect
5. #70479C2semidirect
6. #69818C2semidirect
7. #69812C2semidirect
11. #68671C2semidirect
12. #68644C2semidirect
13. #68642C2semidirect
14. #68637C2semidirect
15. #68622C2semidirect
68. #1917D3semidirect
390. D12#126semidirect
637. C3#27574semidirect

Maximal subgroups

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

subgroupindex
#737702
#736922
C3×#19172
#704832
#704792
#698182
#698122
#686732
#686712
#686442
#686422
#686372
#686222
#275743