Finite Group Databasebeta

← #10852293

Subgroups — #10852293

Order 768.

958 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82457C211yesmaximal, characteristic
2C2×#5022C211yesmaximal, characteristic, complemented
3C2×#4804C211yesmaximal, characteristic
4#76224C211yesmaximal, characteristic, complemented
5C3×#1865C211yesmaximal, characteristic, complemented
6#70587C211yesmaximal, characteristic, complemented
7#70580C211yesmaximal, characteristic, complemented
8#70267C211yesmaximal, characteristic
9#70237C211yesmaximal, characteristic, complemented
10#70221C211yesmaximal, characteristic, complemented
11#69629C211yesmaximal, characteristic, complemented
12#68901C211yesmaximal, characteristic, complemented
13#68868C211yesmaximal, characteristic, complemented
14#68679C211yesmaximal, characteristic, complemented
15#68666C211yesmaximal, characteristic, complemented
16#25953—31nomaximal, Sylow
17C22×#239C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×D3×C8C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×#870C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33#5022—24no—
34#4993—21no—
35#4988—21no—
36#4988C2211yescharacteristic
37#4987—21no—
38#4987C2211yescharacteristic
39#4987C2211yescharacteristic
40#4976—21no—
41#4975C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#807C2211yescharacteristic
48#4939—21no—
49#4939C2211yescharacteristic
50C4×(C2×D3)⋊C4—21no—
51C4×(C2×D3)⋊C4C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53#4804—24no—
54#4676—21no—
55#4652—21no—
56C3×#340C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#479—31no—
63C2×#416—31no—
64#2441—31no—
65#2341—31no—
66#2094—31no—
67#2067—31no—
68#2065—31no—
69#2025—31no—
70#1865—31no—
71#1865D311yescharacteristic, complemented
72#1854—31no—
73#1850—31no—
74#1807—31no—
75#1803—31no—
76#1788—31no—
77C22×C3×Q8C2311yescharacteristic
78C2×#239—22no—
79C2×#239—22no—
80C2×#239—22no—
81C2×#239—41no—
82C2×#239—41no—
83C2×#239Q814yes—
84C22×D12—21no—
85C22×D12C2311yescharacteristic
86C22×C4×D3—21no—
87C22×C4×D3C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×C4×Q8—22no—
91C3×C4×Q8—41no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99Q8×Dic3—22no—
100Q8×Dic3—41no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4D411yescharacteristic
109C2×(C2×D3)⋊C4D411yescharacteristic
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C2×D3×C8—22no—
125C2×D3×C8—22no—
126C2×D3×C8—41no—
127#892—22no—
128#892—41no—
129#870—22no—
130#870—41no—
131C3×#106—41no—
132C3×#103—22no—
133C3×#103—22no—
134C3×#95—21no—
135#835—22no—
136#835D414yes—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95D411yescharacteristic
141C3⋊#95D411yescharacteristic
142#821—22no—
143#821—41no—
144(C2×D3)⋊C8—22no—
145Dic3⋊C8—41no—
146#809—22no—
147#809—41no—
148#808—22no—
149#808—41no—
150#807—22no—
151C4⋊(C3⋊C8)—22no—
152C22×#41—31no—
153C23×C8—31no—
154C2×#125—31no—
155C2×C4×Q8—31no—
156C2×C4×Q8C2×D311yescharacteristic
157C2×#117—31no—
158#479—64no—
159C2×#107—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106C2×D311yescharacteristic
163C2×C4⋊C8—31no—
164C2×C4⋊C8—31no—
165#416—64no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103C2×D311yescharacteristic
169C2×#103C2×D311yescharacteristic
170C2×#102—31no—
171C2×#102—31no—
172C2×#102—31no—
173C2×#102—31no—
174C2×C22⋊C8—31no—
175#399—61no—
176#396—61no—
177#390—31no—
178#390—31no—
179#390—61no—
180#390—61no—
181#389—31no—
182#389—61no—
183#389C2×D311yescharacteristic
184#387—61no—
185#384—31no—
186#380—31no—
187#380—31no—
188#380—61no—
189C4×C22⋊C4—31no—
190C4×C22⋊C4—61no—
191#340—31no—
192#340—31no—
193#340—31no—
194#340C2×D311yescharacteristic
195#336—31no—
196#336C2×D311yescharacteristic
197C23×D3—21no—
198C23×D3C2×D411yescharacteristic
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8C2×D412yes—
203C2×C3×Q8C2×Q812yes—
204C2×C3×Q8#4112yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218#239—28no—
219#239—44no—
220#239—44no—
221#239—44no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—24no—
227C2×D12—21no—
228C2×D12—42no—
229C2×D12—42no—
230C2×D12C2×Q812yes—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—24no—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×C4×D3C2×Q812yes—
239C2×C4×D3#4112yes—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—42no—
253C3×C4⋊C4—81no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—42no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—41no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—81no—
263(C2×D3)⋊C4—81no—
264(C2×D3)⋊C4—81no—
265(C2×D3)⋊C4—81no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—42no—
272C4⋊Dic3C2×D412yes—
273Dic3⋊C4—42no—
274Dic3⋊C4—42no—
275Dic3⋊C4—42no—
276Dic3⋊C4—42no—
277Dic3⋊C4—81no—
278C4×Dic3—21no—
279C4×Dic3—21no—
280C4×Dic3—42no—
281C4×Dic3—42no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8C2×D412yes—
288D3×C8—44no—
289C2×#41—34no—
290C2×#41—62no—
291C2×#41—62no—
292C2×#41—62no—
293C2×#41—121no—
294C2×#41—121no—
295C22×Q8C22×D311yescharacteristic
296C22×D4—31no—
297C22×D4—61no—
298C23×C4—31no—
299C23×C4—61no—
300C22×C8—31no—
301C22×C8—31no—
302C22×C8—62no—
303C22×C8—62no—
304C22×C8—121no—
305C22×C8C22×D311yescharacteristic
306C22×C8C22×D311yescharacteristic
307#125—62no—
308#125—121no—
309C4×Q8—22no—
310C4×Q8—41no—
311C4×Q8—62no—
312C4×Q8—121no—
313#117—62no—
314#117—121no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C22⋊C4—31no—
330C2×C22⋊C4—31no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C42—21no—
339C2×C42—31no—
340C2×C42—61no—
341C2×C42—61no—
342C2×C42C22×D311yescharacteristic
343#107—62no—
344#106—41no—
345#106—62no—
346#106—121no—
347C4⋊C8—62no—
348C4⋊C8—121no—
349#103—22no—
350#103—22no—
351#103—34no—
352#103—62no—
353#102—62no—
354#102—62no—
355#102—121no—
356#102—121no—
357C22⋊C8—62no—
358#95—21no—
359#95—31no—
360#95—31no—
361#95—61no—
362#95—61no—
363#95—61no—
364C23×C3C22×D411yescharacteristic
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—21no—
370C22×D3—21no—
371C22×D3—22no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3C22⋊Q812yes—
379C3×Q8—22no—
380C3×Q8—22no—
381C3×Q8—22no—
382C3×Q8C22⋊Q814yes—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C22×Q811yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4H(5,𝔽2)11yescharacteristic
406C2×C3×C4H(5,𝔽2)11yescharacteristic
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—22no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3C22⋊Q812yes—
424C2×Dic3#13512yes—
425D12—22no—
426D12—22no—
427D12—48no—
428D12—41no—
429D12—41no—
430C4×D3—22no—
431C4×D3—22no—
432C4×D3—22no—
433C4×D3—22no—
434C4×D3—48no—
435C3×C8—41no—
436C3×C8—41no—
437C3⋊C8—22no—
438C3⋊C8—41no—
439C24—31no—
440C24—61no—
441#41—68no—
442#41—124no—
443#41—124no—
444#41—124no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8#23512yes—
449C2×Q8D3×D412yes—
450C2×Q8D3×Q812yes—
451C2×D4—32no—
452C2×D4—64no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—122no—
459C2×D4—122no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—32no—
467C22×C4—32no—
468C22×C4—31no—
469C22×C4—64no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—122no—
478C22×C4—122no—
479C22×C4D3×D411yescharacteristic
480C22×C4D3×D411yescharacteristic
481C22×C4C23×D311yescharacteristic
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—32no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—124no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—32no—
502C4⋊C4—42no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—81no—
509C4⋊C4—122no—
510C4⋊C4—122no—
511C4⋊C4—122no—
512C4⋊C4—122no—
513C4⋊C4—122no—
514C4⋊C4—241no—
515C22⋊C4—62no—
516C22⋊C4—121no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—122no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C22⋊C4—241no—
523C22⋊C4—241no—
524C42—21no—
525C42—21no—
526C42—42no—
527C42—61no—
528C42—61no—
529C42—122no—
530C42—122no—
531C22×C3C2×C22⋊Q811yescharacteristic
532C22×C3D4211yescharacteristic
533C22×C3#54611yescharacteristic
534C22×C3C2×#13511yescharacteristic
535C22×C3#57511yescharacteristic
536C22×C3#57711yescharacteristic
537C22×C3#57711yescharacteristic
538C2×D3—22no—
539C2×D3—22no—
540C2×D3—22no—
541C2×D3—41no—
542C2×D3—41no—
543C2×D3—41no—
544C2×D3—41no—
545C2×D3—42no—
546C2×D3—42no—
547C2×D3—42no—
548C2×D3—81no—
549C2×D3—81no—
550C2×D3—81no—
551C2×D3—81no—
552C3×C4—24no—
553C3×C4—41no—
554C3×C4—41no—
555C3×C4—81no—
556C3×C4—81no—
557C3×C4—81no—
558C3×C4C2×C22⋊Q811yescharacteristic
559C3×C4#54311yescharacteristic
560C3×C4#54611yescharacteristic
561C3×C4#55111yescharacteristic
562Dic3—22no—
563Dic3—22no—
564Dic3—42no—
565Dic3—41no—
566Dic3—81no—
567Dic3—81no—
568C23—32no—
569C23—61no—
570C23—61no—
571C23—61no—
572C23—61no—
573C23—61no—
574C23—61no—
575C23—62no—
576C23—121no—
577C23—121no—
578C23—121no—
579C23—121no—
580C23—121no—
581C23—121no—
582C23C2×D3×D411yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—22no—
586Q8#89614yes—
587D4—62no—
588D4—62no—
589D4—128no—
590D4—121no—
591D4—121no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—32no—
602C2×C4—32no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—62no—
624C2×C4—128no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4C2×#23511yescharacteristic
634C2×C4C2×D3×D411yescharacteristic
635C2×C4#100411yescharacteristic
636C2×C4C2×D3×Q811yescharacteristic
637C2×C4D3×#4111yescharacteristic
638C2×C4#100911yescharacteristic
639C8—41no—
640C8—41no—
641C8—62no—
642C8—121no—
643C2×C3#243711yescharacteristic
644C2×C3#307611yescharacteristic
645C2×C3#308411yescharacteristic
646C2×C3#327111yescharacteristic
647C2×C3#328411yescharacteristic
648C2×C3#330111yescharacteristic
649C2×C3#331711yescharacteristic
650D3—42no—
651D3—82no—
652C22—62no—
653C22—62no—
654C22—62no—
655C22—121no—
656C22—121no—
657C22—121no—
658C22—121no—
659C22—122no—
660C22—122no—
661C22—122no—
662C22—241no—
663C22—241no—
664C22—241no—
665C22—241no—
666C22#555011yescharacteristic
667C22#559011yescharacteristic
668C22D3×C22⋊Q811yescharacteristic
669C22#576211yescharacteristic
670C22D3×#13511yescharacteristic
671C22#577711yescharacteristic
672C22#577811yescharacteristic
673C4—24no—
674C4—41no—
675C4—41no—
676C4—62no—
677C4—62no—
678C4—81no—
679C4—81no—
680C4—81no—
681C4—122no—
682C4—121no—
683C4—241no—
684C4—241no—
685C4C2×#89611yescharacteristic
686C4#555611yescharacteristic
687C4#563411yescharacteristic
688C4#565411yescharacteristic
689C3#2595311yescharacteristic, complemented, Sylow
690C2—122no—
691C2—242no—
692C2#7460711yescharacteristic
693C2#7967911yescharacteristic
694C2#7984411yescharacteristic
695C2#8006511yescharacteristic
696C2#8027011yescharacteristic
697C2#8101411yescharacteristic
698C2#8122911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1865C2
socleC23×C3C22×D4
prime radicalC22×#239C22
hypercenter#1865D3
hypocenterC3#25953
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
2. C2×#5022C2semidirect
4. #76224C2semidirect
5. C3×#1865C2semidirect
6. #70587C2semidirect
7. #70580C2semidirect
9. #70237C2semidirect
10. #70221C2semidirect
11. #69629C2semidirect
12. #68901C2semidirect
13. #68868C2semidirect
14. #68679C2semidirect
15. #68666C2semidirect
71. #1865D3semidirect
689. C3#25953semidirect

Maximal subgroups

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

subgroupindex
#824572
C2×#50222
C2×#48042
#762242
C3×#18652
#705872
#705802
#702672
#702372
#702212
#696292
#689012
#688682
#686792
#686662
#259533