Finite Group Databasebeta

← #10845357

Subgroups — #10845357

Order 768.

850 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82499C211yesmaximal, characteristic
2C2×#5022C211yesmaximal, characteristic, complemented
3C2×C8×D12C211yesmaximal, characteristic
4#73806C211yesmaximal, characteristic
5C3×C4×#103C211yesmaximal, characteristic, complemented
6#70275C211yesmaximal, characteristic, complemented
7#70274C211yesmaximal, characteristic, complemented
8#70270C211yesmaximal, characteristic, complemented
9#70270C211yesmaximal, characteristic, complemented
10#69778C211yesmaximal, characteristic, complemented
11#69468C211yesmaximal, characteristic, complemented
12#68891C211yesmaximal, characteristic, complemented
13#68889C211yesmaximal, characteristic, complemented
14#68869C211yesmaximal, characteristic, complemented
15#68868C211yesmaximal, characteristic, complemented
16#25343—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C2×#946C2211yescharacteristic
20C22×D3×C8C2211yescharacteristic
21C22×D3×C8C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×#870C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32#5022—24no—
33#4991—21no—
34#4987C2211yescharacteristic
35#4987C2211yescharacteristic
36#4987C2211yescharacteristic
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39#4987C2211yescharacteristic
40#4981—21no—
41C2×#808C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4940—21no—
46#4940—21no—
47#4940—21no—
48#4940—21no—
49#4936C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C8×D12—24no—
52#4672—21no—
53#4659—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#479—31no—
63C2×C8×D4—31no—
64#2562—31no—
65#2536—31no—
66#2077—31no—
67#2075—31no—
68#2036—31no—
69#2035—31no—
70#1991—31no—
71#1949—31no—
72#1892—31no—
73#1892—31no—
74#1808—31no—
75#1807—31no—
76C4×#103D311yescharacteristic, complemented
77C22×C3×Q8C2311yescharacteristic
78C22×D12D411yescharacteristic
79C22×C4×D3C2311yescharacteristic
80C22×C4×D3C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92#946—22no—
93#946—22no—
94#946—41no—
95#946—41no—
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×(C2×D3)⋊C4D411yescharacteristic
105C2×(C2×D3)⋊C4D411yescharacteristic
106C2×C4⋊Dic3D411yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C2×D3×C8—22no—
118C2×D3×C8—22no—
119C2×D3×C8—22no—
120C2×D3×C8—22no—
121C2×D3×C8—41no—
122C2×D3×C8—41no—
123C4×D12—22no—
124C4×D12—22no—
125#870—22no—
126#870—22no—
127C3×#103—22no—
128C3×#103—22no—
129C3×C4×C8—22no—
130C3×#95—21no—
131#835—22no—
132#835—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141(C2×D3)⋊C8—22no—
142(C2×D3)⋊C8—22no—
143#808—22no—
144#808—41no—
145#807—22no—
146#807—41no—
147C4⋊(C3⋊C8)D414yes—
148C23×C8—31no—
149C23×C8—31no—
150C2×#125—31no—
151C2×C22⋊Q8—31no—
152C2×C22⋊Q8—31no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4—31no—
155#479—64no—
156C8×D4—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#103C2×D311yescharacteristic
166C2×C22⋊C8—31no—
167C2×C22⋊C8—31no—
168C2×C4×C8C2×D311yescharacteristic
169#400—61no—
170#398—61no—
171#394—61no—
172#394—61no—
173#388—61no—
174#388—61no—
175#388—61no—
176#388—61no—
177#382—31no—
178#380—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380—31no—
184C4×C4⋊C4C2×D311yescharacteristic
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193C23×D3C2×D411yescharacteristic
194C23×D3C2×D411yescharacteristic
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8#4112yes—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×D12—22no—
215C2×D12—22no—
216C2×D12—22no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8#4112yes—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—42no—
242C3×C42—42no—
243C3×C42—81no—
244C3×C42C2×D412yes—
245C3×C42#4112yes—
246(C2×D3)⋊C4—22no—
247(C2×D3)⋊C4—22no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—81no—
255(C2×D3)⋊C4—81no—
256C4⋊Dic3—22no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265Dic3⋊C4—42no—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8C2×D412yes—
274C2×C3⋊C8C2×D412yes—
275D3×C8—44no—
276D3×C8—44no—
277C22×Q8C22×D311yescharacteristic
278C22×D4—31no—
279C23×C4—31no—
280C23×C4—31no—
281C22×C8—31no—
282C22×C8—31no—
283C22×C8—62no—
284C22×C8—62no—
285C22×C8—62no—
286C22×C8—62no—
287C22×C8—121no—
288C22×C8—121no—
289C22×C8C22×D311yescharacteristic
290C22×C8C22×D311yescharacteristic
291#125—62no—
292#125—62no—
293C22⋊Q8—62no—
294C22⋊Q8—62no—
295C22⋊Q8—121no—
296C22⋊Q8—121no—
297C4×Q8—22no—
298C4×Q8—22no—
299C4×D4—62no—
300C4×D4—62no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—31no—
315C2×C22⋊C4—31no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
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×C42—21no—
325C2×C42—21no—
326C2×C42C22×D311yescharacteristic
327#107—62no—
328#107—121no—
329#106—62no—
330#106—121no—
331C4⋊C8—34no—
332#103—22no—
333#103—22no—
334#103—62no—
335#103—62no—
336C22⋊C8—62no—
337C22⋊C8—62no—
338C4×C8—22no—
339#95—21no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3C22×D411yescharacteristic
349C22×D3—21no—
350C22×D3—21no—
351C22×D3—21no—
352C22×D3—21no—
353C22×D3—21no—
354C22×D3—21no—
355C22×D3—21no—
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3#13512yes—
362C22×D3#13512yes—
363C3×Q8—22no—
364C3×Q8—22no—
365C3×Q8—41no—
366C3×Q8—41no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—22no—
372C2×C3×C4—41no—
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#12112yes—
384C2×C3×C4#12112yes—
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4H(5,𝔽2)11yescharacteristic
390C2×C3×C42−1+411yescharacteristic
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409D12—28no—
410C4×D3—41no—
411C4×D3—41no—
412C4×D3—41no—
413C4×D3—41no—
414C4×D3—41no—
415C4×D3—41no—
416C4×D3—41no—
417C4×D3—41no—
418C3×C8—22no—
419C3×C8—41no—
420C3⋊C8—22no—
421C3⋊C8—22no—
422C24—31no—
423C24—31no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#23512yes—
430C2×D4—62no—
431C2×D4—62no—
432C2×D4—62no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4D3×D411yescharacteristic
456C22×C4D3×D411yescharacteristic
457C22×C4C23×D311yescharacteristic
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—32no—
464C2×C8—32no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—124no—
470C2×C8—124no—
471C2×C8#23512yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—62no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—122no—
489C4⋊C4—122no—
490C4⋊C4—122no—
491C4⋊C4—122no—
492C22⋊C4—62no—
493C22⋊C4—62no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—122no—
498C22⋊C4—122no—
499C22⋊C4—122no—
500C22⋊C4—241no—
501C22⋊C4—241no—
502C42—42no—
503C42—81no—
504C42#23512yes—
505C42D3×D412yes—
506C22×C3C2×#12111yescharacteristic
507C22×C3#54611yescharacteristic
508C22×C3#54711yescharacteristic
509C22×C3C2×#13511yescharacteristic
510C22×C3C2×#13511yescharacteristic
511C22×C3#57711yescharacteristic
512C22×C3#57811yescharacteristic
513C2×D3—22no—
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—22no—
517C2×D3—22no—
518C2×D3—22no—
519C2×D3—41no—
520C2×D3—41no—
521C2×D3—41no—
522C2×D3—41no—
523C2×D3—41no—
524C2×D3—41no—
525C2×D3—41no—
526C2×D3—41no—
527C3×C4—22no—
528C3×C4—22no—
529C3×C4—42no—
530C3×C4—81no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4#52911yescharacteristic
534C3×C4#54211yescharacteristic
535C3×C4#54611yescharacteristic
536C3×C4#54811yescharacteristic
537Dic3—41no—
538Dic3—41no—
539Dic3—81no—
540Dic3—81no—
541Dic3—81no—
542Dic3—81no—
543C23—32no—
544C23—32no—
545C23—61no—
546C23—61no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23—61no—
556C23—61no—
557C23C2×D3×D411yescharacteristic
558Q8—22no—
559Q8—22no—
560Q8—41no—
561Q8—41no—
562D4—68no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4#88412yes—
606C2×C4#88412yes—
607C2×C4C2×#23511yescharacteristic
608C2×C4C2×#23511yescharacteristic
609C2×C4C2×#23511yescharacteristic
610C2×C4C2×D3×D411yescharacteristic
611C2×C4#100911yescharacteristic
612C2×C4#101011yescharacteristic
613C8—22no—
614C8—41no—
615C8—62no—
616C8—62no—
617C2×C3#242711yescharacteristic
618C2×C3#305011yescharacteristic
619C2×C3#306311yescharacteristic
620C2×C3#330011yescharacteristic
621C2×C3#330111yescharacteristic
622C2×C3#333611yescharacteristic
623C2×C3#333711yescharacteristic
624D3—42no—
625D3—42no—
626C22—62no—
627C22—62no—
628C22—62no—
629C22—62no—
630C22—62no—
631C22—62no—
632C22—121no—
633C22—121no—
634C22—121no—
635C22—121no—
636C22—121no—
637C22—121no—
638C22—121no—
639C22—121no—
640C22C2×#88411yescharacteristic
641C22#555011yescharacteristic
642C22#555111yescharacteristic
643C22#576211yescharacteristic
644C22#576611yescharacteristic
645C22D3×#13511yescharacteristic
646C22D3×#13511yescharacteristic
647C4—22no—
648C4—22no—
649C4—42no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—121no—
654C4—121no—
655C4—241no—
656C4—241no—
657C4—241no—
658C4—241no—
659C4#547911yescharacteristic
660C4#553411yescharacteristic
661C4#555011yescharacteristic
662C4#555211yescharacteristic
663C3#2534311yescharacteristic, complemented, Sylow
664C2—122no—
665C2—122no—
666C2#7348611yescharacteristic
667C2#7880011yescharacteristic
668C2#7883811yescharacteristic
669C2#7967911yescharacteristic
670C2#7968911yescharacteristic
671C2#7973611yescharacteristic
672C2#7974111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#103C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenterC4×#103D3
hypocenterC3#25343
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
2. C2×#5022C2semidirect
5. C3×C4×#103C2semidirect
6. #70275C2semidirect
7. #70274C2semidirect
8. #70270C2semidirect
9. #70270C2semidirect
10. #69778C2semidirect
11. #69468C2semidirect
12. #68891C2semidirect
13. #68889C2semidirect
14. #68869C2semidirect
15. #68868C2semidirect
76. C4×#103D3semidirect
663. C3#25343semidirect

Maximal subgroups

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

subgroupindex
#824992
C2×#50222
C2×C8×D122
#738062
C3×C4×#1032
#702752
#702742
#702702
#697782
#694682
#688912
#688892
#688692
#688682
#253433