Finite Group Databasebeta

← #10849230

Subgroups — #10849230

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5721C211yesmaximal, characteristic
2C2×#4792C211yesmaximal, characteristic
3C2×#4793C211yesmaximal, characteristic
4#76071C211yesmaximal, characteristic
5C3×#2043C211yesmaximal, characteristic
6#71781C211yesmaximal, characteristic
7#70284C211yesmaximal, characteristic
8#70246C211yesmaximal, characteristic
9#70236C211yesmaximal, characteristic
10#70235C211yesmaximal, characteristic
11#70229C211yesmaximal, characteristic
12Dic3×#103C211yesmaximal, characteristic
13#69631C211yesmaximal, characteristic
14#68683C211yesmaximal, characteristic
15#68619C211yesmaximal, characteristic
16#25651—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×#944C2211yescharacteristic
21C2×#944C2211yescharacteristic
22#5721—24no—
23#5721—24no—
24C2×#890—21no—
25C2×#889—21no—
26C2×#889—21no—
27C2×#888C2211yescharacteristic
28C2×#887—21no—
29C2×C4×Dic6C2211yescharacteristic
30C2×C4×Dic6C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#835C2211yescharacteristic
40C2×#818C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×Dic3⋊C8C2211yescharacteristic
44C2×C8×Dic3C2211yescharacteristic
45#4984C2211yescharacteristic
46#4979—21no—
47#4975—21no—
48Dic3×C4⋊C4C2211yescharacteristic
49#4972C2211yescharacteristic
50C2×#810C2211yescharacteristic
51C2×#810C2211yescharacteristic
52C2×#807C2211yescharacteristic
53#4928C2211yescharacteristic
54#4793—24no—
55#4792—24no—
56#4656—21no—
57#4646—21no—
58C3×#336C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#557—31no—
65C2×#487—31no—
66C2×#477—31no—
67#2567—31no—
68#2043—31no—
69#2043—31no—
70#2043—31no—
71#2043D311yescharacteristic
72#1984—31no—
73#1981—31no—
74#1863—31no—
75#1863—31no—
76#1863—31no—
77#1863—31no—
78C4×#103—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×Q8C2311yescharacteristic
81C22×Dic6—21no—
82C22×Dic6D411yescharacteristic
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4⋊Q8—22no—
87C3×C4⋊Q8—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94Q8×Dic3—22no—
95Q8×Dic3—22no—
96Q8×Dic3—22no—
97Q8×Dic3D414yes—
98#944—22no—
99#944—22no—
100#944—22no—
101#944D414yes—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3D411yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4D411yescharacteristic
118C2×Dic3⋊C4D411yescharacteristic
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3C2311yescharacteristic
124C2×C4×Dic3C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127#890—28no—
128#889—28no—
129#889—28no—
130#888—22no—
131#888—22no—
132#887—28no—
133C4×Dic6—22no—
134C4×Dic6—22no—
135C4×Dic6—22no—
136C4×Dic6—22no—
137C3×#103—22no—
138C3×#103—22no—
139C3×#103—22no—
140C3×#103—22no—
141C3×#95—21no—
142#835—22no—
143#835—22no—
144#835—22no—
145#835—22no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150#818—22no—
151#816—22no—
152#816—22no—
153Dic3⋊C8—22no—
154C8×Dic3—22no—
155#810—22no—
156#810—22no—
157#807—22no—
158#557—64no—
159#557—64no—
160C2×C4⋊Q8—31no—
161C2×C4⋊Q8—31no—
162C2×C4⋊Q8—31no—
163C2×C4⋊Q8C2×D311yescharacteristic
164C2×#125—61no—
165C2×#125—61no—
166C2×#125—61no—
167C2×C4×Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8—31no—
170C2×C4×Q8—31no—
171C2×C4×Q8—61no—
172#487—64no—
173#477—64no—
174C2×#107—31no—
175C2×#107—31no—
176C2×C4⋊C8—31no—
177C2×#103—31no—
178C2×#103—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#103—31no—
183C2×#103—31no—
184C2×#103—31no—
185C2×#103C2×D311yescharacteristic
186C2×#103C2×D311yescharacteristic
187C2×#103C2×D311yescharacteristic
188C2×#103C2×D311yescharacteristic
189C2×C4×C8—31no—
190#395—61no—
191#395—61no—
192#391—61no—
193#389—31no—
194#389—31no—
195#389—31no—
196#389C2×D311yescharacteristic
197#384—61no—
198C4×C4⋊C4—31no—
199#340—31no—
200#340—31no—
201#336—31no—
202#336—31no—
203#336—31no—
204#336C2×D311yescharacteristic
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8C2×D412yes—
216C2×C3×Q8C2×D412yes—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×Dic6—22no—
233C2×Dic6—22no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—42no—
240C2×Dic6—81no—
241C2×Dic6—81no—
242C2×Dic6Q1614yes—
243C2×Dic6C2×D412yes—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—81no—
256C3×C4⋊C4#4112yes—
257C3×C42—21no—
258C3×C42—21no—
259C4⋊Dic3—24no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—22no—
263C4⋊Dic3—24no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—81no—
267C4⋊Dic3C2×D412yes—
268C4⋊Dic3#4112yes—
269Dic3⋊C4—24no—
270Dic3⋊C4—24no—
271Dic3⋊C4—24no—
272Dic3⋊C4—22no—
273Dic3⋊C4—22no—
274Dic3⋊C4—24no—
275Dic3⋊C4—24no—
276Dic3⋊C4—42no—
277Dic3⋊C4—81no—
278Dic3⋊C4—81no—
279C4×Dic3—24no—
280C4×Dic3—24no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—24no—
284C4×Dic3—42no—
285C4×Dic3C2×D412yes—
286C4×Dic3C2×D412yes—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C22×Q8—31no—
294C22×Q8—31no—
295C22×Q8—61no—
296C22×Q8C22×D311yescharacteristic
297C22×Q8C22×D311yescharacteristic
298C22×C8—31no—
299C22×C8—31no—
300C22×C8C22×D311yescharacteristic
301C22×C8C22×D311yescharacteristic
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—34no—
305C4⋊Q8—62no—
306C4⋊Q8—62no—
307C4⋊Q8—62no—
308C4⋊Q8—62no—
309C4⋊Q8—62no—
310#125—68no—
311#125—68no—
312#125—68no—
313C4×Q8—34no—
314C4×Q8—62no—
315C4×Q8—62no—
316C4×Q8—62no—
317C4×Q8—62no—
318C4×Q8—62no—
319C4×Q8—68no—
320C4×Q8—62no—
321C4×Q8—62no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4C22×D311yescharacteristic
343C2×C4⋊C4C22×D311yescharacteristic
344C2×C42—31no—
345C2×C42—31no—
346C2×C42—61no—
347C2×C42—61no—
348C2×C42—61no—
349C2×C42—61no—
350C2×C42C22×D311yescharacteristic
351#107—62no—
352#107—62no—
353C4⋊C8—62no—
354#103—22no—
355#103—22no—
356#103—22no—
357#103—22no—
358#103—62no—
359#103—62no—
360#103—62no—
361#103—62no—
362#103—62no—
363#103—62no—
364#103—62no—
365#103—62no—
366C4×C8—62no—
367#95—21no—
368#95—61no—
369#95—61no—
370#95—61no—
371#95—61no—
372C23×C3C22×D411yescharacteristic
373C3×Q8—22no—
374C3×Q8—22no—
375C3×Q8—22no—
376C3×Q8—22no—
377C3×Q8—41no—
378C3×Q8—41no—
379C3×Q8—41no—
380C3×Q8—41no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
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×C4C2×Q1612yes—
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C42−1+411yescharacteristic
404C2×C3×C42−1+411yescharacteristic
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—21no—
412C2×Dic3—22no—
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×Dic3—41no—
424C2×Dic3#12012yes—
425C2×Dic3#12012yes—
426C2×Dic3C2×Q1612yes—
427C2×Dic3C2×Q1612yes—
428C2×Dic3#13712yes—
429Dic6—28no—
430Dic6—22no—
431Dic6—22no—
432Dic6—41no—
433Dic6—41no—
434Dic6—84no—
435C3×C8—41no—
436C3×C8—41no—
437C3⋊C8—41no—
438C3⋊C8—41no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—34no—
450C2×Q8—32no—
451C2×Q8—62no—
452C2×Q8—62no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—122no—
459C2×Q8—241no—
460C2×Q8—241no—
461C2×Q8D3×D412yes—
462C2×Q8D3×D412yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—31no—
468C22×C4—31no—
469C22×C4—31no—
470C22×C4—31no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4D3×D411yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4C23×D311yescharacteristic
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—32no—
496C4⋊C4—32no—
497C4⋊C4—62no—
498C4⋊C4—62no—
499C4⋊C4—62no—
500C4⋊C4—64no—
501C4⋊C4—64no—
502C4⋊C4—64no—
503C4⋊C4—64no—
504C4⋊C4—64no—
505C4⋊C4—64no—
506C4⋊C4—64no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—81no—
512C4⋊C4—122no—
513C4⋊C4—241no—
514C4⋊C4—241no—
515C4⋊C4—241no—
516C4⋊C4#23712yes—
517C42—21no—
518C42—21no—
519C42—32no—
520C42—32no—
521C42—64no—
522C42—64no—
523C42—64no—
524C42—61no—
525C42—61no—
526C42—122no—
527C22×C3C2×#12011yescharacteristic
528C22×C3#54811yescharacteristic
529C22×C3#54811yescharacteristic
530C22×C3C22×Q1611yescharacteristic
531C22×C3C2×#13711yescharacteristic
532C22×C3#57711yescharacteristic
533C22×C3#57711yescharacteristic
534C3×C4—22no—
535C3×C4—42no—
536C3×C4—42no—
537C3×C4—42no—
538C3×C4—41no—
539C3×C4—81no—
540C3×C4#53211yescharacteristic
541C3×C4#53611yescharacteristic
542C3×C4#54711yescharacteristic
543C3×C4#54711yescharacteristic
544Dic3—22no—
545Dic3—22no—
546Dic3—42no—
547Dic3—42no—
548Dic3—41no—
549Dic3—42no—
550Dic3—81no—
551Dic3#45114yes—
552C23C2×D3×D411yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—41no—
558Q8—41no—
559Q8—41no—
560Q8—41no—
561Q8—68no—
562Q8—62no—
563Q8—62no—
564Q8—121no—
565Q8—121no—
566Q8—244no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—32no—
578C2×C4—32no—
579C2×C4—32no—
580C2×C4—41no—
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—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—62no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—62no—
597C2×C4—61no—
598C2×C4—62no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4D3×Q1612yes—
609C2×C4C2×D3×D411yescharacteristic
610C2×C4C2×D3×D411yescharacteristic
611C2×C4C2×#23711yescharacteristic
612C2×C4D3×#4111yescharacteristic
613C2×C4#101011yescharacteristic
614C2×C4#101011yescharacteristic
615C8—41no—
616C8—41no—
617C8—121no—
618C8—121no—
619C2×C3#243211yescharacteristic
620C2×C3C2×#45111yescharacteristic
621C2×C3#301011yescharacteristic
622C2×C3#336311yescharacteristic
623C2×C3#336311yescharacteristic
624C2×C3#337211yescharacteristic
625C2×C3#337211yescharacteristic
626C22D3×#12011yescharacteristic
627C22#559811yescharacteristic
628C22#567711yescharacteristic
629C22#576211yescharacteristic
630C22C2×D3×Q1611yescharacteristic
631C22D3×#13711yescharacteristic
632C22#577811yescharacteristic
633C4—22no—
634C4—34no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—41no—
639C4—62no—
640C4—62no—
641C4—81no—
642C4—122no—
643C4—122no—
644C4—121no—
645C4—122no—
646C4—241no—
647C4#559611yescharacteristic
648C4#561111yescharacteristic
649C4#564811yescharacteristic
650C4#565511yescharacteristic
651C3#2565111yescharacteristic, complemented, Sylow
652C2#7690411yescharacteristic
653C2#8004111yescharacteristic
654C2D3×#45111yescharacteristic
655C2#8041811yescharacteristic
656C2#8046211yescharacteristic
657C2#8162411yescharacteristic
658C2#8173211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2043C2
socle, prime radicalC23×C3C22×D4
hypercenter#2043D3
hypocenterC3#25651
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2043

Complements

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

NH ≅ G/Ntype
651. C3#25651semidirect

Maximal subgroups

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

subgroupindex
C2×#57212
C2×#47922
C2×#47932
#760712
C3×#20432
#717812
#702842
#702462
#702362
#702352
#702292
Dic3×#1032
#696312
#686832
#686192
#256513