Finite Group Databasebeta

← #10849293

Subgroups — #10849293

Order 768.

1012 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5568C211yesmaximal, characteristic
2C2×#4871C211yesmaximal, characteristic
3C2×#4797C211yesmaximal, characteristic, complemented
4#76079C211yesmaximal, characteristic, complemented
5C3×#1865C211yesmaximal, characteristic, complemented
6#71628C211yesmaximal, characteristic, complemented
7#70588C211yesmaximal, characteristic, complemented
8#70279C211yesmaximal, characteristic
9#70245C211yesmaximal, characteristic
10#70229C211yesmaximal, characteristic, complemented
11#69628C211yesmaximal, characteristic
12#68906C211yesmaximal, characteristic, complemented
13#68782C211yesmaximal, characteristic
14#68630C211yesmaximal, characteristic
15#68620C211yesmaximal, characteristic, complemented
16#25656—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×#944C2211yescharacteristic
20#5568—24no—
21#5568—24no—
22C2×#898—21no—
23C2×#896—21no—
24C2×#893C2211yescharacteristic
25C2×#890C2211yescharacteristic
26C2×#887C2211yescharacteristic
27C2×#876—21no—
28C2×#875C2211yescharacteristic
29C2×#872—21no—
30C2×C4×Dic6C2211yescharacteristic
31C2×C3×#106C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#815C2211yescharacteristic
42C2×Dic3⋊C8C2211yescharacteristic
43#4984C2211yescharacteristic
44#4982—21no—
45Dic3×C4⋊C4C2211yescharacteristic
46#4972—21no—
47#4972C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#809C2211yescharacteristic
51C2×#809C2211yescharacteristic
52C2×#807C2211yescharacteristic
53#4928C2211yescharacteristic
54#4871—24no—
55#4797—24no—
56#4656—21no—
57#4647—21no—
58C3×#340C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#556—31no—
65C2×#491—31no—
66C2×#478—31no—
67#2584—31no—
68#2042—31no—
69#2042—31no—
70#2042—31no—
71#2042—31no—
72#1985—31no—
73#1981—31no—
74#1865—31no—
75#1865D311yescharacteristic, complemented
76#1863—31no—
77#1863—31no—
78#1762—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—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⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42C2311yescharacteristic
95#947—22no—
96#947—22no—
97#944—22no—
98#944D414yes—
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⋊Dic3D411yescharacteristic
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⋊C4D411yescharacteristic
116C2×Dic3⋊C4D411yescharacteristic
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#898—28no—
124#896—28no—
125#893—22no—
126#893D414yes—
127#890—22no—
128#890—22no—
129#887—22no—
130#887—22no—
131#876—28no—
132#875—22no—
133#875—22no—
134#872—28no—
135C4×Dic6—22no—
136C4×Dic6—22no—
137C3×#106—22no—
138C3×#103—22no—
139C3×#103—22no—
140C3×#95—21no—
141#835—22no—
142#835—22no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147#821—22no—
148#821—22no—
149#816—22no—
150#816—22no—
151#815—22no—
152Dic3⋊C8—22no—
153#810—22no—
154#810—22no—
155#809—22no—
156#809—22no—
157#807—22no—
158#556—64no—
159#556—64no—
160C2×C4⋊Q8—31no—
161C2×#126—61no—
162C2×#126—61no—
163C2×#125—31no—
164C2×#124—31no—
165C2×#124—31no—
166C2×C22⋊Q8—61no—
167C2×C4×Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8C2×D311yescharacteristic
170C2×C4×D4—31no—
171C2×#117—61no—
172#491—64no—
173#478—64no—
174C2×#107—31no—
175C2×#106C2×D311yescharacteristic
176C2×C4⋊C8—31no—
177C2×#103—31no—
178C2×#103—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#103—31no—
183C2×#103C2×D311yescharacteristic
184C2×#103C2×D311yescharacteristic
185C2×#102—31no—
186C2×#102—31no—
187C2×#102—31no—
188C2×#102—31no—
189C2×#97—31no—
190#398—61no—
191#395—61no—
192#389—31no—
193#389—31no—
194#389—31no—
195#389—61no—
196#389—61no—
197#389C2×D311yescharacteristic
198C4×C4⋊C4—31no—
199#340—31no—
200#340C2×D311yescharacteristic
201#336—31no—
202#336—31no—
203#336—31no—
204#336C2×D311yescharacteristic
205C23×D3—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8C2×D412yes—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12C2×D412yes—
232C2×C4×D3—24no—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—22no—
241C2×Dic6—41no—
242C2×Dic6—41no—
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—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—81no—
256C3×C4⋊C4C2×D412yes—
257C3×C4⋊C4#4112yes—
258C3×C42—24no—
259C3×C42—21no—
260C3×C42—21no—
261(C2×D3)⋊C4—24no—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—24no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—24no—
266C4⋊Dic3—21no—
267C4⋊Dic3—22no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—42no—
271C4⋊Dic3—42no—
272C4⋊Dic3#4112yes—
273Dic3⋊C4—24no—
274Dic3⋊C4—22no—
275Dic3⋊C4—22no—
276Dic3⋊C4—24no—
277Dic3⋊C4—24no—
278Dic3⋊C4—42no—
279Dic3⋊C4—42no—
280Dic3⋊C4—81no—
281C4×Dic3—24no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C4×Dic3—81no—
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×Q8C22×D311yescharacteristic
296C22×D4—31no—
297C23×C4—61no—
298C22×C8—31no—
299C22×C8—31no—
300C22×C8C22×D311yescharacteristic
301C22×C8C22×D311yescharacteristic
302C4⋊Q8—34no—
303C4⋊Q8—62no—
304#126—68no—
305#126—68no—
306#125—62no—
307#125—62no—
308#124—62no—
309#124—62no—
310#124—62no—
311#124—62no—
312C22⋊Q8—68no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—62no—
316C4×Q8—62no—
317C4×Q8—62no—
318C4×Q8—62no—
319C4×D4—34no—
320C4×D4—62no—
321#117—68no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
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⋊C4C22×D311yescharacteristic
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C4⋊C4C22×D311yescharacteristic
340C2×C22⋊C4—61no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4—61no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C42—21no—
346C2×C42—31no—
347C2×C42—31no—
348C2×C42—61no—
349C2×C42—61no—
350C2×C42C22×D311yescharacteristic
351#107—62no—
352#106—22no—
353C4⋊C8—62no—
354#103—22no—
355#103—22no—
356#103—62no—
357#103—62no—
358#103—62no—
359#103—62no—
360#103—62no—
361#103—62no—
362#102—62no—
363#102—62no—
364#102—62no—
365#102—62no—
366#97—62no—
367#95—21no—
368#95—61no—
369#95—61no—
370#95—61no—
371#95—61no—
372C23×C3C22×D411yescharacteristic
373C22×D3—22no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C3×Q8—22no—
381C3×Q8—22no—
382C3×Q8—41no—
383C3×Q8—41no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—22no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
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×C4—41no—
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C42−1+411yescharacteristic
407C2×C3×C42−1+411yescharacteristic
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—22no—
415C2×Dic3—22no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3#12012yes—
428C2×Dic3#12012yes—
429D12—22no—
430D12—22no—
431D12—41no—
432D12—41no—
433C4×D3—48no—
434Dic6—22no—
435Dic6—22no—
436Dic6—44no—
437Dic6—41no—
438Dic6—41no—
439C3×C8—41no—
440C3×C8—41no—
441C3⋊C8—41no—
442C3⋊C8—41no—
443C24—61no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—32no—
450C2×Q8—62no—
451C2×Q8—61no—
452C2×Q8—61no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—121no—
457C2×Q8—121no—
458C2×Q8D3×D412yes—
459C2×D4—32no—
460C2×D4—61no—
461C2×D4—61no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—31no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—64no—
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—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—32no—
500C4⋊C4—62no—
501C4⋊C4—62no—
502C4⋊C4—62no—
503C4⋊C4—64no—
504C4⋊C4—64no—
505C4⋊C4—64no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—81no—
510C4⋊C4—122no—
511C4⋊C4—122no—
512C4⋊C4—122no—
513C4⋊C4—122no—
514C4⋊C4—241no—
515C4⋊C4#23512yes—
516C4⋊C4D3×D412yes—
517C22⋊C4—64no—
518C22⋊C4—64no—
519C22⋊C4—64no—
520C22⋊C4—64no—
521C22⋊C4—64no—
522C42—24no—
523C42—21no—
524C42—21no—
525C42—32no—
526C42—32no—
527C42—64no—
528C42—61no—
529C42—61no—
530C42—241no—
531C22×C3C2×#12011yescharacteristic
532C22×C3#54811yescharacteristic
533C22×C3#54811yescharacteristic
534C22×C3#57511yescharacteristic
535C22×C3#57511yescharacteristic
536C22×C3#57711yescharacteristic
537C22×C3#57811yescharacteristic
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—42no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C2×D3—81no—
545C3×C4—42no—
546C3×C4—42no—
547C3×C4—41no—
548C3×C4—42no—
549C3×C4—41no—
550C3×C4—81no—
551C3×C4#53211yescharacteristic
552C3×C4#53611yescharacteristic
553C3×C4#54711yescharacteristic
554C3×C4#54711yescharacteristic
555Dic3—22no—
556Dic3—22no—
557Dic3—42no—
558Dic3—42no—
559Dic3—41no—
560Dic3—41no—
561Dic3—81no—
562C23—62no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23C2×D3×D411yescharacteristic
570Q8—22no—
571Q8—22no—
572Q8—41no—
573Q8—41no—
574Q8—62no—
575Q8—62no—
576Q8—124no—
577Q8—121no—
578Q8—121no—
579D4—62no—
580D4—62no—
581D4—121no—
582D4—121no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—32no—
593C2×C4—32no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—62no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—62no—
615C2×C4—128no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4C2×#23511yescharacteristic
624C2×C4C2×D3×D411yescharacteristic
625C2×C4C2×D3×D411yescharacteristic
626C2×C4#100711yescharacteristic
627C2×C4D3×#4111yescharacteristic
628C2×C4#101011yescharacteristic
629C8—41no—
630C8—41no—
631C8—121no—
632C8—121no—
633C2×C3#243211yescharacteristic
634C2×C3#300511yescharacteristic
635C2×C3#301311yescharacteristic
636C2×C3#337611yescharacteristic
637C2×C3#337611yescharacteristic
638C2×C3#337711yescharacteristic
639C2×C3#337711yescharacteristic
640D3—82no—
641C22—122no—
642C22—122no—
643C22—122no—
644C22—241no—
645C22—241no—
646C22—241no—
647C22—241no—
648C22#555211yescharacteristic
649C22D3×#12011yescharacteristic
650C22#563311yescharacteristic
651C22#576211yescharacteristic
652C22#576811yescharacteristic
653C22#577711yescharacteristic
654C22#577911yescharacteristic
655C4—42no—
656C4—41no—
657C4—42no—
658C4—41no—
659C4—42no—
660C4—62no—
661C4—62no—
662C4—81no—
663C4—121no—
664C4—122no—
665C4—121no—
666C4—122no—
667C4—241no—
668C4#550711yescharacteristic
669C4#554811yescharacteristic
670C4#562811yescharacteristic
671C4#565511yescharacteristic
672C3#2565611yescharacteristic, complemented, Sylow
673C2—242no—
674C2#7441111yescharacteristic
675C2#7963011yescharacteristic
676C2#7981811yescharacteristic
677C2#8004211yescharacteristic
678C2#8022711yescharacteristic
679C2#8100511yescharacteristic
680C2#8126711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1865C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1865D3
hypocenterC3#25656
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
3. C2×#4797C2semidirect
4. #76079C2semidirect
5. C3×#1865C2semidirect
6. #71628C2semidirect
7. #70588C2semidirect
10. #70229C2semidirect
12. #68906C2semidirect
15. #68620C2semidirect
75. #1865D3semidirect
672. C3#25656semidirect

Maximal subgroups

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

subgroupindex
C2×#55682
C2×#48712
C2×#47972
#760792
C3×#18652
#716282
#705882
#702792
#702452
#702292
#696282
#689062
#687822
#686302
#686202
#256563