Finite Group Databasebeta

← #10900977

Subgroups — #10900977

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5517C211yesmaximal, characteristic
2C2×C3×#485C211yesmaximal, characteristic
3#82810C211yesmaximal, characteristic
4C2×#5045C211yesmaximal, characteristic
5#71575C211yesmaximal, characteristic
6#70618C211yesmaximal, characteristic, complemented
7#70617C211yesmaximal, characteristic, complemented
8#70566C211yesmaximal, characteristic, complemented
9#70565C211yesmaximal, characteristic, complemented
10#70232C211yesmaximal, characteristic, complemented
11#70232C211yesmaximal, characteristic, complemented
12#70231C211yesmaximal, characteristic, complemented
13#70231C211yesmaximal, characteristic, complemented
14#69926C211yesmaximal, characteristic
15#69910C211yesmaximal, characteristic
16#29645—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×C4×Dic3C2211yescharacteristic
21#5517—24no—
22#5517—24no—
23C2×#890C2211yescharacteristic
24C2×#890C2211yescharacteristic
25C2×#888C2211yescharacteristic
26C2×#888C2211yescharacteristic
27C2×#878—21no—
28C2×#878—21no—
29C2×C4×Dic6—21no—
30C2×C4×Dic6—21no—
31C3×#485—24no—
32C2×C3×#107C2211yescharacteristic
33C2×C3×#106C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C2×C3×#103C2211yescharacteristic
36C2×C3×C22⋊C8C2211yescharacteristic
37#5229C2211yescharacteristic
38#5229C2211yescharacteristic
39C2×#835C2211yescharacteristic
40C2×#835C2211yescharacteristic
41#5212C2211yescharacteristic
42C2×#830C2211yescharacteristic
43#5045—24no—
44#4978—21no—
45#4978—21no—
46#4972C2211yescharacteristic
47#4972C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4946—21no—
51#4943—21no—
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4529C2211yescharacteristic
64C2×#533—31no—
65C2×#485—31no—
66C2×#485D311yescharacteristic
67#2855—31no—
68#2059—31no—
69#2059—31no—
70#2058—31no—
71#2057—31no—
72#2026—31no—
73#2026—31no—
74#2024—31no—
75#2024—31no—
76#1825—31no—
77#1825—31no—
78#1817—31no—
79C22×C3×Q8C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C23×Dic3D411yescharacteristic
82C23×Dic3D411yescharacteristic
83C22×Dic6—21no—
84C22×Dic6—21no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×#117—22no—
90C3×#117D414yes—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×#217—21no—
99C2×#217—21no—
100#926—22no—
101#926D414yes—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—22no—
116C2×C4×Dic3—22no—
117C2×C4×Dic3D411yescharacteristic
118C2×C4×Dic3D411yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#890—22no—
124#890—22no—
125#890—22no—
126#890—22no—
127#888—22no—
128#888—22no—
129#888—22no—
130#888—22no—
131#878—28no—
132#878—28no—
133C4×Dic6—28no—
134C4×Dic6—28no—
135C3×#107—22no—
136C3×#106—22no—
137C3×#103—22no—
138C3×#103—22no—
139C3×C22⋊C8—22no—
140#835—22no—
141#835—22no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149#830—22no—
150#808—22no—
151#807—22no—
152#533—64no—
153#533—64no—
154C2×C4⋊Q8—31no—
155C2×C4⋊Q8—31no—
156C2×#125—31no—
157C2×#125—31no—
158C2×C22⋊Q8—61no—
159C2×C22⋊Q8—61no—
160C2×C22⋊Q8C2×D311yescharacteristic
161C2×C4×Q8—61no—
162C2×C4×Q8—61no—
163C2×#117—31no—
164C2×#117C2×D311yescharacteristic
165C22×C42—31no—
166#485—24no—
167#485—64no—
168C2×#107—31no—
169C2×#107C2×D311yescharacteristic
170C2×#106—31no—
171C2×#106C2×D311yescharacteristic
172C2×#103—31no—
173C2×#103—31no—
174C2×#103C2×D311yescharacteristic
175C2×#103C2×D311yescharacteristic
176C2×C22⋊C8—31no—
177C2×C22⋊C8C2×D311yescharacteristic
178#391—31no—
179#391—31no—
180#389—31no—
181#389—31no—
182#387—61no—
183#387—61no—
184#385—61no—
185#381—61no—
186#380—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194#340—31no—
195#336—31no—
196#336—31no—
197#336—31no—
198#336—31no—
199C24×C3C2×D411yescharacteristic
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D412yes—
217C22×C3×C4C2411yescharacteristic
218C22×Dic3—22no—
219C22×Dic3—22no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—22no—
223C22×Dic3—22no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—41no—
227C22×Dic3—41no—
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×Dic6—24no—
233C2×Dic6—24no—
234C2×Dic6—42no—
235C2×Dic6—42no—
236C2×Dic6—42no—
237C2×Dic6—42no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4C2×D412yes—
250C3×C4⋊C4C2×D412yes—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—42no—
253C3×C42—24no—
254#217—24no—
255#217—42no—
256C4⋊Dic3—24no—
257C4⋊Dic3—24no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3C2×D412yes—
265C4⋊Dic3C2×D412yes—
266Dic3⋊C4—24no—
267Dic3⋊C4—24no—
268Dic3⋊C4—24no—
269Dic3⋊C4—24no—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272C4×Dic3—22no—
273C4×Dic3—22no—
274C4×Dic3—24no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C4×Dic3—21no—
278C4×Dic3—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C22×Q8—61no—
286C22×Q8—61no—
287C22×Q8C22×D311yescharacteristic
288C23×C4—31no—
289C23×C4—31no—
290C23×C4C22×D311yescharacteristic
291C22×C8—31no—
292C22×C8—31no—
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295C4⋊Q8—62no—
296C4⋊Q8—62no—
297C4⋊Q8—62no—
298C4⋊Q8—62no—
299#125—62no—
300#125—62no—
301#125—62no—
302#125—62no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305C22⋊Q8—68no—
306C22⋊Q8—68no—
307C4×Q8—68no—
308C4×Q8—68no—
309#117—22no—
310#117—34no—
311#117—62no—
312#117#6714yes—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
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—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C42—21no—
334C2×C42—31no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—61no—
339C2×C42—62no—
340C2×C42—62no—
341#107—22no—
342#107—62no—
343#106—22no—
344#106—62no—
345#103—22no—
346#103—22no—
347#103—62no—
348#103—62no—
349C22⋊C8—22no—
350C22⋊C8—62no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3#12012yes—
365C23×C3C22×D411yescharacteristic
366C3×Q8—41no—
367C3×Q8—41no—
368C3×Q8—41no—
369C3×Q8—41no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4#12012yes—
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C42−1+411yescharacteristic
392C2×C3×C42−1+411yescharacteristic
393C2×C3×C42−1+411yescharacteristic
394C2×C3×C42−1+411yescharacteristic
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—44no—
408C2×Dic3—44no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3#13512yes—
420C2×Dic3#13712yes—
421Dic6—48no—
422Dic6—48no—
423C3×C8—41no—
424C3×C8—41no—
425C3⋊C8—41no—
426C3⋊C8—41no—
427C24D3×D411yescharacteristic
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—64no—
435C2×Q8—64no—
436C2×Q8—122no—
437C2×Q8—122no—
438C2×Q8—122no—
439C2×Q8—122no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—31no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—121no—
461C22×C4—121no—
462C22×C4D3×D411yescharacteristic
463C22×C4D3×D412yes—
464C22×C4C2×#6712yes—
465C22×C4C23×D311yescharacteristic
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—32no—
483C4⋊C4—32no—
484C4⋊C4—42no—
485C4⋊C4—64no—
486C4⋊C4—64no—
487C4⋊C4—64no—
488C4⋊C4—64no—
489C4⋊C4—64no—
490C4⋊C4—64no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—122no—
498C4⋊C4—122no—
499C4⋊C4C2×#6712yes—
500C4⋊C4C2×#6712yes—
501C22⋊C4—24no—
502C22⋊C4—42no—
503C22⋊C4—64no—
504C22⋊C4—122no—
505C42—24no—
506C42—62no—
507C42—62no—
508C42—64no—
509C42—61no—
510C42—61no—
511C42—61no—
512C42—61no—
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3C2×#12011yescharacteristic
521C22×C3#53611yescharacteristic
522C22×C3#53611yescharacteristic
523C22×C3C2×#13511yescharacteristic
524C22×C3C2×#13711yescharacteristic
525C22×C3#57511yescharacteristic
526C22×C3#57511yescharacteristic
527C3×C4—22no—
528C3×C4—42no—
529C3×C4—42no—
530C3×C4—81no—
531C3×C4—81no—
532C3×C4#53611yescharacteristic
533C3×C4#53611yescharacteristic
534C3×C4#53611yescharacteristic
535C3×C4#53611yescharacteristic
536Dic3—22no—
537Dic3—22no—
538Dic3—42no—
539Dic3—41no—
540Dic3—42no—
541Dic3—41no—
542Dic3—81no—
543Dic3—81no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23#93712yes—
551C23C2×D3×D411yescharacteristic
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—41no—
556Q8—128no—
557Q8—128no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—32no—
567C2×C4—32no—
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—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—124no—
591C2×C4—124no—
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#93712yes—
603C2×C4C2×D3×D411yescharacteristic
604C2×C4#101011yescharacteristic
605C2×C4#101011yescharacteristic
606C2×C4#101011yescharacteristic
607C2×C4#101011yescharacteristic
608C2×C4C22×#6711yescharacteristic
609C8—41no—
610C8—41no—
611C8—121no—
612C8—121no—
613C2×C3—42no—
614C2×C3#261411yescharacteristic
615C2×C3#300211yescharacteristic
616C2×C3#300711yescharacteristic
617C2×C3#323111yescharacteristic
618C2×C3#323311yescharacteristic
619C2×C3#323711yescharacteristic
620C2×C3#324011yescharacteristic
621C22—22no—
622C22—22no—
623C22—22no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22#564811yescharacteristic
629C22#564811yescharacteristic
630C22D3×#13511yescharacteristic
631C22#576811yescharacteristic
632C22D3×#13711yescharacteristic
633C22#577711yescharacteristic
634C22C2×#93711yescharacteristic
635C4—22no—
636C4—42no—
637C4—42no—
638C4—62no—
639C4—62no—
640C4—81no—
641C4—81no—
642C4—121no—
643C4—122no—
644C4—121no—
645C4—122no—
646C4—241no—
647C4—241no—
648C4#564811yescharacteristic
649C4#564811yescharacteristic
650C4#582511yescharacteristic
651C4#582511yescharacteristic
652C3#2964511yescharacteristic, complemented, Sylow
653C2—42no—
654C2#8150011yescharacteristic
655C2#8150111yescharacteristic
656C2#8150211yescharacteristic
657C2#8150311yescharacteristic
658C2#8286411yescharacteristic
659C2#8391111yescharacteristic
660C2#8398111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#485C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#485D3
hypocenterC3#29645
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#485

Complements

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

NH ≅ G/Ntype
6. #70618C2semidirect
7. #70617C2semidirect
8. #70566C2semidirect
9. #70565C2semidirect
10. #70232C2semidirect
11. #70232C2semidirect
12. #70231C2semidirect
13. #70231C2semidirect
652. C3#29645semidirect

Maximal subgroups

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

subgroupindex
C2×#55172
C2×C3×#4852
#828102
C2×#50452
#715752
#706182
#706172
#705662
#705652
#702322
#702312
#699262
#699102
#296453