Finite Group Databasebeta

← #10901409

Subgroups — #10901409

Order 768.

1056 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5479C211yesmaximal, characteristic
2C3×#2851C211yesmaximal, characteristic, complemented
3C2×#5063C211yesmaximal, characteristic, complemented
4C2×#5066C211yesmaximal, characteristic, complemented
5#71530C211yesmaximal, characteristic, complemented
6#69628C211yesmaximal, characteristic, complemented
7#69628C211yesmaximal, characteristic, complemented
8#69606C211yesmaximal, characteristic, complemented
9#69606C211yesmaximal, characteristic, complemented
10#69559C211yesmaximal, characteristic, complemented
11#69556C211yesmaximal, characteristic, complemented
12#69527C211yesmaximal, characteristic, complemented
13#69526C211yesmaximal, characteristic, complemented
14#69524C211yesmaximal, characteristic
15#69523C211yesmaximal, characteristic
16#29693—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C2×#930C2211yescharacteristic
20C2×#929—21no—
21C2×C4×#67—21no—
22C2×#924C2211yescharacteristic
23#5479—24no—
24#5479—24no—
25C2×#876—21no—
26C2×#875C2211yescharacteristic
27C2×C4×D12C2211yescharacteristic
28C2×#872—21no—
29C2×#870C2211yescharacteristic
30C2×C4×Dic6C2211yescharacteristic
31C3×#389C2211yescharacteristic
32C3×#389C2211yescharacteristic
33C3×#387—21no—
34C3×#387—21no—
35C3×#385—21no—
36C3×#385—21no—
37C3×#382C2211yescharacteristic
38C3×#380C2211yescharacteristic
39C3×#380C2211yescharacteristic
40C2×#830C2211yescharacteristic
41C2×#830C2211yescharacteristic
42#5066—24no—
43#5063—24no—
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#809C2211yescharacteristic
49C2×#809C2211yescharacteristic
50C2×#809C2211yescharacteristic
51C2×#809C2211yescharacteristic
52C2×#808C2211yescharacteristic
53C2×#808C2211yescharacteristic
54C2×#807C2211yescharacteristic
55C2×#807C2211yescharacteristic
56C2×C4⋊(C3⋊C8)C2211yescharacteristic
57C2×C4⋊(C3⋊C8)C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4524C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#529—31no—
65C2×#479—31no—
66C2×#478—31no—
67#2851D311yescharacteristic, complemented
68#2059—31no—
69#2059—31no—
70#2042—31no—
71#2042—31no—
72#2036—31no—
73#2035—31no—
74#2034—31no—
75#2033—31no—
76#1805—31no—
77#1804—31no—
78#1797—31no—
79C23×C3×C4D411yescharacteristic
80C23×C3×C4D411yescharacteristic
81C23×C3×C4C2311yescharacteristic
82C22×#67—21no—
83C22×D12C2311yescharacteristic
84C22×C4×D3—21no—
85C22×Dic6C2311yescharacteristic
86C3×#125—22no—
87C3×#125—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C42—22no—
99C2×C3×C42—22no—
100C2×C3×C42D411yescharacteristic
101C2×C3×C42D411yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#217—21no—
105#930—22no—
106#930—22no—
107#929—28no—
108C4×#67—28no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115#924—22no—
116#924—22no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×C4×Dic3—21no—
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126#876—28no—
127#875—22no—
128#875—22no—
129C4×D12—22no—
130C4×D12D414yes—
131#872—28no—
132#870—22no—
133#870—22no—
134C4×Dic6—22no—
135C4×Dic6D414yes—
136C3×#95—21no—
137C3×#95—21no—
138C3×#95—21no—
139C3×#95—21no—
140C3×#95—21no—
141C3×#95—21no—
142#830—41no—
143#830—41no—
144#810—22no—
145#810—22no—
146#810—41no—
147#810—41no—
148#809—22no—
149#809—22no—
150#809—41no—
151#809—41no—
152#808—22no—
153#808—22no—
154#807—22no—
155#807—22no—
156C4⋊(C3⋊C8)—22no—
157C4⋊(C3⋊C8)—22no—
158#529—64no—
159#529—64no—
160C2×#126—61no—
161C2×#125—31no—
162C2×#125C2×D311yescharacteristic
163C2×#124—31no—
164C2×#123—61no—
165C2×C22⋊Q8—31no—
166C2×#121—31no—
167C2×C4×Q8—31no—
168C2×C4×D4—31no—
169C2×C4×D4—61no—
170C2×#117—61no—
171C22×C42C2×D311yescharacteristic
172#479—64no—
173#478—64no—
174C2×#107—31no—
175C2×#107—31no—
176C2×#106—31no—
177C2×#106—31no—
178C2×C4⋊C8—31no—
179C2×C4⋊C8—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#103—31no—
183C2×#103—31no—
184C2×#102—31no—
185C2×#102—31no—
186C2×#102—31no—
187C2×#102—31no—
188C2×C22⋊C8—31no—
189C2×C22⋊C8—31no—
190#389C2×D311yescharacteristic
191#389C2×D311yescharacteristic
192#387—21no—
193#387—21no—
194#385—21no—
195#385—21no—
196#382C2×D311yescharacteristic
197#380C2×D311yescharacteristic
198#380C2×D311yescharacteristic
199#340—31no—
200#340—31no—
201#340—31no—
202#340—31no—
203#336—31no—
204#336—31no—
205C24×C3C2×D411yescharacteristic
206C23×D3—21no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—41no—
221C22×C3×C4—41no—
222C22×C3×C4—41no—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2411yescharacteristic
229C2×#67—24no—
230C2×#67—42no—
231C2×#67—42no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234C22×Dic3—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12C2×D412yes—
241C2×C4×D3—24no—
242C2×C4×D3—42no—
243C2×C4×D3—42no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6C2×D412yes—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—42no—
259C3×C4⋊C4—42no—
260C3×C4⋊C4—42no—
261C3×C4⋊C4—42no—
262C3×C22⋊C4—42no—
263C3×C22⋊C4—42no—
264C3×C42—22no—
265C3×C42—22no—
266C3×C42—21no—
267C3×C42—21no—
268C3×C42C2×D412yes—
269C3×C42C2×D412yes—
270#217—24no—
271(C2×D3)⋊C4—24no—
272(C2×D3)⋊C4—24no—
273(C2×D3)⋊C4—24no—
274(C2×D3)⋊C4—24no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3C2×D412yes—
278C4⋊Dic3C2×D412yes—
279Dic3⋊C4—24no—
280Dic3⋊C4—24no—
281Dic3⋊C4—24no—
282Dic3⋊C4—24no—
283C4×Dic3—24no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C22×Q8—31no—
297C22×D4—31no—
298C22×D4—61no—
299C23×C4—61no—
300C23×C4#6711yescharacteristic
301C23×C4#6711yescharacteristic
302C23×C4C22×D311yescharacteristic
303C22×C8—31no—
304C22×C8—31no—
305C22×C8—31no—
306C22×C8—31no—
307#126—68no—
308#125—22no—
309#125—22no—
310#125—62no—
311#125—62no—
312#124—62no—
313#124—62no—
314#123—68no—
315C22⋊Q8—62no—
316C22⋊Q8—62no—
317#121—62no—
318#121—62no—
319C4×Q8—34no—
320C4×Q8—62no—
321C4×D4—34no—
322C4×D4—68no—
323C4×D4—62no—
324#117—68no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
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⋊C4C22×D311yescharacteristic
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C4⋊C4C22×D311yescharacteristic
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4—61no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—61no—
349C2×C42#6711yescharacteristic
350C2×C42#6711yescharacteristic
351C2×C42C22×D311yescharacteristic
352C2×C42C22×D311yescharacteristic
353#107—62no—
354#107—62no—
355#106—62no—
356#106—62no—
357C4⋊C8—62no—
358C4⋊C8—62no—
359#103—62no—
360#103—62no—
361#103—121no—
362#103—121no—
363#102—62no—
364#102—62no—
365#102—121no—
366#102—121no—
367C22⋊C8—121no—
368C22⋊C8—121no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—21no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3#13512yes—
382C23×C3C22×D411yescharacteristic
383C22×D3—22no—
384C22×D3—41no—
385C22×D3—41no—
386C22×D3—41no—
387C22×D3—41no—
388C22×D3—41no—
389C22×D3—41no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—44no—
404C2×C3×C4—44no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4#12012yes—
422C2×C3×C4#12012yes—
423C2×C3×C4C22×D411yescharacteristic
424C2×C3×C4C22×D411yescharacteristic
425C2×C3×C4H(5,𝔽2)11yescharacteristic
426C2×C3×C42−1+411yescharacteristic
427C2×C3×C42−1+411yescharacteristic
428C2×C3×C42−1+411yescharacteristic
429#67—48no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439D12—22no—
440D12—22no—
441D12—41no—
442D12—41no—
443C4×D3—48no—
444Dic6—22no—
445Dic6—22no—
446Dic6—41no—
447Dic6—41no—
448C3⋊C8—41no—
449C3⋊C8—41no—
450C3⋊C8—41no—
451C3⋊C8—41no—
452C24—61no—
453C24C2×#6711yescharacteristic
454C2×Q8—32no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×D4—32no—
461C2×D4—64no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—122no—
468C2×D4—122no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—61no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—64no—
489C22×C4—122no—
490C22×C4—122no—
491C22×C4C2×#6711yescharacteristic
492C22×C4C2×#6711yescharacteristic
493C22×C4C2×#6711yescharacteristic
494C22×C4C2×#6711yescharacteristic
495C22×C4C2×#6711yescharacteristic
496C22×C4C23×D311yescharacteristic
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—61no—
502C2×C8—61no—
503C2×C8—61no—
504C2×C8—61no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—32no—
518C4⋊C4—32no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—64no—
524C4⋊C4—64no—
525C4⋊C4—64no—
526C4⋊C4—64no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—64no—
532C22⋊C4—64no—
533C22⋊C4—64no—
534C22⋊C4—64no—
535C22⋊C4—64no—
536C42—22no—
537C42—22no—
538C42—21no—
539C42—21no—
540C42—64no—
541C42D3×D412yes—
542C42D3×D412yes—
543C22×C3—22no—
544C22×C3—22no—
545C22×C3—22no—
546C22×C3—41no—
547C22×C3—41no—
548C22×C3—41no—
549C22×C3—41no—
550C22×C3C2×#12011yescharacteristic
551C22×C3#53611yescharacteristic
552C22×C3#53711yescharacteristic
553C22×C3C2×#13511yescharacteristic
554C22×C3#57511yescharacteristic
555C22×C3#57511yescharacteristic
556C22×C3#57511yescharacteristic
557C2×D3—42no—
558C2×D3—42no—
559C2×D3—42no—
560C2×D3—81no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—81no—
564C3×C4—22no—
565C3×C4—22no—
566C3×C4—41no—
567C3×C4—41no—
568C3×C4—41no—
569C3×C4—81no—
570C3×C4—81no—
571C3×C4—81no—
572C3×C4—81no—
573C3×C4#53611yescharacteristic
574C3×C4#53611yescharacteristic
575C3×C4#53711yescharacteristic
576C3×C4#53711yescharacteristic
577Dic3—42no—
578Dic3—42no—
579Dic3—42no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—62no—
587C23—121no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23#95012yes—
594C23C22×#6711yescharacteristic
595Q8—62no—
596Q8—62no—
597Q8—121no—
598Q8—121no—
599D4—62no—
600D4—62no—
601D4—128no—
602D4—121no—
603D4—121no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—62no—
636C2×C4—62no—
637C2×C4—62no—
638C2×C4—128no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4#93712yes—
646C2×C4#93712yes—
647C2×C4C2×D3×D411yescharacteristic
648C2×C4C2×D3×D411yescharacteristic
649C2×C4#100711yescharacteristic
650C2×C4#100711yescharacteristic
651C2×C4#100911yescharacteristic
652C2×C4#101011yescharacteristic
653C8—121no—
654C8—121no—
655C8—121no—
656C8—121no—
657C2×C3—42no—
658C2×C3#261611yescharacteristic
659C2×C3#300211yescharacteristic
660C2×C3#300511yescharacteristic
661C2×C3#323211yescharacteristic
662C2×C3#324011yescharacteristic
663C2×C3#325011yescharacteristic
664C2×C3#325611yescharacteristic
665D3—82no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—122no—
674C22—122no—
675C22—122no—
676C22—241no—
677C22—241no—
678C22—241no—
679C22—241no—
680C22C2×#93711yescharacteristic
681C22#581611yescharacteristic
682C22C2×#95011yescharacteristic
683C22#582211yescharacteristic
684C22#582211yescharacteristic
685C22#582211yescharacteristic
686C22#583311yescharacteristic
687C4—22no—
688C4—22no—
689C4—41no—
690C4—41no—
691C4—41no—
692C4—81no—
693C4—81no—
694C4—81no—
695C4—81no—
696C4—122no—
697C4—122no—
698C4—122no—
699C4#562811yescharacteristic
700C4#562811yescharacteristic
701C4#562911yescharacteristic
702C4#562911yescharacteristic
703C3#2969311yescharacteristic, complemented, Sylow
704C2—42no—
705C2—242no—
706C2#8282011yescharacteristic
707C2#8292611yescharacteristic
708C2#8292811yescharacteristic
709C2#8301811yescharacteristic
710C2#8314911yescharacteristic
711C2#8315111yescharacteristic
712C2#8336211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2851C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2851D3
hypocenterC3#29693
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
2. C3×#2851C2semidirect
3. C2×#5063C2semidirect
4. C2×#5066C2semidirect
5. #71530C2semidirect
6. #69628C2semidirect
7. #69628C2semidirect
8. #69606C2semidirect
9. #69606C2semidirect
10. #69559C2semidirect
11. #69556C2semidirect
12. #69527C2semidirect
13. #69526C2semidirect
67. #2851D3semidirect
703. C3#29693semidirect

Maximal subgroups

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

subgroupindex
C2×#54792
C3×#28512
C2×#50632
C2×#50662
#715302
#696282
#696062
#695592
#695562
#695272
#695262
#695242
#695232
#296933