Finite Group Databasebeta

← #10849266

Subgroups — #10849266

Order 768.

1018 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×#5584C211yesmaximal, characteristic
2C3×#2584C211yesmaximal, characteristic, complemented
3C2×#5063C211yesmaximal, characteristic, complemented
4C2×#5068C211yesmaximal, characteristic
5#71634C211yesmaximal, characteristic, complemented
6#69633C211yesmaximal, characteristic, complemented
7#69633C211yesmaximal, characteristic, complemented
8#69632C211yesmaximal, characteristic, complemented
9#69631C211yesmaximal, characteristic, complemented
10#69628C211yesmaximal, characteristic
11#69628C211yesmaximal, characteristic
12#69626C211yesmaximal, characteristic
13#69625C211yesmaximal, characteristic
14#69488C211yesmaximal, characteristic
15C4×#810C211yesmaximal, characteristic, complemented
16#25654—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×#946—21no—
20#5584—24no—
21#5584—24no—
22C2×#898—21no—
23C2×#898—21no—
24C2×#892—21no—
25C2×#875C2211yescharacteristic
26C2×#875C2211yescharacteristic
27C2×C4×D12C2211yescharacteristic
28C2×#869C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C2×C4×Dic6C2211yescharacteristic
31C3×#398—21no—
32C3×#395—21no—
33C3×#389—21no—
34C3×#389—21no—
35C3×#389C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×C4×C4⋊C4C2211yescharacteristic
40#5068—24no—
41#5063—24no—
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#809C2211yescharacteristic
51C2×#809C2211yescharacteristic
52C2×#809C2211yescharacteristic
53C2×#809C2211yescharacteristic
54C2×#808C2211yescharacteristic
55C2×#807C2211yescharacteristic
56C2×C4⋊(C3⋊C8)C2211yescharacteristic
57C2×C4×C3⋊C8C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62#4524C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#556—31no—
65C2×#488—31no—
66C2×#479—31no—
67#2584D311yescharacteristic, complemented
68#2043—31no—
69#2043—31no—
70#2042—31no—
71#2042—31no—
72#1984—31no—
73#1983—31no—
74#1865—31no—
75#1864—31no—
76#1863—31no—
77#1862—31no—
78C4×#103—31no—
79C22×C3×Q8D411yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×Dic6C2311yescharacteristic
83C22×Dic6C2311yescharacteristic
84C3×#125—22no—
85C3×#125—22no—
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⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C42—21no—
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2311yescharacteristic
105#946—28no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122#898—28no—
123#898—28no—
124#892—28no—
125#875—22no—
126#875—22no—
127#875—22no—
128#875D414yes—
129C4×D12—22no—
130C4×D12—22no—
131#869—22no—
132#869—22no—
133C4×Dic6—22no—
134C4×Dic6—22no—
135C4×Dic6—22no—
136C4×Dic6D414yes—
137C3×#95—21no—
138C3×#95—21no—
139C3×#95—21no—
140C3×#95—21no—
141C3×#95—21no—
142#810—22no—
143#810—22no—
144#810—22no—
145#810—22no—
146#810—22no—
147#810—22no—
148#810—22no—
149#810—22no—
150#809—22no—
151#809—22no—
152#809—22no—
153#809—22no—
154#808—22no—
155#807—22no—
156C4⋊(C3⋊C8)—22no—
157C4×C3⋊C8—22no—
158#556—64no—
159#556—64no—
160C2×C4⋊Q8—31no—
161C2×#126—61no—
162C2×#126—61no—
163C2×#125C2×D311yescharacteristic
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#488—64no—
173#479—64no—
174C2×#107—31no—
175C2×#106—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×#102—31no—
186C2×#102—31no—
187C2×#102—31no—
188C2×#102—31no—
189C2×C4×C8—31no—
190#398—21no—
191#395—21no—
192#389—21no—
193#389—21no—
194#389C2×D311yescharacteristic
195#389C2×D311yescharacteristic
196#389C2×D311yescharacteristic
197#389C2×D311yescharacteristic
198C4×C4⋊C4C2×D311yescharacteristic
199#340—31no—
200#340—31no—
201#336—31no—
202#336—31no—
203#336—31no—
204#336—31no—
205C23×D3—21no—
206C2×C3×Q8—22no—
207C2×C3×Q8—22no—
208C2×C3×Q8—41no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2411yescharacteristic
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12C2×D412yes—
229C2×C4×D3—24no—
230C2×C4×D3—42no—
231C2×C4×D3—42no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—21no—
242C2×Dic6C2×D412yes—
243C2×Dic6C2×D412yes—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—42no—
254C3×C4⋊C4—42no—
255C3×C4⋊C4—42no—
256C3×C4⋊C4—42no—
257C3×C4⋊C4—81no—
258C3×C4⋊C4—81no—
259C3×C4⋊C4#4112yes—
260C3×C4⋊C4#4112yes—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—21no—
264C3×C42—21no—
265C3×C42—81no—
266C3×C42C2×D412yes—
267C3×C42C2×D412yes—
268(C2×D3)⋊C4—24no—
269(C2×D3)⋊C4—24no—
270(C2×D3)⋊C4—24no—
271(C2×D3)⋊C4—24no—
272(C2×D3)⋊C4—24no—
273C4⋊Dic3—24no—
274C4⋊Dic3—24no—
275C4⋊Dic3—21no—
276C4⋊Dic3C2×D412yes—
277Dic3⋊C4—24no—
278Dic3⋊C4—24no—
279C4×Dic3—24no—
280C4×Dic3—24no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
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—
293C22×Q8—31no—
294C22×Q8—31no—
295C22×Q8#6711yescharacteristic
296C22×D4—31no—
297C23×C4—61no—
298C22×C8—31no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8—31no—
302C4⋊Q8—62no—
303C4⋊Q8—62no—
304#126—68no—
305#126—68no—
306#125—22no—
307#125—22no—
308#124—34no—
309#124—62no—
310#124—62no—
311#124—62no—
312C22⋊Q8—68no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—34no—
316C4×Q8—62no—
317C4×Q8—62no—
318C4×Q8—62no—
319C4×D4—62no—
320C4×D4—62no—
321#117—68no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4#6711yescharacteristic
334C2×C4⋊C4#6711yescharacteristic
335C2×C4⋊C4#6711yescharacteristic
336C2×C4⋊C4C22×D311yescharacteristic
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—61no—
347C2×C42—61no—
348C2×C42C22×D311yescharacteristic
349C2×C42C22×D311yescharacteristic
350C2×C42C22×D311yescharacteristic
351#107—62no—
352#106—62no—
353C4⋊C8—62no—
354#103—62no—
355#103—62no—
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—
366C4×C8—62no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
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—44no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
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—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4#12012yes—
409C2×C3×C4#12012yes—
410C2×C3×C4#13512yes—
411C2×C3×C4C22×D411yescharacteristic
412C2×C3×C4C22×D411yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
414C2×C3×C4C2×#4111yescharacteristic
415C2×C3×C42−1+411yescharacteristic
416C2×C3×C42−1+411yescharacteristic
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—22no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426D12—22no—
427D12—22no—
428D12—41no—
429D12—41no—
430C4×D3—48no—
431Dic6—22no—
432Dic6—22no—
433Dic6—22no—
434Dic6—22no—
435Dic6—41no—
436Dic6—41no—
437Dic6—41no—
438Dic6—41no—
439C3⋊C8—41no—
440C3⋊C8—41no—
441C3⋊C8—41no—
442C3⋊C8—41no—
443C24—61no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—32no—
447C2×Q8—32no—
448C2×Q8—41no—
449C2×Q8—61no—
450C2×Q8—61no—
451C2×Q8—61no—
452C2×Q8—61no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—61no—
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—64no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—122no—
474C22×C4—122no—
475C22×C4C2×#6711yescharacteristic
476C22×C4C2×#6711yescharacteristic
477C22×C4C2×#6711yescharacteristic
478C22×C4C2×#6711yescharacteristic
479C22×C4C2×#6711yescharacteristic
480C22×C4C2×#6711yescharacteristic
481C22×C4C23×D311yescharacteristic
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—32no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—64no—
509C4⋊C4—64no—
510C4⋊C4—64no—
511C4⋊C4—64no—
512C4⋊C4—61no—
513C4⋊C4—81no—
514C4⋊C4—81no—
515C4⋊C4#23912yes—
516C4⋊C4#23912yes—
517C22⋊C4—64no—
518C22⋊C4—64no—
519C22⋊C4—64no—
520C22⋊C4—64no—
521C22⋊C4—64no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—64no—
527C42—64no—
528C42—81no—
529C42D3×D412yes—
530C42D3×D412yes—
531C22×C3C2×#12011yescharacteristic
532C22×C3#54811yescharacteristic
533C22×C3#54811yescharacteristic
534C22×C3C2×#13511yescharacteristic
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—22no—
546C3×C4—22no—
547C3×C4—22no—
548C3×C4—42no—
549C3×C4—42no—
550C3×C4—41no—
551C3×C4—41no—
552C3×C4—41no—
553C3×C4—81no—
554C3×C4—81no—
555C3×C4#53211yescharacteristic
556C3×C4#53611yescharacteristic
557C3×C4#54711yescharacteristic
558C3×C4#54711yescharacteristic
559Dic3—42no—
560Dic3—42no—
561Dic3—42no—
562C23—62no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23C22×#6711yescharacteristic
570Q8—44no—
571Q8—62no—
572Q8—62no—
573Q8—62no—
574Q8—62no—
575Q8—121no—
576Q8—121no—
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—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—128no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4#93712yes—
621C2×C4#93712yes—
622C2×C4#95012yes—
623C2×C4C2×D3×D411yescharacteristic
624C2×C4C2×D3×D411yescharacteristic
625C2×C4C2×#23911yescharacteristic
626C2×C4C2×#23911yescharacteristic
627C2×C4#100711yescharacteristic
628C2×C4#100711yescharacteristic
629C8—121no—
630C8—121no—
631C8—121no—
632C8—121no—
633C2×C3#243211yescharacteristic
634C2×C3#300211yescharacteristic
635C2×C3#301311yescharacteristic
636C2×C3#336011yescharacteristic
637C2×C3#336411yescharacteristic
638C2×C3#337611yescharacteristic
639C2×C3#337711yescharacteristic
640D3—82no—
641C22—122no—
642C22—122no—
643C22—122no—
644C22—241no—
645C22—241no—
646C22—241no—
647C22—241no—
648C22C2×#93711yescharacteristic
649C22#581711yescharacteristic
650C22#581711yescharacteristic
651C22C2×#95011yescharacteristic
652C22#582211yescharacteristic
653C22#583611yescharacteristic
654C22#583811yescharacteristic
655C4—22no—
656C4—22no—
657C4—22no—
658C4—42no—
659C4—42no—
660C4—41no—
661C4—41no—
662C4—41no—
663C4—81no—
664C4—81no—
665C4—122no—
666C4—122no—
667C4—122no—
668C4#572411yescharacteristic
669C4#572511yescharacteristic
670C4#572611yescharacteristic
671C4#572611yescharacteristic
672C3#2565411yescharacteristic, complemented, Sylow
673C2—242no—
674C2#8292611yescharacteristic
675C2#8325511yescharacteristic
676C2#8342711yescharacteristic
677C2#8363711yescharacteristic
678C2#8364511yescharacteristic
679C2#8368911yescharacteristic
680C2#8370011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2584C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2584D3
hypocenterC3#25654
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2584

Complements

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

NH ≅ G/Ntype
2. C3×#2584C2semidirect
3. C2×#5063C2semidirect
5. #71634C2semidirect
6. #69633C2semidirect
7. #69633C2semidirect
8. #69632C2semidirect
9. #69631C2semidirect
15. C4×#810C2semidirect
67. #2584D3semidirect
672. C3#25654semidirect

Maximal subgroups

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

subgroupindex
C2×#55842
C3×#25842
C2×#50632
C2×#50682
#716342
#696332
#696322
#696312
#696282
#696262
#696252
#694882
C4×#8102
#256543