Finite Group Databasebeta

← #10849350

Subgroups — #10849350

Order 768.

1048 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5671C211yesmaximal, characteristic
2C2×#4793C211yesmaximal, characteristic, complemented
3C2×#4765C211yesmaximal, characteristic
4#76060C211yesmaximal, characteristic, complemented
5C3×#2042C211yesmaximal, characteristic
6#71722C211yesmaximal, characteristic
7#70284C211yesmaximal, characteristic, complemented
8#70246C211yesmaximal, characteristic, complemented
9#70220C211yesmaximal, characteristic, complemented
10#70168C211yesmaximal, characteristic
11#70167C211yesmaximal, characteristic
12#70162C211yesmaximal, characteristic
13#69631C211yesmaximal, characteristic, complemented
14#68683C211yesmaximal, characteristic, complemented
15#68620C211yesmaximal, characteristic, complemented
16#25661—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C2×#936C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22#5671—24no—
23#5671—24no—
24C2×#888—21no—
25C2×#888C2211yescharacteristic
26C2×#879—21no—
27C2×#879—21no—
28C2×#878—21no—
29C2×#877—21no—
30C2×C4×Dic6C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C2×C3×#102C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#832C2211yescharacteristic
40C2×#832C2211yescharacteristic
41C2×#818C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×#816C2211yescharacteristic
44C2×#815C2211yescharacteristic
45C2×Dic3⋊C8C2211yescharacteristic
46#4984C2211yescharacteristic
47#4982—21no—
48Dic3×C4⋊C4C2211yescharacteristic
49#4972C2211yescharacteristic
50C2×#810C2211yescharacteristic
51C2×#810C2211yescharacteristic
52C2×#807C2211yescharacteristic
53#4928C2211yescharacteristic
54#4793—24no—
55#4765—24no—
56#4657—21no—
57#4646—21no—
58C3×#336C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#556—31no—
65C2×#489—31no—
66C2×#477—31no—
67#2546—31no—
68#2043—31no—
69#2043—31no—
70#2042—31no—
71#2042D311yescharacteristic
72#1985—31no—
73#1983—31no—
74#1863—31no—
75#1863—31no—
76#1862—31no—
77#1862—31no—
78#1763—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×Dic3—21no—
82C22×Dic6—21no—
83C22×Dic6D411yescharacteristic
84C22×Dic6C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×#124—22no—
88C3×#124—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98Q8×Dic3—22no—
99Q8×Dic3D414yes—
100#944—22no—
101#944—22no—
102#936—22no—
103#936D414yes—
104D4×Dic3—22no—
105D4×Dic3—22no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3D411yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4D411yescharacteristic
119C2×Dic3⋊C4D411yescharacteristic
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#888—22no—
128#888—22no—
129#888—44no—
130#879—28no—
131#879—28no—
132#878—28no—
133#877—28no—
134C4×Dic6—22no—
135C4×Dic6—22no—
136C4×Dic6—22no—
137C4×Dic6—22no—
138C3×#103—22no—
139C3×#103—22no—
140C3×#102—22no—
141C3×#102—22no—
142C3×#95—21no—
143#835—22no—
144#835—22no—
145#832—22no—
146#832—22no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150C3⋊#95—21no—
151#818—22no—
152#816—22no—
153#816—22no—
154#815—22no—
155Dic3⋊C8—22no—
156#810—22no—
157#810—22no—
158#807—22no—
159#556—64no—
160#556—64no—
161C2×C4⋊Q8—31no—
162C2×C4⋊Q8—31no—
163C2×C4⋊Q8—61no—
164C2×#126—61no—
165C2×#126—61no—
166C2×#124—31no—
167C2×#124C2×D311yescharacteristic
168C2×C22⋊Q8—61no—
169C2×C4×Q8—31no—
170C2×C4×Q8—31no—
171C2×C4×Q8—31no—
172C2×C4×D4—31no—
173C2×#117—61no—
174#489—64no—
175#477—64no—
176C2×#107—31no—
177C2×#107—31no—
178C2×C4⋊C8—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×#102—31no—
188C2×#102—31no—
189C2×#102C2×D311yescharacteristic
190C2×#102C2×D311yescharacteristic
191C2×#97—31no—
192#398—61no—
193#398—61no—
194#391—61no—
195#389—31no—
196#389—31no—
197#389—31no—
198#389C2×D311yescharacteristic
199C4×C4⋊C4—31no—
200#340—31no—
201#340—31no—
202#336—31no—
203#336—31no—
204#336—31no—
205#336C2×D311yescharacteristic
206C24×C3—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8C2×D412yes—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4C2×D412yes—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C22×Dic3—24no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—42no—
231C22×Dic3—42no—
232C22×Dic3C2×D411yescharacteristic
233C22×Dic3C2×D411yescharacteristic
234C22×Dic3C2×D411yescharacteristic
235C22×Dic3C2×D411yescharacteristic
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—22no—
242C2×Dic6—22no—
243C2×Dic6—41no—
244C2×Dic6—42no—
245C2×Dic6—42no—
246C2×Dic6—42no—
247C2×Dic6C2×D412yes—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—42no—
258C3×C4⋊C4#4112yes—
259C3×C22⋊C4—24no—
260C3×C22⋊C4—24no—
261C3×C42—21no—
262C3×C42—21no—
263#217—24no—
264#217—24no—
265#217—24no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—22no—
269C4⋊Dic3—24no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—42no—
273C4⋊Dic3C2×D412yes—
274C4⋊Dic3#4112yes—
275Dic3⋊C4—24no—
276Dic3⋊C4—24no—
277Dic3⋊C4—24no—
278Dic3⋊C4—22no—
279Dic3⋊C4—22no—
280Dic3⋊C4—42no—
281Dic3⋊C4—42no—
282Dic3⋊C4—42no—
283C4×Dic3—24no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C4×Dic3—24no—
287C4×Dic3—42no—
288C4×Dic3C2×D412yes—
289C4×Dic3C2×D412yes—
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×Q8—31no—
298C22×Q8—61no—
299C22×Q8C22×D311yescharacteristic
300C22×D4C22×D311yescharacteristic
301C23×C4—61no—
302C22×C8—31no—
303C22×C8—31no—
304C22×C8C22×D311yescharacteristic
305C22×C8C22×D311yescharacteristic
306C4⋊Q8—62no—
307C4⋊Q8—62no—
308C4⋊Q8—62no—
309C4⋊Q8—62no—
310C4⋊Q8—124no—
311#126—68no—
312#126—68no—
313#124—22no—
314#124—22no—
315#124—34no—
316#124—62no—
317C22⋊Q8—68no—
318C4×Q8—34no—
319C4×Q8—62no—
320C4×Q8—62no—
321C4×Q8—62no—
322C4×Q8—62no—
323C4×Q8—62no—
324C4×D4—62no—
325C4×D4—62no—
326#117—68no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—31no—
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×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—61no—
347C2×C22⋊C4—61no—
348C2×C22⋊C4—61no—
349C2×C42—31no—
350C2×C42—31no—
351C2×C42—61no—
352C2×C42—61no—
353C2×C42—61no—
354C2×C42C22×D311yescharacteristic
355#107—62no—
356#107—62no—
357C4⋊C8—62no—
358#103—22no—
359#103—22no—
360#103—62no—
361#103—62no—
362#103—62no—
363#103—62no—
364#103—62no—
365#103—62no—
366#102—22no—
367#102—22no—
368#102—62no—
369#102—62no—
370#97—62no—
371#95—21no—
372#95—61no—
373#95—61no—
374#95—61no—
375#95—61no—
376C23×C3—22no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3—41no—
380C23×C3—41no—
381C23×C3—41no—
382C23×C3—41no—
383C23×C3C22×D411yescharacteristic
384C3×Q8—22no—
385C3×Q8—22no—
386C3×Q8—41no—
387C3×Q8—41no—
388C3×D4—22no—
389C3×D4—22no—
390C3×D4—41no—
391C3×D4—41no—
392C2×C3×C4—21no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
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×C4C22×D411yescharacteristic
408C2×C3×C4C22×D411yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C42−1+411yescharacteristic
412C2×C3×C42−1+411yescharacteristic
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—22no—
420C2×Dic3—21no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—48no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3#12012yes—
436C2×Dic3#12012yes—
437C2×Dic3AGL(1,ℤ/8)12yes—
438Dic6—22no—
439Dic6—22no—
440Dic6—44no—
441Dic6—41no—
442Dic6—41no—
443Dic6—84no—
444C3×C8—41no—
445C3×C8—41no—
446C3⋊C8—41no—
447C3⋊C8—41no—
448C24—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—32no—
455C2×Q8—62no—
456C2×Q8—62no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×Q8—61no—
461C2×Q8—61no—
462C2×Q8—121no—
463C2×Q8—122no—
464C2×Q8—122no—
465C2×Q8—122no—
466C2×Q8D3×D412yes—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4D3×D412yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—64no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—122no—
486C22×C4—122no—
487C22×C4D3×D411yescharacteristic
488C22×C4D3×D411yescharacteristic
489C22×C4C23×D311yescharacteristic
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—61no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—32no—
506C4⋊C4—32no—
507C4⋊C4—42no—
508C4⋊C4—62no—
509C4⋊C4—62no—
510C4⋊C4—62no—
511C4⋊C4—64no—
512C4⋊C4—64no—
513C4⋊C4—64no—
514C4⋊C4—64no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—61no—
518C4⋊C4—61no—
519C4⋊C4—122no—
520C4⋊C4—122no—
521C4⋊C4—122no—
522C4⋊C4—122no—
523C4⋊C4#23712yes—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—64no—
527C22⋊C4—64no—
528C22⋊C4—64no—
529C42—21no—
530C42—21no—
531C42—32no—
532C42—32no—
533C42—64no—
534C42—64no—
535C42—61no—
536C42—61no—
537C42—122no—
538C22×C3—42no—
539C22×C3—42no—
540C22×C3—42no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3—81no—
544C22×C3—81no—
545C22×C3C2×#12011yescharacteristic
546C22×C3#54811yescharacteristic
547C22×C3#54811yescharacteristic
548C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
549C22×C3#57511yescharacteristic
550C22×C3#57811yescharacteristic
551C22×C3#57811yescharacteristic
552C3×C4—42no—
553C3×C4—42no—
554C3×C4—41no—
555C3×C4—41no—
556C3×C4—81no—
557C3×C4#53211yescharacteristic
558C3×C4#53611yescharacteristic
559C3×C4#54711yescharacteristic
560C3×C4#54711yescharacteristic
561Dic3—22no—
562Dic3—22no—
563Dic3—22no—
564Dic3—42no—
565Dic3—42no—
566Dic3—42no—
567Dic3—41no—
568Dic3—81no—
569C23—22no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23C2×D3×D411yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—41no—
580Q8—41no—
581Q8—62no—
582Q8—62no—
583Q8—124no—
584Q8—121no—
585Q8—121no—
586Q8—244no—
587D4—22no—
588D4—22no—
589D4—41no—
590D4—41no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—32no—
600C2×C4—32no—
601C2×C4—32no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—62no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—62no—
619C2×C4—61no—
620C2×C4—62no—
621C2×C4—128no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4C2×D3×D411yescharacteristic
632C2×C4C2×D3×D411yescharacteristic
633C2×C4C2×#23711yescharacteristic
634C2×C4D3×#4111yescharacteristic
635C2×C4#101011yescharacteristic
636C2×C4#101011yescharacteristic
637C8—41no—
638C8—41no—
639C8—121no—
640C8—121no—
641C2×C3—82no—
642C2×C3#243211yescharacteristic
643C2×C3#300611yescharacteristic
644C2×C3#301111yescharacteristic
645C2×C3#336711yescharacteristic
646C2×C3#336711yescharacteristic
647C2×C3#337711yescharacteristic
648C2×C3#337711yescharacteristic
649C22—42no—
650C22—42no—
651C22—42no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22D3×#12011yescharacteristic
657C22#559811yescharacteristic
658C22#567711yescharacteristic
659C22#576811yescharacteristic
660C22#577111yescharacteristic
661C22D3×AGL(1,ℤ/8)11yescharacteristic
662C22#577911yescharacteristic
663C4—41no—
664C4—42no—
665C4—42no—
666C4—41no—
667C4—62no—
668C4—62no—
669C4—62no—
670C4—81no—
671C4—122no—
672C4—121no—
673C4—122no—
674C4—122no—
675C4—241no—
676C4#559611yescharacteristic
677C4#561111yescharacteristic
678C4#564811yescharacteristic
679C4#565511yescharacteristic
680C3#2566111yescharacteristic, complemented, Sylow
681C2—82no—
682C2#7690411yescharacteristic
683C2#8008411yescharacteristic
684C2#8023011yescharacteristic
685C2#8051811yescharacteristic
686C2#8058611yescharacteristic
687C2#8166311yescharacteristic
688C2#8167011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2042C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2042D3
hypocenterC3#25661
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2042

Complements

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

NH ≅ G/Ntype
2. C2×#4793C2semidirect
4. #76060C2semidirect
7. #70284C2semidirect
8. #70246C2semidirect
9. #70220C2semidirect
13. #69631C2semidirect
14. #68683C2semidirect
15. #68620C2semidirect
680. C3#25661semidirect

Maximal subgroups

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

subgroupindex
C2×#56712
C2×#47932
C2×#47652
#760602
C3×#20422
#717222
#702842
#702462
#702202
#701682
#701672
#701622
#696312
#686832
#686202
#256613