Finite Group Databasebeta

← #10855279

Subgroups — #10855279

Order 768.

904 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×#4818C211yesmaximal, characteristic
2C2×#4788C211yesmaximal, characteristic
3#76503C211yesmaximal, characteristic, complemented
4#74360C211yesmaximal, characteristic
5C3×#2072C211yesmaximal, characteristic, complemented
6#70284C211yesmaximal, characteristic, complemented
7#70237C211yesmaximal, characteristic, complemented
8#69687C211yesmaximal, characteristic
9#69591C211yesmaximal, characteristic, complemented
10#68896C211yesmaximal, characteristic
11#68895C211yesmaximal, characteristic
12#68850C211yesmaximal, characteristic, complemented
13#68809C211yesmaximal, characteristic, complemented
14#68672C211yesmaximal, characteristic
15#68610C211yesmaximal, characteristic, complemented
16#26166—31nomaximal, Sylow
17C2×Q8×Dic3C2211yescharacteristic
18C2×#944C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×D3×C4⋊C4—21no—
24C2×C4×D12C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#384C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34#4989—21no—
35#4989—21no—
36#4987—21no—
37#4984C2211yescharacteristic
38#4973C2211yescharacteristic
39#4972C2211yescharacteristic
40#4972C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C4×C4⋊Dic3C2211yescharacteristic
47#4929C2211yescharacteristic
48#4818—24no—
49#4788—24no—
50#4661—21no—
51#4659—21no—
52#4653—21no—
53#4653—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#461—31no—
63C2×C4×SD16—31no—
64#2446—31no—
65D4×C4⋊C4—31no—
66#2096—31no—
67#2088—31no—
68#2072D311yescharacteristic, complemented
69#2043—31no—
70#1933—31no—
71#1916—31no—
72#1877—31no—
73#1877—31no—
74#1877—31no—
75#1865—31no—
76#1864—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91Q8×Dic3—22no—
92Q8×Dic3—41no—
93#944—22no—
94#944—41no—
95C2×#215—22no—
96C2×#215—22no—
97C2×#215D414yes—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C2×#204—22no—
117C2×#204—22no—
118C2×#204—22no—
119#893—22no—
120#893—22no—
121#893—41no—
122#893—41no—
123D3×C4⋊C4—44no—
124C4×D12—22no—
125C4×D12—22no—
126C3×#103—22no—
127C3×#103—22no—
128C3×#95—21no—
129C3×#95—21no—
130#835—22no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#821—22no—
141#816—22no—
142Dic3⋊C8—22no—
143C8×Dic3—22no—
144#810—22no—
145#810—41no—
146#809—22no—
147#808—22no—
148#808—41no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×C4⋊Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×C4×D4—31no—
156C22×C4⋊C4—61no—
157#461—64no—
158C4×SD16—64no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103C2×D311yescharacteristic
167C2×#103C2×D311yescharacteristic
168C2×#102—31no—
169C2×#102—31no—
170C2×C4×C8—31no—
171#400—61no—
172#398—61no—
173#391—31no—
174#391—31no—
175#391C2×D311yescharacteristic
176#389—31no—
177#389—31no—
178#389—31no—
179#387—61no—
180#387—61no—
181#385—61no—
182#385—61no—
183#384C2×D311yescharacteristic
184#380—61no—
185C4×C4⋊C4—31no—
186#340—31no—
187#340—31no—
188#340C2×D311yescharacteristic
189#340C2×D311yescharacteristic
190#336—31no—
191#336—31no—
192#336—31no—
193#336C2×D311yescharacteristic
194C23×D3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8C2×D412yes—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12C2×D412yes—
219C2×D12#4112yes—
220C2×D12#4112yes—
221C2×C4×D3—24no—
222C2×C4×D3—42no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×C4×D3—42no—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—81no—
229C2×C4×D3—81no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4—81no—
250C3×C42—21no—
251C3×C42—21no—
252#215—28no—
253(C2×D3)⋊C4—24no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—41no—
260C4⋊Dic3—42no—
261C4⋊Dic3—42no—
262C4⋊Dic3SD1614yescomplemented
263C4⋊Dic3#4112yes—
264Dic3⋊C4—42no—
265Dic3⋊C4—42no—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—42no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8C2×D412yes—
279#204—44no—
280C22×Q8—31no—
281C22×Q8C22×D311yescharacteristic
282C22×D4—31no—
283C23×C4—61no—
284C23×C4—61no—
285C23×C4—61no—
286C2×SD16—34no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—62no—
290C2×SD16—62no—
291C2×SD16—62no—
292C22×C8—31no—
293C22×C8—31no—
294C22×C8C22×D311yescharacteristic
295C22×C8C22×D311yescharacteristic
296C4⋊Q8—62no—
297C4⋊Q8—121no—
298C4×Q8—62no—
299C4×Q8—121no—
300C4×D4—62no—
301C4×D4—62no—
302C4×D4—62no—
303C4×D4—62no—
304C4×D4—121no—
305C4×D4—121no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—124no—
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C42—31no—
326C2×C42—31no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329#106—62no—
330#106—121no—
331C4⋊C8—62no—
332#103—22no—
333#103—22no—
334#103—62no—
335#103—62no—
336#103—62no—
337#103—121no—
338#102—62no—
339#102—62no—
340C4×C8—62no—
341#95—21no—
342#95—21no—
343#95—31no—
344#95—31no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352C23×C3C22×D411yescharacteristic
353C22×D3—22no—
354C22×D3—22no—
355C22×D3—22no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C3×Q8—22no—
361C3×Q8—22no—
362C3×Q8—41no—
363C3×Q8—41no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4C2×SD1612yes—
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C2×C3×C42−1+411yescharacteristic
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—22no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3C2×SD1612yes—
408C2×Dic3C2×SD1612yes—
409D12—22no—
410D12—22no—
411D12—22no—
412D12#12314yes—
413C4×D3—48no—
414C4×D3—84no—
415C4×D3—84no—
416Dic6—41no—
417Dic6—41no—
418Dic6—41no—
419Dic6—41no—
420C3×C8—41no—
421C3×C8—41no—
422C3⋊C8—22no—
423C3⋊C8—41no—
424C24—61no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8D3×D412yes—
437C2×D4—32no—
438C2×D4—32no—
439C2×D4—32no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—64no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—122no—
456C22×C4—122no—
457C22×C4—122no—
458C22×C4—122no—
459C22×C4—122no—
460C22×C4—122no—
461C22×C4—241no—
462C22×C4—241no—
463C22×C4D3×D411yescharacteristic
464C22×C4D3×D411yescharacteristic
465C22×C4C23×D311yescharacteristic
466SD16—68no—
467SD16—124no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—32no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—34no—
487C4⋊C4—32no—
488C4⋊C4—42no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—81no—
493C4⋊C4—121no—
494C4⋊C4—122no—
495C4⋊C4—122no—
496C4⋊C4—122no—
497C4⋊C4—122no—
498C4⋊C4—122no—
499C4⋊C4—122no—
500C22⋊C4—64no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C42—21no—
504C42—21no—
505C42—61no—
506C42—61no—
507C42—61no—
508C42—61no—
509C42—122no—
510C22×C3#52511yescharacteristic
511C22×C3D4211yescharacteristic
512C22×C3#54811yescharacteristic
513C22×C3C22×SD1611yescharacteristic
514C22×C3#57511yescharacteristic
515C22×C3#57711yescharacteristic
516C22×C3#57811yescharacteristic
517C2×D3—24no—
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—42no—
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—42no—
524C3×C4—22no—
525C3×C4—41no—
526C3×C4—42no—
527C3×C4—81no—
528C3×C4—81no—
529C3×C4—81no—
530C3×C4C2×#12311yescharacteristic
531C3×C4#53911yescharacteristic
532C3×C4#54711yescharacteristic
533C3×C4#55311yescharacteristic
534Dic3—22no—
535Dic3—22no—
536Dic3—42no—
537Dic3—41no—
538Dic3—41no—
539Dic3—81no—
540Dic3—81no—
541C23—62no—
542C23—62no—
543C23—62no—
544C23—121no—
545C23—121no—
546C23—121no—
547C23—121no—
548C23C2×D3×D411yescharacteristic
549Q8—22no—
550Q8—22no—
551Q8—41no—
552Q8—41no—
553Q8—121no—
554Q8—121no—
555Q8—121no—
556Q8—121no—
557D4—34no—
558D4—62no—
559D4—62no—
560D4—62no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—32no—
568C2×C4—32no—
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—41no—
579C2×C4—41no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—62no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—128no—
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—244no—
601C2×C4—244no—
602C2×C4D3×SD1612yes—
603C2×C4C2×D3×D411yescharacteristic
604C2×C4D3×#4111yescharacteristic
605C2×C4D3×#4111yescharacteristic
606C2×C4D3×#4111yescharacteristic
607C2×C4#100911yescharacteristic
608C2×C4#101011yescharacteristic
609C8—41no—
610C8—41no—
611C8—62no—
612C8—121no—
613C2×C3#244311yescharacteristic
614C2×C3#310111yescharacteristic
615C2×C3#312011yescharacteristic
616C2×C3#326911yescharacteristic
617C2×C3D4×SD1611yescharacteristic
618C2×C3#335911yescharacteristic
619C2×C3#337611yescharacteristic
620D3—44no—
621C22—64no—
622C22—122no—
623C22—122no—
624C22—122no—
625C22—122no—
626C22—122no—
627C22—122no—
628C22#560911yescharacteristic
629C22#565611yescharacteristic
630C22#567711yescharacteristic
631C22C2×D3×SD1611yescharacteristic
632C22#576611yescharacteristic
633C22#577511yescharacteristic
634C22#577711yescharacteristic
635C4—22no—
636C4—41no—
637C4—42no—
638C4—62no—
639C4—62no—
640C4—81no—
641C4—81no—
642C4—81no—
643C4—122no—
644C4—121no—
645C4—121no—
646C4—241no—
647C4—241no—
648C4D3×#12311yescharacteristic
649C4#565511yescharacteristic
650C4#570711yescharacteristic
651C4#570911yescharacteristic
652C3#2616611yescharacteristic, complemented, Sylow
653C2—124no—
654C2#7767311yescharacteristic
655C2#8040611yescharacteristic
656C2#8050011yescharacteristic
657C2#8141211yescharacteristic
658C2#8145611yescharacteristic
659C2#8162011yescharacteristic
660C2#8167611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2072C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2072D3
hypocenterC3#26166
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2072

Complements

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

NH ≅ G/Ntype
3. #76503C2semidirect
5. C3×#2072C2semidirect
6. #70284C2semidirect
7. #70237C2semidirect
9. #69591C2semidirect
12. #68850C2semidirect
13. #68809C2semidirect
15. #68610C2semidirect
68. #2072D3semidirect
262. C4⋊Dic3SD16semidirect
652. C3#26166semidirect

Maximal subgroups

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

subgroupindex
C2×#48182
C2×#47882
#765032
#743602
C3×#20722
#702842
#702372
#696872
#695912
#688962
#688952
#688502
#688092
#686722
#686102
#261663