Finite Group Databasebeta

← #10858112

Subgroups — #10858112

Order 768.

896 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82636C211yesmaximal, characteristic
2#82249C211yesmaximal, characteristic, complemented
3C2×#4995C211yesmaximal, characteristic, complemented
4C2×#4763C211yesmaximal, characteristic, complemented
5C3×#1875C211yesmaximal, characteristic
6#71682C211yesmaximal, characteristic
7#70194C211yesmaximal, characteristic, complemented
8#70184C211yesmaximal, characteristic, complemented
9#70169C211yesmaximal, characteristic, complemented
10#69951C211yesmaximal, characteristic, complemented
11#69928C211yesmaximal, characteristic, complemented
12#68828C211yesmaximal, characteristic, complemented
13#68815C211yesmaximal, characteristic, complemented
14#68809C211yesmaximal, characteristic, complemented
15#68796C211yesmaximal, characteristic, complemented
16#26366—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×D4×Dic3—21no—
20C2×D4×Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C2×#890C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27#5221C2211yescharacteristic
28#5220—21no—
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31#5212C2211yescharacteristic
32C4×#217C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37#4995—24no—
38#4984C2211yescharacteristic
39#4984C2211yescharacteristic
40#4975C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4951—21no—
48#4950—21no—
49#4949—21no—
50#4948—21no—
51#4933—21no—
52#4928—21no—
53#4763—24no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#478—31no—
63C2×#428—31no—
64#2441—31no—
65#2361—31no—
66#2094—31no—
67#2088—31no—
68#2067—31no—
69#2059—31no—
70#1948—31no—
71#1918—31no—
72#1895—31no—
73#1875D311yescharacteristic
74#1864—31no—
75#1853—31no—
76#1851—31no—
77C22×C3×D4—21no—
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C23×Dic3—21no—
81C23×Dic3—21no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×#121—22no—
85C3×#121Q814yes—
86C3×#117—22no—
87C3×#117—41no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C42—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217D411yescharacteristic
99C2×#217D411yescharacteristic
100D4×Dic3—22no—
101D4×Dic3—41no—
102D4×Dic3—44no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—41no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C2×#208—22no—
121C2×#208—22no—
122C2×#208—22no—
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#890—22no—
126#890—41no—
127C3×#102—22no—
128C3×#102—41no—
129C3×C22⋊C8—22no—
130#832—22no—
131#832—22no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139#817—22no—
140#817—41no—
141Dic3⋊C8—22no—
142Dic3⋊C8—22no—
143#808—22no—
144#808—22no—
145#807—22no—
146#807D414yescomplemented
147C22×M16—31no—
148C2×#125—31no—
149C2×#121C2×D311yescharacteristic
150C2×C4×D4—31no—
151C2×C4×D4—61no—
152C2×#117C2×D311yescharacteristic
153C22×C4⋊C4—31no—
154#478—64no—
155#428—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#106—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×C4⋊C8—31no—
163C2×C4⋊C8—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×C22⋊C8C2×D311yescharacteristic
169#399—61no—
170#396—61no—
171#390—31no—
172#389—31no—
173#389—31no—
174#389—31no—
175#389—61no—
176#388—61no—
177#387—61no—
178#386—61no—
179#385—61no—
180#384—31no—
181#380—31no—
182C4×C4⋊C4—31no—
183C4×C22⋊C4—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340C2×D311yescharacteristic
189#340C2×D311yescharacteristic
190#336—31no—
191#336—31no—
192C24×C3—21no—
193C24×C3C2×D411yescharacteristic
194C2×C3×D4—24no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—42no—
201C2×C3×D4—42no—
202C2×C3×D4C2×Q812yes—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×Q812yes—
212C22×C3×C4#4112yes—
213C22×C3×C4C2411yescharacteristic
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—42no—
219C22×Dic3—42no—
220C22×Dic3—42no—
221C22×Dic3—42no—
222C22×Dic3—81no—
223C22×Dic3—81no—
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4C2×D412yes—
239C3×C4⋊C4C2×Q812yes—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—42no—
242C3×C42—42no—
243#217—22no—
244#217—41no—
245#217—42no—
246#217—42no—
247#217—81no—
248#217—81no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—42no—
257C4⋊Dic3—42no—
258C4⋊Dic3—42no—
259C4⋊Dic3—42no—
260C4⋊Dic3C2×D412yes—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—81no—
264Dic3⋊C4—81no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—42no—
270#208—28no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8C2×D412yes—
276C2×C3⋊C8#4112yes—
277C22×D4—21no—
278C22×D4C22×D311yescharacteristic
279C23×C4—61no—
280C23×C4—61no—
281C23×C4C22×D311yescharacteristic
282C2×M16—62no—
283C2×M16—62no—
284C2×M16—62no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289#125—62no—
290#125—121no—
291#121—22no—
292#121Dic614yes—
293C4×D4—62no—
294C4×D4—121no—
295C4×D4—124no—
296#117—22no—
297#117—41no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—62no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—121no—
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—31no—
318C2×C22⋊C4—31no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C42—21no—
324C2×C42—31no—
325C2×C42—31no—
326C2×C42—61no—
327#107—34no—
328#107—62no—
329#106—62no—
330#106—62no—
331#106—62no—
332#106—121no—
333C4⋊C8—62no—
334C4⋊C8—62no—
335#102—22no—
336#102—41no—
337#102—62no—
338#102—62no—
339C22⋊C8—22no—
340#95—31no—
341#95—31no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—22no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3C22⋊Q812yes—
361C23×C3C22×D411yescharacteristic
362C3×D4—22no—
363C3×D4—22no—
364C3×D4—48no—
365C3×D4—41no—
366C3×D4—41no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
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×C4—41no—
382C2×C3×C4C22⋊Q812yes—
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C22×Q811yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4H(5,𝔽2)11yescharacteristic
388C2×C3×C42−1+411yescharacteristic
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—22no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—84no—
413C2×Dic3—84no—
414C2×Dic3AGL(1,ℤ/8)12yes—
415C3×C8—41no—
416C3×C8—41no—
417C3⋊C8—22no—
418C3⋊C8—22no—
419C24—21no—
420C24D3×D411yescharacteristic
421C2×D4—24no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4C2×Dic612yes—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—122no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4—241no—
450C22×C4—241no—
451C22×C4C2×Dic612yes—
452C22×C4D3×D411yescharacteristic
453C22×C4#23712yes—
454C22×C4C23×D311yescharacteristic
455M16—68no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—32no—
463C2×C8—32no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—32no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—122no—
481C4⋊C4—122no—
482C4⋊C4—122no—
483C4⋊C4—122no—
484C4⋊C4—122no—
485C4⋊C4—122no—
486C4⋊C4—241no—
487C4⋊C4—241no—
488C4⋊C4C2×Dic612yes—
489C4⋊C4D3×D412yes—
490C22⋊C4—24no—
491C22⋊C4—42no—
492C22⋊C4—62no—
493C22⋊C4—122no—
494C22⋊C4—121no—
495C22⋊C4—122no—
496C22⋊C4—241no—
497C22⋊C4—241no—
498C42—42no—
499C42—61no—
500C42—61no—
501C42—61no—
502C42—61no—
503C42—122no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—81no—
515C22×C3—81no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3C2×C22⋊Q811yescharacteristic
519C22×C3D4211yescharacteristic
520C22×C3#54711yescharacteristic
521C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
522C22×C3#57511yescharacteristic
523C22×C3#57711yescharacteristic
524C22×C3#57711yescharacteristic
525C3×C4—22no—
526C3×C4—42no—
527C3×C4—42no—
528C3×C4—81no—
529C3×C4#52511yescharacteristic
530C3×C4#53911yescharacteristic
531C3×C4D4×Q811yescharacteristic
532C3×C4#55411yescharacteristic
533Dic3—22no—
534Dic3—41no—
535Dic3—41no—
536Dic3—41no—
537Dic3—42no—
538Dic3—81no—
539Dic3—81no—
540Dic3—81no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23#87812yes—
555C23C2×D3×D411yescharacteristic
556D4—22no—
557D4—22no—
558D4—48no—
559D4—41no—
560D4—41no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—32no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—61no—
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—121no—
590C2×C4—121no—
591C2×C4—121no—
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×C4#87812yes—
603C2×C4C22×Dic611yescharacteristic
604C2×C4C2×D3×D411yescharacteristic
605C2×C4C2×#23711yescharacteristic
606C2×C4#100411yescharacteristic
607C2×C4D3×#4111yescharacteristic
608C2×C4#101011yescharacteristic
609C8—41no—
610C8—41no—
611C8—62no—
612C8—62no—
613C2×C3—42no—
614C2×C3—82no—
615C2×C3#244911yescharacteristic
616C2×C3#307711yescharacteristic
617C2×C3#308411yescharacteristic
618C2×C3#327111yescharacteristic
619C2×C3#327211yescharacteristic
620C2×C3#334511yescharacteristic
621C2×C3#335111yescharacteristic
622C22—22no—
623C22—22no—
624C22—22no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22C2×#87811yescharacteristic
637C22#559011yescharacteristic
638C22#565511yescharacteristic
639C22#577011yescharacteristic
640C22#577011yescharacteristic
641C22D3×AGL(1,ℤ/8)11yescharacteristic
642C22#577311yescharacteristic
643C4—22no—
644C4—42no—
645C4—42no—
646C4—62no—
647C4—81no—
648C4—121no—
649C4—121no—
650C4—121no—
651C4—122no—
652C4—241no—
653C4—241no—
654C4—241no—
655C4D4×Dic611yescharacteristic
656C4#553911yescharacteristic
657C4#560211yescharacteristic
658C4#560611yescharacteristic
659C3#2636611yescharacteristic, complemented, Sylow
660C2—42no—
661C2—82no—
662C2#7470911yescharacteristic
663C2#7882411yescharacteristic
664C2#7891711yescharacteristic
665C2#8009811yescharacteristic
666C2#8012411yescharacteristic
667C2#8139111yescharacteristic
668C2#8139611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1875C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#1875D3
hypocenterC3#26366
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
2. #82249C2semidirect
3. C2×#4995C2semidirect
4. C2×#4763C2semidirect
7. #70194C2semidirect
8. #70184C2semidirect
9. #70169C2semidirect
10. #69951C2semidirect
11. #69928C2semidirect
12. #68828C2semidirect
13. #68815C2semidirect
14. #68809C2semidirect
15. #68796C2semidirect
146. #807D4semidirect
659. C3#26366semidirect

Maximal subgroups

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

subgroupindex
#826362
#822492
C2×#49952
C2×#47632
C3×#18752
#716822
#701942
#701842
#701692
#699512
#699282
#688282
#688152
#688092
#687962
#263663