Finite Group Databasebeta

← #10831760

Subgroups — #10831760

Order 768.

1064 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5539C211yesmaximal, characteristic
2C2×#4854C211yesmaximal, characteristic, complemented
3C2×#4757C211yesmaximal, characteristic
4#76076C211yesmaximal, characteristic, complemented
5C3×#1864C211yesmaximal, characteristic
6#71612C211yesmaximal, characteristic
7#70566C211yesmaximal, characteristic, complemented
8#70171C211yesmaximal, characteristic
9#70169C211yesmaximal, characteristic
10#70154C211yesmaximal, characteristic
11#69606C211yesmaximal, characteristic, complemented
12#68911C211yesmaximal, characteristic, complemented
13#68806C211yesmaximal, characteristic, complemented
14#68690C211yesmaximal, characteristic, complemented
15#68620C211yesmaximal, characteristic, complemented
16#24343—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C2×#924—21no—
22C22×Dic3⋊C4—21no—
23#5539—24no—
24#5539—24no—
25C2×#890C2211yescharacteristic
26C2×#888C2211yescharacteristic
27C2×#887C2211yescharacteristic
28C2×#878—21no—
29C2×#878—21no—
30C2×#870C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×#106C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#815C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41#4980—21no—
42#4979—21no—
43#4977C2211yescharacteristic
44#4973C2211yescharacteristic
45#4972C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4929C2211yescharacteristic
50#4854—24no—
51#4757—24no—
52#4658—21no—
53#4652—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#554—31no—
65C2×#475—31no—
66C2×#440—31no—
67#2581—31no—
68#2059—31no—
69#2059—31no—
70#2026—31no—
71#2026—31no—
72#1993—31no—
73#1985—31no—
74#1877—31no—
75#1877—31no—
76#1864—31no—
77#1864D311yescharacteristic
78#1804—31no—
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C23×Dic3—21no—
82C23×Dic3—21no—
83C22×Dic6—21no—
84C22×Dic6D411yescharacteristic
85C22×Dic6C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C42C2311yescharacteristic
96C2×#217—21no—
97C2×#217—21no—
98D4×Dic3—22no—
99D4×Dic3—22no—
100D4×Dic3—22no—
101D4×Dic3D414yes—
102C2×#214—22no—
103C2×#214—22no—
104C2×#214Q814yescomplemented
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3D411yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110#924—28no—
111C2×Dic3⋊C4—28no—
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×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#890—22no—
126#890—22no—
127#888—22no—
128#888D414yes—
129#887—22no—
130#887—22no—
131#878—28no—
132#878—28no—
133#870—22no—
134#870—22no—
135C4×Dic6—22no—
136C4×Dic6—22no—
137C3×#106—22no—
138C3×#102—22no—
139C3×#102—22no—
140C3×#95—21no—
141#832—22no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149#816—22no—
150#816—22no—
151#815—22no—
152Dic3⋊C8—22no—
153#810—22no—
154#808—22no—
155#807—22no—
156C22×SD16—31no—
157#554—64no—
158#554—64no—
159C2×C4⋊Q8—31no—
160C2×#125—31no—
161C2×#125—31no—
162C2×C22⋊Q8—61no—
163C2×C22⋊Q8—61no—
164C2×C22⋊Q8—61no—
165C2×C4×Q8—31no—
166C2×C4×Q8—31no—
167C2×C4×D4—31no—
168C2×C4×D4—31no—
169C2×C4×D4C2×D311yescharacteristic
170C22×C4⋊C4—61no—
171#475—64no—
172#440—64no—
173C2×#107—31no—
174C2×#106—31no—
175C2×#106C2×D311yescharacteristic
176C2×C4⋊C8—31no—
177C2×#103—31no—
178C2×#103—31no—
179C2×#103—31no—
180C2×#102—31no—
181C2×#102C2×D311yescharacteristic
182C2×#102C2×D311yescharacteristic
183C2×#97—31no—
184#400—61no—
185#395—61no—
186#391—31no—
187#391—31no—
188#389—31no—
189#389C2×D311yescharacteristic
190#387—61no—
191#387—61no—
192#382—31no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340C2×D311yescharacteristic
199#336—31no—
200#336—31no—
201#336—31no—
202#336C2×D311yescharacteristic
203C24×C3—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4C2×D412yes—
208C2×C3×D4C2×Q812yes—
209C2×C3×D4#4112yes—
210C22×C3×C4—21no—
211C22×C3×C4—24no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—81no—
215C22×C3×C4—81no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—24no—
223C22×Dic3—24no—
224C22×Dic3—21no—
225C22×Dic3—42no—
226C22×Dic3—42no—
227C22×Dic3—81no—
228C22×Dic3—81no—
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C22×Dic3C2×D411yescharacteristic
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—22no—
239C2×Dic6—24no—
240C2×Dic6—41no—
241C2×Dic6—41no—
242C2×Dic6—81no—
243C2×Dic6—81no—
244C2×Dic6C2×Q812yes—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—42no—
257C3×C4⋊C4C2×D412yes—
258C3×C22⋊C4—24no—
259C3×C42—21no—
260C3×C42—21no—
261#217—24no—
262#217—24no—
263#214—28no—
264C4⋊Dic3—24no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—22no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3C2×D412yes—
271C4⋊Dic3C2×D412yes—
272Dic3⋊C4—24no—
273Dic3⋊C4—24no—
274Dic3⋊C4—24no—
275Dic3⋊C4—22no—
276Dic3⋊C4—22no—
277Dic3⋊C4—24no—
278Dic3⋊C4—42no—
279Dic3⋊C4—42no—
280C4×Dic3—21no—
281C4×Dic3—21no—
282C4×Dic3—42no—
283C4×Dic3C2×D412yes—
284C4×Dic3C2×D412yes—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8C2×Q812yes—
291C22×Q8—31no—
292C22×Q8—31no—
293C22×Q8—61no—
294C22×D4C22×D311yescharacteristic
295C23×C4—21no—
296C23×C4—61no—
297C23×C4—61no—
298C2×SD16—34no—
299C2×SD16—62no—
300C2×SD16—62no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305C4⋊Q8—34no—
306C4⋊Q8—62no—
307#125—62no—
308#125—62no—
309#125—62no—
310#125—62no—
311C22⋊Q8—68no—
312C22⋊Q8—68no—
313C22⋊Q8—68no—
314C4×Q8—62no—
315C4×Q8—62no—
316C4×Q8—62no—
317C4×Q8—62no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—34no—
321C4×D4—62no—
322C4×D4—62no—
323C4×D4—62no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—68no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4C22×D311yescharacteristic
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C22⋊C4—21no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C42—31no—
346C2×C42—31no—
347C2×C42—61no—
348C2×C42C22×D311yescharacteristic
349#107—62no—
350#106—22no—
351#106—62no—
352C4⋊C8—62no—
353#103—62no—
354#103—62no—
355#103—62no—
356#102—22no—
357#102—22no—
358#102—62no—
359#97—62no—
360#95—21no—
361#95—61no—
362#95—61no—
363#95—61no—
364#95—61no—
365#95—61no—
366#95—61no—
367#95—61no—
368C23×C3—22no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3C22×D411yescharacteristic
376C3×D4—22no—
377C3×D4—22no—
378C3×D4—22no—
379C3×D4C22⋊Q814yes—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—84no—
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C22×Q811yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C42−1+411yescharacteristic
401C2×C3×C42−1+411yescharacteristic
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—48no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—84no—
426C2×Dic3#12012yes—
427C2×Dic3#12012yes—
428Dic6—22no—
429Dic6—22no—
430Dic6—44no—
431Dic6—41no—
432Dic6—41no—
433Dic6—84no—
434C3×C8—41no—
435C3×C8—41no—
436C3⋊C8—22no—
437C3⋊C8—41no—
438C24—21no—
439C2×Q8—32no—
440C2×Q8—62no—
441C2×Q8—64no—
442C2×Q8—61no—
443C2×Q8—61no—
444C2×Q8—61no—
445C2×Q8—61no—
446C2×Q8—61no—
447C2×Q8—121no—
448C2×Q8—121no—
449C2×Q8—241no—
450C2×Q8—241no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4#23512yes—
455C2×D4D3×D412yes—
456C2×D4D3×Q812yes—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—64no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—64no—
470C22×C4—61no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—122no—
474C22×C4—122no—
475C22×C4—241no—
476C22×C4—241no—
477C22×C4D3×D411yescharacteristic
478C22×C4D3×D411yescharacteristic
479C22×C4C23×D311yescharacteristic
480SD16—68no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—32no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—32no—
499C4⋊C4—32no—
500C4⋊C4—42no—
501C4⋊C4—62no—
502C4⋊C4—62no—
503C4⋊C4—62no—
504C4⋊C4—64no—
505C4⋊C4—64no—
506C4⋊C4—64no—
507C4⋊C4—64no—
508C4⋊C4—64no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—122no—
514C4⋊C4—122no—
515C4⋊C4D3×D412yes—
516C22⋊C4—24no—
517C22⋊C4—64no—
518C22⋊C4—64no—
519C42—21no—
520C42—21no—
521C42—32no—
522C42—32no—
523C42—61no—
524C42—61no—
525C42—122no—
526C22×C3—24no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3C2×#12011yescharacteristic
534C22×C3#54811yescharacteristic
535C22×C3D4×Q811yescharacteristic
536C22×C3#57511yescharacteristic
537C22×C3#57511yescharacteristic
538C22×C3#57711yescharacteristic
539C22×C3#57711yescharacteristic
540C3×C4—41no—
541C3×C4—42no—
542C3×C4—41no—
543C3×C4—81no—
544C3×C4—81no—
545C3×C4C2×C22⋊Q811yescharacteristic
546C3×C4#53611yescharacteristic
547C3×C4#54811yescharacteristic
548C3×C4D4×Q811yescharacteristic
549Dic3—22no—
550Dic3—22no—
551Dic3—42no—
552Dic3—41no—
553Dic3—42no—
554Dic3—42no—
555Dic3—41no—
556Dic3—81no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23C2×D3×D411yescharacteristic
565Q8—62no—
566Q8—62no—
567Q8—124no—
568Q8—121no—
569Q8—121no—
570Q8—244no—
571D4—22no—
572D4—22no—
573D4—22no—
574D4#89614yes—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—32no—
583C2×C4—32no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—62no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—84no—
606C2×C4—128no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—244no—
617C2×C4C2×#23511yescharacteristic
618C2×C4C2×D3×D411yescharacteristic
619C2×C4C2×D3×D411yescharacteristic
620C2×C4C2×D3×Q811yescharacteristic
621C2×C4#101011yescharacteristic
622C2×C4#101011yescharacteristic
623C8—41no—
624C8—41no—
625C8—62no—
626C8—121no—
627C2×C3—44no—
628C2×C3#240911yescharacteristic
629C2×C3#300511yescharacteristic
630C2×C3#300911yescharacteristic
631C2×C3#337611yescharacteristic
632C2×C3#337611yescharacteristic
633C2×C3#338811yescharacteristic
634C2×C3#338811yescharacteristic
635C22—24no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22#555211yescharacteristic
643C22D3×#12011yescharacteristic
644C22#563211yescharacteristic
645C22#576211yescharacteristic
646C22#576811yescharacteristic
647C22#577311yescharacteristic
648C22#577511yescharacteristic
649C4—41no—
650C4—42no—
651C4—41no—
652C4—62no—
653C4—62no—
654C4—81no—
655C4—81no—
656C4—121no—
657C4—122no—
658C4—122no—
659C4—121no—
660C4—122no—
661C4—241no—
662C4C2×#89611yescharacteristic
663C4#555211yescharacteristic
664C4#563211yescharacteristic
665C4#564811yescharacteristic
666C3#2434311yescharacteristic, complemented, Sylow
667C2—44no—
668C2#7440511yescharacteristic
669C2#7963011yescharacteristic
670C2#7985411yescharacteristic
671C2#8008311yescharacteristic
672C2#8014811yescharacteristic
673C2#8105711yescharacteristic
674C2#8116011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1864C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1864D3
hypocenterC3#24343
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. C2×#4854C2semidirect
4. #76076C2semidirect
7. #70566C2semidirect
11. #69606C2semidirect
12. #68911C2semidirect
13. #68806C2semidirect
14. #68690C2semidirect
15. #68620C2semidirect
104. C2×#214Q8semidirect
666. C3#24343semidirect

Maximal subgroups

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

subgroupindex
C2×#55392
C2×#48542
C2×#47572
#760762
C3×#18642
#716122
#705662
#701712
#701692
#701542
#696062
#689112
#688062
#686902
#686202
#243433