Finite Group Databasebeta

← #10835909

Subgroups — #10835909

Order 768.

814 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
1C2×#5182C211yesmaximal, characteristic, complemented
2#82786C211yesmaximal, characteristic
3#82744C211yesmaximal, characteristic
4C2×#4720C211yesmaximal, characteristic
5C3×#2001C211yesmaximal, characteristic
6#72066C211yesmaximal, characteristic, complemented
7#71700C211yesmaximal, characteristic
8#71694C211yesmaximal, characteristic
9#70259C211yesmaximal, characteristic, complemented
10#70255C211yesmaximal, characteristic, complemented
11#70229C211yesmaximal, characteristic, complemented
12#70222C211yesmaximal, characteristic, complemented
13#69937C211yesmaximal, characteristic
14#68851C211yesmaximal, characteristic, complemented
15#68798C211yesmaximal, characteristic, complemented
16#24664—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×#944C2211yescharacteristic
22C22×#216C2211yescharacteristic
23C2×#926C2211yescharacteristic
24C2×#924C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5212C2211yescharacteristic
34C2×#830C2211yescharacteristic
35#5182—24no—
36C2×#818C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#815C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43#4984—21no—
44#4984C2211yescharacteristic
45#4982—21no—
46#4982—21no—
47#4980—21no—
48#4977C2211yescharacteristic
49Dic3×C4⋊C4C2211yescharacteristic
50#4973C2211yescharacteristic
51C2×#810C2211yescharacteristic
52C2×#808C2211yescharacteristic
53#4952—21no—
54#4952—21no—
55#4951—21no—
56#4951—21no—
57#4720—24no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#462—31no—
63C2×#433—31no—
64#2621—31no—
65#2323—31no—
66#2036—31no—
67#2035—31no—
68#2001—31no—
69#2001D311yescharacteristic
70#1981—31no—
71#1971—31no—
72#1969—31no—
73#1949—31no—
74#1935—31no—
75#1933—31no—
76#1805—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C2×C3×Q16—22no—
82C2×C3×Q16—22no—
83C2×C3×Q16—41no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—41no—
89C3×C22⋊Q8—41no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217D411yescharacteristic
99C2×#217D411yescharacteristic
100Q8×Dic3—22no—
101Q8×Dic3—22no—
102#944—22no—
103#944—22no—
104C2×#216—22no—
105C2×#216—22no—
106C2×#216—41no—
107#926—22no—
108#926—22no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3D411yescharacteristic
112C2×C4⋊Dic3D411yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116#924—41no—
117#924—41no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3C2311yescharacteristic
124C2×C4×Dic3C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C3×#103—22no—
128C3×#103—41no—
129C3×C22⋊C8—22no—
130#835—22no—
131#835—41no—
132#835—41no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#830—22no—
145#818—41no—
146#816—22no—
147#816—41no—
148#815—22no—
149Dic3⋊C8—22no—
150Dic3⋊C8D414yes—
151C8×Dic3—22no—
152#810—22no—
153#808—41no—
154C22×Q16—31no—
155C22×Q16C2×D311yescharacteristic
156C2×C4⋊Q8—31no—
157C2×C22⋊Q8—31no—
158C2×C22⋊Q8C2×D311yescharacteristic
159C2×C22⋊Q8C2×D311yescharacteristic
160C2×C4×Q8—31no—
161C2×#117—31no—
162#462—64no—
163#433—64no—
164C2×#107—31no—
165C2×#106—31no—
166C2×C4⋊C8—31no—
167C2×C4⋊C8—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103—31no—
173C2×#103—31no—
174C2×#103C2×D311yescharacteristic
175C2×#103C2×D311yescharacteristic
176C2×C22⋊C8—31no—
177C2×C22⋊C8C2×D311yescharacteristic
178C2×#97—31no—
179C2×C4×C8—31no—
180#398—61no—
181#398—61no—
182#397—61no—
183#397—61no—
184#391—31no—
185#391—31no—
186#391—31no—
187#389—31no—
188#389—61no—
189#388—61no—
190#388—61no—
191#387—61no—
192#382—31no—
193#380—31no—
194C4×C4⋊C4—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340C2×D311yescharacteristic
199C24×C3C2×D411yescharacteristic
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8#4112yes—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C3×Q16—44no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8C2×D412yes—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4#4112yes—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—42no—
253#217—22no—
254#217—22no—
255#217—42no—
256#217—81no—
257#216—44no—
258C4⋊Dic3—22no—
259C4⋊Dic3—22no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—42no—
267C4⋊Dic3—81no—
268Dic3⋊C4—42no—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C4×Dic3—42no—
274C4×Dic3—81no—
275C4×Dic3C2×D412yes—
276C4×Dic3#4112yes—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8C2×D412yes—
283C22×Q8—31no—
284C22×Q8C22×D311yescharacteristic
285C22×Q8C22×D311yescharacteristic
286C23×C4C22×D311yescharacteristic
287C2×Q16—22no—
288C2×Q16—22no—
289C2×Q16—41no—
290C2×Q16—62no—
291C2×Q16—62no—
292C2×Q16—121no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297C4⋊Q8—62no—
298C4⋊Q8—62no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301C22⋊Q8—41no—
302C22⋊Q8—41no—
303C22⋊Q8—121no—
304C22⋊Q8—121no—
305C4×Q8—62no—
306C4×Q8—62no—
307#117—62no—
308#117—62no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—31no—
326C2×C22⋊C4—31no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C42—31no—
330C2×C42—31no—
331C2×C42—61no—
332C2×C42—61no—
333#107—121no—
334#106—121no—
335C4⋊C8—34no—
336C4⋊C8—62no—
337#103—22no—
338#103—41no—
339#103—62no—
340#103—62no—
341#103—62no—
342#103—121no—
343#103—121no—
344#103—121no—
345C22⋊C8—22no—
346C22⋊C8—62no—
347#97—62no—
348C4×C8—62no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3C22×D411yescharacteristic
368C3×Q8—22no—
369C3×Q8—22no—
370C3×Q8—41no—
371C3×Q8—41no—
372C3×Q8—41no—
373C3×Q8—41no—
374C3×Q8—41no—
375C3×Q8—41no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
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×C4C22×D411yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4H(5,𝔽2)11yescharacteristic
400C2×C3×C4H(5,𝔽2)11yescharacteristic
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3#12112yes—
424C2×Dic3#12112yes—
425Dic6—41no—
426Dic6—41no—
427Dic6—41no—
428Dic6—41no—
429C3×C8—22no—
430C3×C8—41no—
431C3⋊C8—22no—
432C3⋊C8—41no—
433C24D3×D411yescharacteristic
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—61no—
446C2×Q8—61no—
447C2×Q8—61no—
448C2×Q8—61no—
449C2×Q8—61no—
450C2×Q8—61no—
451C2×Q8#23712yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4D3×D411yescharacteristic
471C22×C4C23×D311yescharacteristic
472Q16—44no—
473Q16—124no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—32no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8D3×D412yes—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—62no—
492C4⋊C4—62no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—122no—
500C4⋊C4—122no—
501C4⋊C4—122no—
502C4⋊C4—122no—
503C4⋊C4—241no—
504C4⋊C4#23712yes—
505C22⋊C4—24no—
506C22⋊C4—42no—
507C22⋊C4—62no—
508C22⋊C4—62no—
509C22⋊C4—122no—
510C22⋊C4—241no—
511C42—32no—
512C42—32no—
513C42—61no—
514C42—61no—
515C42—122no—
516C42—241no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—41no—
524C22×C3C2×#12111yescharacteristic
525C22×C3#52511yescharacteristic
526C22×C3#53511yescharacteristic
527C22×C3#57711yescharacteristic
528C22×C3#57811yescharacteristic
529C22×C3#57811yescharacteristic
530C22×C3#57811yescharacteristic
531C3×C4—42no—
532C3×C4—41no—
533C3×C4—42no—
534C3×C4—81no—
535C3×C4—81no—
536C3×C4#54611yescharacteristic
537C3×C4#55011yescharacteristic
538C3×C4#55311yescharacteristic
539C3×C4#55511yescharacteristic
540Dic3—22no—
541Dic3—22no—
542Dic3—41no—
543Dic3—41no—
544Dic3—81no—
545Dic3—81no—
546Dic3—81no—
547Dic3—81no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23C2×D3×D411yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—41no—
560Q8—41no—
561Q8—41no—
562Q8—41no—
563Q8—41no—
564Q8—121no—
565Q8—121no—
566Q8—121no—
567Q8—121no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—32no—
574C2×C4—32no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4C2×D3×D411yescharacteristic
612C2×C4C2×#23711yescharacteristic
613C2×C4C2×#23711yescharacteristic
614C2×C4#100411yescharacteristic
615C2×C4#100411yescharacteristic
616C2×C4D3×#4111yescharacteristic
617C8—22no—
618C8—41no—
619C8—62no—
620C8—121no—
621C2×C3—81no—
622C2×C3#241411yescharacteristic
623C2×C3#305911yescharacteristic
624C2×C3#306311yescharacteristic
625C2×C3#312011yescharacteristic
626C2×C3#312711yescharacteristic
627C2×C3#322311yescharacteristic
628C2×C3#322511yescharacteristic
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22#558611yescharacteristic
637C22#559211yescharacteristic
638C22D3×#12111yescharacteristic
639C22#576611yescharacteristic
640C22#577111yescharacteristic
641C22#577811yescharacteristic
642C22#577911yescharacteristic
643C4—42no—
644C4—41no—
645C4—42no—
646C4—62no—
647C4—62no—
648C4—81no—
649C4—81no—
650C4—121no—
651C4—121no—
652C4—241no—
653C4—241no—
654C4—241no—
655C4—241no—
656C4#559111yescharacteristic
657C4#560111yescharacteristic
658C4#561611yescharacteristic
659C4#571811yescharacteristic
660C3#2466411yescharacteristic, complemented, Sylow
661C2—81no—
662C2#7667011yescharacteristic
663C2#8017711yescharacteristic
664C2#8020311yescharacteristic
665C2#8021711yescharacteristic
666C2#8027111yescharacteristic
667C2#8076011yescharacteristic
668C2#8090711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2001C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2001D3
hypocenterC3#24664
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

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

NH ≅ G/Ntype
1. C2×#5182C2semidirect
6. #72066C2semidirect
9. #70259C2semidirect
10. #70255C2semidirect
11. #70229C2semidirect
12. #70222C2semidirect
14. #68851C2semidirect
15. #68798C2semidirect
660. C3#24664semidirect

Maximal subgroups

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

subgroupindex
C2×#51822
#827862
#827442
C2×#47202
C3×#20012
#720662
#717002
#716942
#702592
#702552
#702292
#702222
#699372
#688512
#687982
#246643