Finite Group Databasebeta

← #11712889

Subgroups — #11712889

Order 768.

811 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 811 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5912C211yesmaximal, characteristic
2C2×#5400C211yesmaximal, characteristic, complemented
3C2×C8×S4C211yesmaximal, characteristic, complemented
4A4×C22⋊C8C211yesmaximal, characteristic, complemented
5#73141C211yesmaximal, characteristic, complemented
6(C2×S4)⋊C8C211yesmaximal, characteristic, complemented
7(C2×S4)⋊C8C211yesmaximal, characteristic, complemented
8#22398—31nomaximal, Sylow
9C22×C4×A4C2211yescharacteristic
10C2×GL(2,ℤ/4)C411yescharacteristic
11C2×GL(2,ℤ/4)C411yescharacteristic
12#5912—22no—
13#5912—22no—
14C2×C4⋊S4C411yescharacteristic
15C2×C4×S4C2211yescharacteristic
16C2×C4×S4C2211yescharacteristic
17C2×A4⋊Q8C411yescharacteristic
18C2×C8×A4C2211yescharacteristic
19C2×C8×A4C2211yescharacteristic
20C2×A4⋊C8C2211yescharacteristic
21C2×A4⋊C8C2211yescharacteristic
22#5400—22no—
23C8×S4—22no—
24#4727—41nomaximal
25C2×#525—31no—
26C2×#435—31no—
27C2×C8×D4—31no—
28C2×#408—31no—
29C22×C22⋊C8D311yescharacteristic, complemented
30#1955—31no—
31#1954—31no—
32#1870—31no—
33#1869—31no—
34#1869—31no—
35#1869—31no—
36#1798—31no—
37#1780—31no—
38#1779—31no—
39#1750—31no—
40C23×A4C2×C411yescharacteristic
41C22×S4C2×C411yescharacteristic
42C22×S4C2×C411yescharacteristic
43C2×#235—41no—
44C2×C4×A4—21no—
45C2×C4×A4—21no—
46C2×C4×A4C2×C411yescharacteristic
47C2×C4×A4C2311yescharacteristic
48GL(2,ℤ/4)—22no—
49GL(2,ℤ/4)—22no—
50C2×SL(2,ℤ/4)C2×C411yescharacteristic
51C2×SL(2,ℤ/4)C2×C411yescharacteristic
52C4⋊S4—22no—
53C4×S4—21no—
54C4×S4—21no—
55C4×S4D412yes—
56C4×S4D412yes—
57A4⋊Q8—22no—
58C2×ΓL(1,𝔽25)—41no—
59C2×D3×C8—41no—
60C8×A4—21no—
61C8×A4—21no—
62A4⋊C8—21no—
63A4⋊C8—21no—
64C3×C22⋊C8—41no—
65#830—41no—
66(C2×D3)⋊C8—41no—
67(C2×D3)⋊C8—41no—
68C22×#41—31no—
69C24×C4C2×D311yescharacteristic
70C22×M16—31no—
71C23×C8—31no—
72C23×C8C2×D311yescharacteristic
73C23×C8C2×D311yescharacteristic
74#525—62no—
75#525—62no—
76#525—62no—
77#525—62no—
78C2×#123—31no—
79C2×#123—31no—
80C2×C22⋊Q8—31no—
81C2×C22⋊Q8—31no—
82C2×#121—31no—
83C2×#121—31no—
84C2×#120—31no—
85C2×#120—31no—
86C2×C4×D4—31no—
87C2×C4×D4—31no—
88C2×C4×D4—31no—
89C2×C4×D4—31no—
90C2×#117—31no—
91#435—62no—
92#435—62no—
93C8×D4—62no—
94C8×D4—62no—
95C2×C4⋊C8—31no—
96C2×C4⋊C8—31no—
97#408—61no—
98#408—61no—
99#408—61no—
100#408—61no—
101C2×C22⋊C8—31no—
102C2×C22⋊C8—31no—
103C2×C22⋊C8—31no—
104C2×C22⋊C8—31no—
105C2×C22⋊C8—31no—
106C2×C22⋊C8—31no—
107C2×C22⋊C8—31no—
108C2×C22⋊C8—31no—
109C2×C22⋊C8—31no—
110C2×C22⋊C8—31no—
111C2×C22⋊C8—31no—
112C2×C22⋊C8—31no—
113C2×C22⋊C8—31no—
114C2×C22⋊C8—31no—
115C2×C22⋊C8—31no—
116C2×C22⋊C8—31no—
117C2×#97—31no—
118C2×C4×C8—31no—
119#336—31no—
120#336—31no—
121#336—31no—
122#336—31no—
123C22×A4—21no—
124C22×A4—21no—
125C22×A4—21no—
126C22×A4C22×C411yescharacteristic
127C2×S4—21no—
128C2×S4—21no—
129C2×S4—21no—
130C2×S4—21no—
131C2×S4—21no—
132C2×S4C22⋊C412yes—
133C22×C3×C4—41no—
134C2×#67—41no—
135C2×#67—41no—
136#235—82no—
137#235—82no—
138C2×D12—41no—
139C2×C4×D3—41no—
140C2×C4×D3—41no—
141C2×Dic6—41no—
142C4×A4—21no—
143C4×A4C2×D411yescharacteristic
144C4×A4C2×D411yescharacteristic
145SL(2,ℤ/4)—21no—
146SL(2,ℤ/4)C22⋊C412yes—
147C2×C3×C8—41no—
148C2×C3×C8—41no—
149C2×C3⋊C8—41no—
150C2×C3⋊C8—41no—
151ΓL(1,𝔽25)—82no—
152D3×C8—82no—
153C25C4×D311yescharacteristic
154C2×#41—62no—
155C2×#41—62no—
156C2×#41—61no—
157C2×#41—61no—
158C2×#41—61no—
159C2×#41—61no—
160C2×#41—121no—
161C2×#41—121no—
162C22×Q8—31no—
163C22×D4—31no—
164C22×D4—31no—
165C22×D4—31no—
166C22×D4—31no—
167C22×D4—31no—
168C22×D4—31no—
169C22×D4—31no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—31no—
173C23×C4—31no—
174C23×C4—31no—
175C23×C4—31no—
176C23×C4—31no—
177C23×C4—31no—
178C23×C4—61no—
179C23×C4—61no—
180C23×C4C4×D311yescharacteristic
181C23×C4C22×D311yescharacteristic
182C2×M16—61no—
183C2×M16—62no—
184C2×M16—61no—
185C2×M16—121no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—31no—
189C22×C8—31no—
190C22×C8—31no—
191C22×C8—31no—
192C22×C8—31no—
193C22×C8—31no—
194C22×C8—31no—
195C22×C8—31no—
196C22×C8—61no—
197C22×C8—61no—
198C22×C8—61no—
199C22×C8—62no—
200C22×C8—61no—
201C22×C8—121no—
202#123—62no—
203#123—62no—
204#123—62no—
205#123—62no—
206C22⋊Q8—62no—
207C22⋊Q8—62no—
208C22⋊Q8—62no—
209C22⋊Q8—62no—
210#121—62no—
211#121—62no—
212#121—62no—
213#121—62no—
214#120—62no—
215#120—62no—
216#120—62no—
217#120—62no—
218C4×D4—32no—
219C4×D4—32no—
220C4×D4—32no—
221C4×D4—32no—
222C4×D4—62no—
223C4×D4—62no—
224C4×D4—62no—
225C4×D4—62no—
226C4×D4—61no—
227C4×D4—61no—
228C4×D4—61no—
229C4×D4—61no—
230#117—62no—
231#117—62no—
232C2×C4⋊C4—31no—
233C2×C4⋊C4—31no—
234C2×C4⋊C4—31no—
235C2×C4⋊C4—31no—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—31no—
238C2×C22⋊C4—31no—
239C2×C22⋊C4—31no—
240C2×C22⋊C4—31no—
241C2×C22⋊C4—31no—
242C2×C22⋊C4—31no—
243C2×C22⋊C4—31no—
244C2×C22⋊C4—31no—
245C2×C22⋊C4—31no—
246C2×C22⋊C4—31no—
247C2×C22⋊C4—31no—
248C2×C42—31no—
249C2×C42—31no—
250C4⋊C8—62no—
251C4⋊C8—62no—
252C22⋊C8—31no—
253C22⋊C8—31no—
254C22⋊C8—31no—
255C22⋊C8—61no—
256C22⋊C8—61no—
257C22⋊C8—61no—
258C22⋊C8—61no—
259C22⋊C8—61no—
260C22⋊C8—61no—
261C22⋊C8—62no—
262C22⋊C8—61no—
263C22⋊C8—62no—
264C22⋊C8—61no—
265C22⋊C8—61no—
266C22⋊C8—61no—
267C22⋊C8—61no—
268C22⋊C8—61no—
269C22⋊C8—61no—
270C22⋊C8—121no—
271C22⋊C8—121no—
272C22⋊C8—121no—
273C22⋊C8—121no—
274C22⋊C8S411yescharacteristic, complemented
275#97—62no—
276C4×C8—62no—
277C23×C3—41no—
278C22×D3—41no—
279C22×D3—41no—
280C2×A4—41no—
281C2×A4C2×C22⋊C411yescharacteristic
282C2×A4#13111yescharacteristic
283C2×A4#13111yescharacteristic
284S4—22no—
285S4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—81no—
289C2×C3×C4—81no—
290#67—82no—
291#67—82no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294D12—82no—
295C4×D3—42no—
296C4×D3—42no—
297C4×D3—81no—
298C4×D3—81no—
299Dic6—82no—
300C3×C8—81no—
301C3×C8—81no—
302C3⋊C8—81no—
303C3⋊C8—81no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—31no—
308C24—31no—
309C24—31no—
310C24—31no—
311C24—61no—
312C24—61no—
313C24—61no—
314C24C2×C4×D311yescharacteristic
315#41—122no—
316#41—122no—
317#41—122no—
318#41—122no—
319#41—122no—
320#41—122no—
321#41—122no—
322#41—122no—
323C2×Q8—62no—
324C2×Q8—61no—
325C2×Q8—61no—
326C2×Q8—121no—
327C2×D4—32no—
328C2×D4—32no—
329C2×D4—62no—
330C2×D4—62no—
331C2×D4—61no—
332C2×D4—61no—
333C2×D4—61no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—61no—
337C2×D4—62no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—61no—
342C2×D4—61no—
343C2×D4—62no—
344C2×D4—61no—
345C2×D4—61no—
346C2×D4—61no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4—61no—
350C2×D4—61no—
351C2×D4—61no—
352C2×D4—121no—
353C2×D4—121no—
354C2×D4—121no—
355C2×D4—121no—
356C2×D4—121no—
357C22×C4—21no—
358C22×C4—31no—
359C22×C4—31no—
360C22×C4—31no—
361C22×C4—31no—
362C22×C4—32no—
363C22×C4—32no—
364C22×C4—31no—
365C22×C4—31no—
366C22×C4—31no—
367C22×C4—31no—
368C22×C4—31no—
369C22×C4—31no—
370C22×C4—31no—
371C22×C4—31no—
372C22×C4—31no—
373C22×C4—31no—
374C22×C4—31no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—121no—
400C22×C4—121no—
401C22×C4D3×D411yescharacteristic
402C22×C4D3×D411yescharacteristic
403C22×C4C2×S411yescharacteristic
404M16—122no—
405M16—122no—
406C2×C8—31no—
407C2×C8—31no—
408C2×C8—61no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—122no—
429C2×C8—122no—
430C2×C8C2×S411yescharacteristic
431C2×C8C2×S411yescharacteristic
432C4⋊C4—32no—
433C4⋊C4—32no—
434C4⋊C4—62no—
435C4⋊C4—62no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—62no—
439C4⋊C4—62no—
440C22⋊C4—32no—
441C22⋊C4—32no—
442C22⋊C4—32no—
443C22⋊C4—32no—
444C22⋊C4—62no—
445C22⋊C4—62no—
446C22⋊C4—62no—
447C22⋊C4—62no—
448C22⋊C4—61no—
449C22⋊C4—61no—
450C22⋊C4—62no—
451C22⋊C4—61no—
452C22⋊C4—62no—
453C22⋊C4—61no—
454C42—32no—
455C42—32no—
456C42—61no—
457C42—61no—
458C22×C3—41no—
459C22×C3—81no—
460C22×C3—81no—
461C22×C3—81no—
462C2×D3—42no—
463C2×D3—81no—
464C2×D3—81no—
465C2×D3—81no—
466C2×D3—81no—
467C2×D3—81no—
468A4#40811yescharacteristic, complemented
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—81no—
472Dic3—42no—
473Dic3—81no—
474C23—31no—
475C23—31no—
476C23—31no—
477C23—31no—
478C23—32no—
479C23—32no—
480C23—31no—
481C23—41no—
482C23—61no—
483C23—61no—
484C23—61no—
485C23—61no—
486C23—61no—
487C23—61no—
488C23—61no—
489C23—61no—
490C23—61no—
491C23—61no—
492C23—61no—
493C23—61no—
494C23—61no—
495C23—61no—
496C23—61no—
497C23—61no—
498C23—61no—
499C23—61no—
500C23—61no—
501C23—61no—
502C23—61no—
503C23—61no—
504C23—61no—
505C23—61no—
506C23—121no—
507C23—121no—
508C23—121no—
509C23—122no—
510C23D3×C22⋊C411yescharacteristic
511C23#90111yescharacteristic
512C23#90711yescharacteristic
513C23C4×S411yescharacteristic
514Q8—122no—
515Q8—122no—
516D4—62no—
517D4—62no—
518D4—62no—
519D4—62no—
520D4—122no—
521D4—122no—
522D4—124no—
523D4—122no—
524D4—122no—
525D4—122no—
526D4—121no—
527D4—121no—
528D4—121no—
529D4—124no—
530D4—122no—
531D4—121no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—31no—
535C2×C4—31no—
536C2×C4—31no—
537C2×C4—31no—
538C2×C4—31no—
539C2×C4—31no—
540C2×C4—31no—
541C2×C4—32no—
542C2×C4—32no—
543C2×C4—32no—
544C2×C4—31no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4C4×S411yescharacteristic
582C2×C4C22×S411yescharacteristic
583C8—21no—
584C8—21no—
585C8—61no—
586C8—61no—
587C8—121no—
588C8—121no—
589C8—121no—
590C8—121no—
591C2×C3—41no—
592C2×C3—41no—
593C2×C3—41no—
594C2×C3—161no—
595D3—82no—
596D3—161no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—31no—
601C22—31no—
602C22—31no—
603C22—31no—
604C22—31no—
605C22—31no—
606C22—31no—
607C22—31no—
608C22—31no—
609C22—61no—
610C22—62no—
611C22—62no—
612C22—62no—
613C22—62no—
614C22—61no—
615C22—62no—
616C22—61no—
617C22—61no—
618C22—121no—
619C22—121no—
620C22—121no—
621C22—121no—
622C22—121no—
623C22—121no—
624C22—121no—
625C22—121no—
626C22—121no—
627C22—121no—
628C22—121no—
629C22—121no—
630C22—121no—
631C22—121no—
632C22—121no—
633C22—121no—
634C22—121no—
635C22#472711yescharacteristic, complemented
636C22C2×C4×S411yescharacteristic
637C4—21no—
638C4—31no—
639C4—31no—
640C4—62no—
641C4—62no—
642C4—61no—
643C4—62no—
644C4—121no—
645C4—121no—
646C4—121no—
647C4D4×S411yescharacteristic
648C4D4×S411yescharacteristic
649C3—41noSylow
650C2—31no—
651C2—31no—
652C2—31no—
653C2—31no—
654C2—41no—
655C2—121no—
656C2—122no—
657C2—241no—
658C2C22⋊C4×S411yescharacteristic
659C2#8543711yescharacteristic
660C2#8544211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC22×A4C22×C4
FittingC22×C22⋊C8D3
socleC24C2×C4×D3
prime radicalC2×#5912C2
cosocleC2×C4×A4C23
hypercenterC22⋊C8S4
hypocenterA4#408
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#5400C2semidirect
3. C2×C8×S4C2semidirect
4. A4×C22⋊C8C2semidirect
5. #73141C2semidirect
6. (C2×S4)⋊C8C2semidirect
7. (C2×S4)⋊C8C2semidirect
29. C22×C22⋊C8D3semidirect
274. C22⋊C8S4semidirect
468. A4#408semidirect
635. C22#4727semidirect

Maximal subgroups

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

subgroupindex
C2×#59122
C2×#54002
C2×C8×S42
A4×C22⋊C82
#731412
(C2×S4)⋊C82
#223983
#47274