Finite Group Databasebeta

← #11712891

Subgroups — #11712891

Order 768.

698 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C4×D4×A4C211yesmaximal, characteristic
2C2×#5409C211yesmaximal, characteristic, complemented
3C22×A4⋊C8C211yesmaximal, characteristic, complemented
4#73141C211yesmaximal, characteristic, complemented
5#73141C211yesmaximal, characteristic, complemented
6C4⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
7#73044C211yesmaximal, characteristic, complemented
8#22399—31nomaximal, Sylow
9C2×D4×A4C411yescharacteristic
10C22×C4×A4C2211yescharacteristic
11C22×C4×A4C2211yescharacteristic
12A4×C4⋊C4C411yescharacteristic
13A4×C22⋊C4C411yescharacteristic
14A4×C22⋊C4C411yescharacteristic
15C42×A4C2211yescharacteristic
16#5409—22no—
17C2×A4⋊C8—22no—
18C2×A4⋊C8C2211yescharacteristic
19C2×A4⋊C8C2211yescharacteristic
20C2×A4⋊C8C2211yescharacteristic
21C2×A4⋊C8C2211yescharacteristic
22#5011—41nomaximal
23C22×C4×D4D311yescharacteristic
24C2×#435—31no—
25C2×#435—31no—
26C2×#407—31no—
27C22×C22⋊C8—31no—
28#1955—31no—
29#1955—31no—
30#1870—31no—
31#1870—31no—
32#1869—31no—
33#1869—31no—
34#1797—31no—
35#1779—31no—
36#1779—31no—
37#1749—31no—
38C23×A4C2×C411yescharacteristic
39C23×A4C2×C411yescharacteristic
40D4×A4—22no—
41C2×C4×A4—21no—
42C2×C4×A4—21no—
43C2×C4×A4—21no—
44C2×C4×A4—21no—
45C2×C4×A4C2×C411yescharacteristic
46C2×C4×A4C2×C411yescharacteristic
47C2×C4×A4C2×C411yescharacteristic
48C2×C4×A4C2×C411yescharacteristic
49C2×C4×A4C2311yescharacteristic
50C3×C4×D4—41no—
51C2×#208—41no—
52C22×C3⋊C8—41no—
53A4⋊C8—21no—
54A4⋊C8—21no—
55A4⋊C8—21no—
56A4⋊C8D412yescomplemented
57#830—41no—
58#830—41no—
59C4⋊(C3⋊C8)—41no—
60#803—41no—
61C23×D4Dic311yescharacteristic
62C24×C4C2×D311yescharacteristic
63C24×C4C2×D311yescharacteristic
64C22×M16—31no—
65C22×M16—31no—
66C23×C8—31no—
67C23×C8—31no—
68C2×C4×D4—31no—
69C2×C4×D4—31no—
70C2×C4×D4—31no—
71C2×C4×D4—31no—
72C2×C4×D4—31no—
73C2×C4×D4—31no—
74C2×C4×D4—31no—
75C2×C4×D4—31no—
76C22×C4⋊C4Dic311yescharacteristic
77C22×C22⋊C4Dic311yescharacteristic
78C22×C22⋊C4Dic311yescharacteristic
79C22×C42C2×D311yescharacteristic
80#435—61no—
81#435—61no—
82#435—61no—
83#435—61no—
84#435—61no—
85#435—61no—
86#435—61no—
87#435—61no—
88C2×C4⋊C8—31no—
89C2×C4⋊C8—31no—
90#407—62no—
91#407—62no—
92C2×C22⋊C8—31no—
93C2×C22⋊C8—31no—
94C2×C22⋊C8—31no—
95C2×C22⋊C8—31no—
96C2×C22⋊C8—31no—
97C2×C22⋊C8—31no—
98C2×C22⋊C8—31no—
99C2×C22⋊C8—31no—
100C2×C22⋊C8—31no—
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—62no—
109C2×C22⋊C8—62no—
110C2×#97—31no—
111C2×#97—31no—
112#336—31no—
113#336—31no—
114#336—31no—
115#336—31no—
116C22×A4—21no—
117C22×A4—21no—
118C22×A4—21no—
119C22×A4—21no—
120C22×A4—21no—
121C22×A4M1612yescomplemented
122C22×A4C22×C411yescharacteristic
123C2×C3×D4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C4×A4—21no—
127C4×A4—21no—
128C4×A4—21no—
129C4×A4—21no—
130C4×A4C2×D411yescharacteristic
131C4×A4#4111yescharacteristic
132C3×C4⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42—41no—
136#208—82no—
137C2×C3⋊C8—41no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8—41no—
141C2×C3⋊C8—82no—
142C25C2×Dic311yescharacteristic
143C25C2×Dic311yescharacteristic
144C22×D4—22no—
145C22×D4—31no—
146C22×D4—31no—
147C22×D4—31no—
148C22×D4—31no—
149C22×D4—62no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—31no—
155C23×C4—31no—
156C23×C4—31no—
157C23×C4—31no—
158C23×C4—31no—
159C23×C4—31no—
160C23×C4—31no—
161C23×C4—31no—
162C23×C4—61no—
163C23×C4—61no—
164C23×C4—61no—
165C23×C4—61no—
166C23×C4C2×Dic311yescharacteristic
167C23×C4C2×Dic311yescharacteristic
168C23×C4C2×Dic311yescharacteristic
169C23×C4C2×Dic311yescharacteristic
170C23×C4C22×D311yescharacteristic
171C2×M16—62no—
172C2×M16—62no—
173C2×M16—61no—
174C2×M16—61no—
175C2×M16—61no—
176C2×M16—61no—
177C2×M16—121no—
178C2×M16—121no—
179C22×C8—31no—
180C22×C8—31no—
181C22×C8—31no—
182C22×C8—31no—
183C22×C8—31no—
184C22×C8—31no—
185C22×C8—31no—
186C22×C8—31no—
187C22×C8—62no—
188C22×C8—62no—
189C22×C8—61no—
190C22×C8—61no—
191C22×C8—61no—
192C22×C8—61no—
193C22×C8—121no—
194C22×C8—121no—
195C4×D4—31no—
196C4×D4—31no—
197C4×D4—31no—
198C4×D4—31no—
199C4×D4—31no—
200C4×D4—31no—
201C4×D4—31no—
202C4×D4—61no—
203C4×D4—61no—
204C4×D4—61no—
205C4×D4—61no—
206C4×D4—61no—
207C4×D4—61no—
208C4×D4—61no—
209C4×D4S411yescharacteristic
210C2×C4⋊C4—31no—
211C2×C4⋊C4—31no—
212C2×C4⋊C4—31no—
213C2×C4⋊C4—31no—
214C2×C22⋊C4—31no—
215C2×C22⋊C4—31no—
216C2×C22⋊C4—31no—
217C2×C22⋊C4—31no—
218C2×C22⋊C4—31no—
219C2×C22⋊C4—31no—
220C2×C22⋊C4—31no—
221C2×C22⋊C4—31no—
222C2×C42—31no—
223C2×C42—31no—
224C2×C42—31no—
225C2×C42—31no—
226C4⋊C8—61no—
227C4⋊C8—61no—
228C4⋊C8—61no—
229C4⋊C8—61no—
230C22⋊C8—32no—
231C22⋊C8—32no—
232C22⋊C8—62no—
233C22⋊C8—62no—
234C22⋊C8—62no—
235C22⋊C8—62no—
236C22⋊C8—61no—
237C22⋊C8—61no—
238C22⋊C8—61no—
239C22⋊C8—61no—
240C22⋊C8—61no—
241C22⋊C8—61no—
242C22⋊C8—61no—
243C22⋊C8—61no—
244C22⋊C8—61no—
245C22⋊C8—61no—
246C22⋊C8—61no—
247C22⋊C8—61no—
248C22⋊C8—61no—
249C22⋊C8—61no—
250C22⋊C8—121no—
251C22⋊C8—121no—
252C22⋊C8—121no—
253C22⋊C8—121no—
254#97—61no—
255#97—61no—
256#97—61no—
257#97—61no—
258C23×C3—41no—
259C23×C3—41no—
260C2×A4—22no—
261C2×A4—41no—
262C2×A4C4×D411yescharacteristic
263C2×A4C2×M1611yescharacteristic
264C2×A4#13111yescharacteristic
265C3×D4—82no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—81no—
272C2×C3×C4—81no—
273C2×C3×C4—81no—
274C2×C3×C4—81no—
275C3⋊C8—42no—
276C3⋊C8—81no—
277C3⋊C8—81no—
278C3⋊C8—81no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—31no—
285C24—31no—
286C24—31no—
287C24—32no—
288C24—31no—
289C24—61no—
290C24—61no—
291C24—61no—
292C24—61no—
293C24—61no—
294C24#20812yescomplemented
295C24C22×Dic311yescharacteristic
296C2×D4—34no—
297C2×D4—31no—
298C2×D4—31no—
299C2×D4—31no—
300C2×D4—61no—
301C2×D4—62no—
302C2×D4—62no—
303C2×D4—62no—
304C2×D4—62no—
305C2×D4—62no—
306C2×D4—62no—
307C2×D4—62no—
308C2×D4SL(2,ℤ/4)11yescharacteristic
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—31no—
314C22×C4—31no—
315C22×C4—31no—
316C22×C4—31no—
317C22×C4—31no—
318C22×C4—31no—
319C22×C4—31no—
320C22×C4—31no—
321C22×C4—31no—
322C22×C4—31no—
323C22×C4—31no—
324C22×C4—31no—
325C22×C4—31no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—31no—
329C22×C4—31no—
330C22×C4—31no—
331C22×C4—61no—
332C22×C4—61no—
333C22×C4—61no—
334C22×C4—61no—
335C22×C4—61no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4—61no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4—121no—
362C22×C4—121no—
363C22×C4—121no—
364C22×C4—121no—
365C22×C4D3×D411yescharacteristic
366C22×C4#23711yescharacteristic
367C22×C4C2×S411yescharacteristic
368C22×C4C2×S411yescharacteristic
369M16—122no—
370M16—122no—
371M16—122no—
372M16—122no—
373C2×C8—32no—
374C2×C8—32no—
375C2×C8—61no—
376C2×C8—61no—
377C2×C8—61no—
378C2×C8—61no—
379C2×C8—61no—
380C2×C8—61no—
381C2×C8—61no—
382C2×C8—61no—
383C2×C8—61no—
384C2×C8—61no—
385C2×C8—61no—
386C2×C8—61no—
387C2×C8—61no—
388C2×C8—61no—
389C2×C8—61no—
390C2×C8—61no—
391C2×C8—61no—
392C2×C8—61no—
393C2×C8—61no—
394C2×C8—61no—
395C2×C8—61no—
396C2×C8—61no—
397C2×C8—122no—
398C2×C8—122no—
399C2×C8—122no—
400C2×C8—122no—
401C4⋊C4—31no—
402C4⋊C4—31no—
403C4⋊C4—31no—
404C4⋊C4—61no—
405C4⋊C4SL(2,ℤ/4)11yescharacteristic
406C22⋊C4—31no—
407C22⋊C4—31no—
408C22⋊C4—31no—
409C22⋊C4—31no—
410C22⋊C4—31no—
411C22⋊C4—31no—
412C22⋊C4—61no—
413C22⋊C4—61no—
414C22⋊C4SL(2,ℤ/4)11yescharacteristic
415C22⋊C4SL(2,ℤ/4)11yescharacteristic
416C42—31no—
417C42—31no—
418C42—31no—
419C42—61no—
420C42C2×S411yescharacteristic
421C22×C3—42no—
422C22×C3—41no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3—81no—
426C22×C3—81no—
427C22×C3—81no—
428A4#43511yescharacteristic, complemented
429C3×C4—41no—
430C3×C4—41no—
431C3×C4—81no—
432C3×C4—81no—
433C3×C4—81no—
434C3×C4—81no—
435C23—22no—
436C23—31no—
437C23—31no—
438C23—31no—
439C23—31no—
440C23—32no—
441C23—32no—
442C23—32no—
443C23—31no—
444C23—32no—
445C23—31no—
446C23—41no—
447C23—61no—
448C23—61no—
449C23—61no—
450C23—61no—
451C23—61no—
452C23—61no—
453C23—61no—
454C23—61no—
455C23—61no—
456C23—61no—
457C23—61no—
458C23—62no—
459C23—61no—
460C23—61no—
461C23—61no—
462C23—61no—
463C23—62no—
464C23—62no—
465C23—61no—
466C23—61no—
467C23—61no—
468C23—61no—
469C23—61no—
470C23—121no—
471C23—121no—
472C23—121no—
473C23—122no—
474C23—122no—
475C23C2×#20811yescharacteristic
476C23D4×Dic311yescharacteristic
477C23#94811yescharacteristic
478C23C2×SL(2,ℤ/4)11yescharacteristic
479C23C2×SL(2,ℤ/4)11yescharacteristic
480D4—22no—
481D4—62no—
482D4—62no—
483D4—64no—
484D4—62no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—31no—
490C2×C4—31no—
491C2×C4—31no—
492C2×C4—31no—
493C2×C4—31no—
494C2×C4—31no—
495C2×C4—31no—
496C2×C4—32no—
497C2×C4—31no—
498C2×C4—31no—
499C2×C4—31no—
500C2×C4—31no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4C2×SL(2,ℤ/4)11yescharacteristic
534C2×C4C2×SL(2,ℤ/4)11yescharacteristic
535C2×C4C2×SL(2,ℤ/4)11yescharacteristic
536C2×C4C2×SL(2,ℤ/4)11yescharacteristic
537C2×C4C22×S411yescharacteristic
538C8—62no—
539C8—62no—
540C8—121no—
541C8—121no—
542C8—121no—
543C8—121no—
544C8—121no—
545C8—121no—
546C2×C3—41no—
547C2×C3—41no—
548C2×C3—41no—
549C2×C3—82no—
550C2×C3—161no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—31no—
557C22—31no—
558C22—31no—
559C22—31no—
560C22—31no—
561C22—31no—
562C22—31no—
563C22—31no—
564C22—31no—
565C22—32no—
566C22—62no—
567C22—62no—
568C22—62no—
569C22—62no—
570C22—61no—
571C22—62no—
572C22—62no—
573C22—62no—
574C22—62no—
575C22—61no—
576C22—61no—
577C22—61no—
578C22—61no—
579C22—61no—
580C22—121no—
581C22—121no—
582C22—121no—
583C22—121no—
584C22—121no—
585C22—121no—
586C22—121no—
587C22—121no—
588C22#501111yescharacteristic, complemented
589C22#540912yescomplemented
590C22C22×SL(2,ℤ/4)11yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—31no—
596C4—31no—
597C4—61no—
598C4—61no—
599C4—61no—
600C4—61no—
601C4D4×S411yescharacteristic
602C4#591411yescharacteristic
603C3—41noSylow
604C2—22no—
605C2—31no—
606C2—31no—
607C2—31no—
608C2—31no—
609C2—41no—
610C2—62no—
611C2—121no—
612C2C2×#540911yescharacteristic
613C2D4×SL(2,ℤ/4)11yescharacteristic
614C2#8551111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC22×A4C22×C4
FittingC22×C4×D4D3
socleC24C22×Dic3
prime radicalC2×D4×A4C4
cosocleC2×C4×A4C23
hypercenterC4×D4S4
hypocenterA4#435
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#5409C2semidirect
3. C22×A4⋊C8C2semidirect
4. #73141C2semidirect
5. #73141C2semidirect
6. C4⋊(A4⋊C8)C2semidirect
7. #73044C2semidirect
56. A4⋊C8D4semidirect
121. C22×A4M16semidirect
294. C24#208semidirect
428. A4#435semidirect
588. C22#5011semidirect
589. C22#5409semidirect

Maximal subgroups

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

subgroupindex
C4×D4×A42
C2×#54092
C22×A4⋊C82
#731412
C4⋊(A4⋊C8)2
#730442
#223993
#50114