Finite Group Databasebeta

← #11713113

Subgroups — #11713113

Order 768.

579 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 579 conjugacy classes of subgroups — too many to draw. Only its 65 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×#5400C211yesmaximal, characteristic, complemented
2C2×C8×S4C211yesmaximal, characteristic, complemented
3C4×C4⋊S4C211yesmaximal, characteristic
4A4×C4⋊C8C211yesmaximal, characteristic, complemented
5(C2×S4)⋊C8C211yesmaximal, characteristic, complemented
6(C2×S4)⋊C8C211yesmaximal, characteristic, complemented
7#73044C211yesmaximal, characteristic, complemented
8#22987—31nomaximal, Sylow
9C2×C4⋊S4C411yescharacteristic
10C2×C4×S4C2211yescharacteristic
11C2×C4×S4C2211yescharacteristic
12C2×C8×A4C2211yescharacteristic
13C2×C8×A4C2211yescharacteristic
14C42×A4C2211yescharacteristic
15(C2×S4)⋊C4C411yescharacteristic
16(C2×S4)⋊C4C411yescharacteristic
17C4⋊SL(2,ℤ/4)C411yescharacteristic
18C2×A4⋊C8C2211yescharacteristic
19C2×A4⋊C8C2211yescharacteristic
20#5400—22no—
21C8×S4—22no—
22#4836—41nomaximal
23C2×#435—31no—
24C2×#435—31no—
25C2×C8×D4—31no—
26C22×C4⋊C8D311yescharacteristic, complemented
27C4×#121—31no—
28#1955—31no—
29#1940—31no—
30#1924—31no—
31#1923—31no—
32#1869—31no—
33#1869—31no—
34#1851—31no—
35#1850—31no—
36#1771—31no—
37#1749—31no—
38C22×S4C2×C411yescharacteristic
39C22×S4C2×C411yescharacteristic
40C2×C4×A4C2×C411yescharacteristic
41C2×C4×A4C2×C411yescharacteristic
42C2×C4×A4C2311yescharacteristic
43C2×SL(2,ℤ/4)C2×C411yescharacteristic
44C2×SL(2,ℤ/4)C2×C411yescharacteristic
45C4⋊S4—22no—
46C4×S4—21no—
47C4×S4—21no—
48C4×S4—21no—
49C4×S4—21no—
50C2×ΓL(1,𝔽25)—41no—
51C2×D3×C8—41no—
52C4×D12—41no—
53C8×A4—21no—
54C8×A4—21no—
55A4⋊C8—21no—
56A4⋊C8D412yescomplemented
57C3×C4⋊C8—41no—
58(C2×D3)⋊C8—41no—
59(C2×D3)⋊C8—41no—
60#803—41no—
61C22×M16—31no—
62C23×C8—31no—
63C23×C8C2×D311yescharacteristic
64C23×C8C2×D311yescharacteristic
65C2×#121—31no—
66C2×C4×D4—31no—
67C2×C4×D4—31no—
68C2×C4×D4—31no—
69C22×C42C2×D311yescharacteristic
70#435—62no—
71#435—62no—
72#435—61no—
73#435—61no—
74#435—61no—
75#435—61no—
76C8×D4—62no—
77C8×D4—62no—
78C2×C4⋊C8—31no—
79C2×C4⋊C8—31no—
80C2×C4⋊C8—31no—
81C2×C4⋊C8—31no—
82C2×C4⋊C8—31no—
83C2×C4⋊C8—31no—
84C2×C22⋊C8—31no—
85C2×C22⋊C8—31no—
86C2×C22⋊C8—31no—
87C2×C22⋊C8—31no—
88C2×C22⋊C8—31no—
89C2×C22⋊C8—31no—
90C2×C22⋊C8—31no—
91C2×C22⋊C8—31no—
92C2×#97—31no—
93C2×#97—31no—
94C2×#97—31no—
95C2×C4×C8—31no—
96#390—31no—
97#389—31no—
98#388—31no—
99#388—31no—
100#386—31no—
101#386—31no—
102#380—31no—
103C4×C4⋊C4—31no—
104C4×C22⋊C4—31no—
105C4×C22⋊C4—31no—
106#336—31no—
107#336—31no—
108#336—31no—
109#336—31no—
110#336—31no—
111#336—31no—
112C22×A4C22×C411yescharacteristic
113C2×S4—21no—
114C2×S4—21no—
115C2×S4—21no—
116C2×S4—21no—
117C2×S4—21no—
118C2×S4M1612yescomplemented
119C2×D12—41no—
120C2×C4×D3—41no—
121C2×C4×D3—41no—
122C4×A4—21no—
123C4×A4—21no—
124C4×A4C2×D411yescharacteristic
125C4×A4#4111yescharacteristic
126SL(2,ℤ/4)—21no—
127SL(2,ℤ/4)—21no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C3×C42—41no—
131(C2×D3)⋊C4—41no—
132(C2×D3)⋊C4—41no—
133C4⋊Dic3—41no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136ΓL(1,𝔽25)—82no—
137D3×C8—82no—
138C22×D4—31no—
139C22×D4—31no—
140C22×D4—31no—
141C23×C4—31no—
142C23×C4—31no—
143C23×C4C4×D311yescharacteristic
144C23×C4C4×D311yescharacteristic
145C23×C4C22×D311yescharacteristic
146C2×M16—62no—
147C2×M16—61no—
148C2×M16—61no—
149C2×M16—121no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—31no—
153C22×C8—31no—
154C22×C8—31no—
155C22×C8—31no—
156C22×C8—31no—
157C22×C8—31no—
158C22×C8—31no—
159C22×C8—31no—
160C22×C8—62no—
161C22×C8—61no—
162C22×C8—61no—
163C22×C8—61no—
164C22×C8—61no—
165C22×C8—121no—
166#121—62no—
167#121—62no—
168C4×D4—61no—
169C4×D4—61no—
170C4×D4—61no—
171C4×D4—61no—
172C4×D4—61no—
173C4×D4—61no—
174C4×D4—61no—
175C4×D4—61no—
176C4×D4—61no—
177C4×D4—61no—
178C4×D4—61no—
179C4×D4—61no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—31no—
185C2×C22⋊C4—31no—
186C2×C22⋊C4—31no—
187C2×C22⋊C4—31no—
188C2×C22⋊C4—31no—
189C2×C22⋊C4—31no—
190C2×C22⋊C4—31no—
191C2×C22⋊C4—31no—
192C2×C22⋊C4—31no—
193C2×C22⋊C4—31no—
194C2×C22⋊C4—31no—
195C2×C42—31no—
196C2×C42—31no—
197C2×C42—31no—
198C2×C42—31no—
199C2×C42—31no—
200C2×C42—31no—
201C4⋊C8—31no—
202C4⋊C8—31no—
203C4⋊C8—31no—
204C4⋊C8—61no—
205C4⋊C8—62no—
206C4⋊C8—62no—
207C4⋊C8S411yescharacteristic, complemented
208C22⋊C8—32no—
209C22⋊C8—32no—
210C22⋊C8—62no—
211C22⋊C8—62no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217C22⋊C8—61no—
218C22⋊C8—121no—
219C22⋊C8—121no—
220#97—61no—
221#97—61no—
222#97—61no—
223#97—62no—
224#97—61no—
225C4×C8—62no—
226#95—31no—
227#95—31no—
228#95—31no—
229#95—31no—
230#95—31no—
231#95—31no—
232C22×D3—41no—
233C22×D3—41no—
234C2×A4C4×D411yescharacteristic
235C2×A4C2×M1611yescharacteristic
236C2×A4#13111yescharacteristic
237S4—22no—
238S4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244D12—82no—
245C4×D3—81no—
246C4×D3—81no—
247C4×D3—81no—
248C4×D3—81no—
249C3×C8—81no—
250C3×C8—81no—
251C3⋊C8—42no—
252C3⋊C8—81no—
253C24—31no—
254C24—31no—
255C24C2×C4×D311yescharacteristic
256C2×D4—32no—
257C2×D4—32no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—62no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—61no—
271C2×D4—121no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—31no—
275C22×C4—31no—
276C22×C4—31no—
277C22×C4—31no—
278C22×C4—31no—
279C22×C4—31no—
280C22×C4—31no—
281C22×C4—31no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—31no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4—61no—
302C22×C4D3×D411yescharacteristic
303C22×C4#23911yescharacteristic
304M16—122no—
305M16—122no—
306C2×C8—32no—
307C2×C8—32no—
308C2×C8—31no—
309C2×C8—31no—
310C2×C8—61no—
311C2×C8—61no—
312C2×C8—61no—
313C2×C8—61no—
314C2×C8—61no—
315C2×C8—61no—
316C2×C8—61no—
317C2×C8—61no—
318C2×C8—61no—
319C2×C8—61no—
320C2×C8—61no—
321C2×C8—61no—
322C2×C8—61no—
323C2×C8—61no—
324C2×C8—61no—
325C2×C8—61no—
326C2×C8—61no—
327C2×C8—61no—
328C2×C8—122no—
329C2×C8—122no—
330C2×C8C2×S411yescharacteristic
331C2×C8C2×S411yescharacteristic
332C4⋊C4—61no—
333C4⋊C4—61no—
334C4⋊C4—61no—
335C4⋊C4—61no—
336C4⋊C4—61no—
337C4⋊C4—62no—
338C4⋊C4—61no—
339C4⋊C4—121no—
340C22⋊C4—62no—
341C22⋊C4—62no—
342C22⋊C4—61no—
343C22⋊C4—61no—
344C22⋊C4—61no—
345C22⋊C4—61no—
346C22⋊C4—61no—
347C22⋊C4—61no—
348C22⋊C4—62no—
349C22⋊C4—61no—
350C22⋊C4—61no—
351C22⋊C4—62no—
352C22⋊C4—61no—
353C22⋊C4—61no—
354C22⋊C4—61no—
355C22⋊C4—61no—
356C42—31no—
357C42—31no—
358C42—31no—
359C42—61no—
360C42—61no—
361C42—61no—
362C42—61no—
363C42—61no—
364C42C2×S411yescharacteristic
365C22×C3—41no—
366C2×D3—42no—
367C2×D3—81no—
368C2×D3—81no—
369C2×D3—81no—
370C2×D3—81no—
371C2×D3—81no—
372A4#43511yescharacteristic, complemented
373C3×C4—41no—
374C3×C4—41no—
375C3×C4—81no—
376C3×C4—81no—
377Dic3—81no—
378Dic3—81no—
379C23—31no—
380C23—31no—
381C23—31no—
382C23—31no—
383C23—32no—
384C23—61no—
385C23—61no—
386C23—61no—
387C23—61no—
388C23—61no—
389C23—61no—
390C23—61no—
391C23—61no—
392C23—61no—
393C23—61no—
394C23—61no—
395C23—61no—
396C23—61no—
397C23#89311yescharacteristic
398C23#90111yescharacteristic
399C23D3×M1611yescharacteristic
400D4—62no—
401D4—62no—
402D4—62no—
403D4—62no—
404D4—122no—
405D4—122no—
406D4—121no—
407D4—121no—
408D4—121no—
409D4—121no—
410C2×C4—31no—
411C2×C4—31no—
412C2×C4—31no—
413C2×C4—31no—
414C2×C4—31no—
415C2×C4—31no—
416C2×C4—31no—
417C2×C4—32no—
418C2×C4—32no—
419C2×C4—31no—
420C2×C4—31no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—121no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4C4×S411yescharacteristic
455C2×C4C4×S411yescharacteristic
456C2×C4C22×S411yescharacteristic
457C8—21no—
458C8—21no—
459C8—62no—
460C8—61no—
461C8—61no—
462C8—62no—
463C8—121no—
464C8—121no—
465C2×C3—41no—
466C2×C3—41no—
467C2×C3—41no—
468D3—82no—
469D3—161no—
470C22—31no—
471C22—31no—
472C22—31no—
473C22—31no—
474C22—31no—
475C22—31no—
476C22—31no—
477C22—31no—
478C22—31no—
479C22—61no—
480C22—62no—
481C22—62no—
482C22—62no—
483C22—121no—
484C22—121no—
485C22—121no—
486C22—121no—
487C22—121no—
488C22—121no—
489C22—121no—
490C22—121no—
491C22—121no—
492C22—121no—
493C22—121no—
494C22#483611yescharacteristic, complemented
495C22C2×C4×S411yescharacteristic
496C4—21no—
497C4—21no—
498C4—31no—
499C4—31no—
500C4—62no—
501C4—61no—
502C4—61no—
503C4—121no—
504C4—121no—
505C4—121no—
506C4—121no—
507C4—121no—
508C4D4×S411yescharacteristic
509C4#591911yescharacteristic
510C3—41noSylow
511C2—31no—
512C2—31no—
513C2—31no—
514C2—31no—
515C2—122no—
516C2—241no—
517C2#8530811yescharacteristic
518C2#8543711yescharacteristic
519C2M16×S411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC22×A4C22×C4
FittingC22×C4⋊C8D3
socleC24C2×C4×D3
prime radicalC2×C4⋊S4C4
cosocleC2×C4×A4C23
hypercenterC4⋊C8S4
hypocenterA4#435
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#5400C2semidirect
2. C2×C8×S4C2semidirect
4. A4×C4⋊C8C2semidirect
5. (C2×S4)⋊C8C2semidirect
6. (C2×S4)⋊C8C2semidirect
7. #73044C2semidirect
26. C22×C4⋊C8D3semidirect
56. A4⋊C8D4semidirect
118. C2×S4M16semidirect
207. C4⋊C8S4semidirect
372. A4#435semidirect
494. C22#4836semidirect

Maximal subgroups

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

subgroupindex
C2×#54002
C2×C8×S42
C4×C4⋊S42
A4×C4⋊C82
(C2×S4)⋊C82
#730442
#229873
#48364