Finite Group Databasebeta

← #11715693

Subgroups — #11715693

Order 768.

964 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 964 conjugacy classes of subgroups — too many to draw. Only its 27 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
1Q8×C22⋊A4C211yesmaximal, characteristic, complemented
2C4⋊#1020C211yesmaximal, characteristic
3#85561C211yesmaximal, characteristic, complemented
4#32879—31nomaximal, Sylow
5C2×#1020—21no—
6C4×C22⋊A4—21no—
7C4×C22⋊A4C2211yescharacteristic
8#5417—43nomaximal
9C24×Q8D311yescharacteristic, complemented
10C2×#448—33no—
11C2×#448—33no—
12#2847—31no—
13#1787—33no—
14#1787—33no—
15#1779—31no—
16C2×C22⋊A4D411yescharacteristic
17#1020—41no—
18Q8×A4—43no—
19Q8×A4—81no—
20C4⋊S4—43no—
21A4⋊C8—43no—
22C23×Q8—33no—
23C23×Q8—33no—
24C23×Q8—63no—
25C23×Q8—61no—
26C23×Q8—61no—
27C23×Q8—62no—
28C24×C4—21no—
29C24×C4C2×D311yescharacteristic
30C22×SD16—33no—
31C22×SD16—31no—
32C2×#121—33no—
33C2×#120—61no—
34#448—63no—
35#448—66no—
36#448—63no—
37#448—66no—
38#448—63no—
39#448—63no—
40C2×#103—33no—
41C2×#103—36no—
42C2×#103—33no—
43C2×C22⋊C8—33no—
44C2×C22⋊C8—33no—
45#386—63no—
46#380—33no—
47C22⋊A4SD1611yescharacteristic, complemented
48C2×S4—83no—
49C4×A4—43no—
50C4×A4—81no—
51C4×A4—83no—
52C4×A4—82no—
53#215—161no—
54C25#6711yescharacteristic
55C22×Q8—21no—
56C22×Q8—33no—
57C22×Q8—31no—
58C22×Q8—33no—
59C22×Q8—36no—
60C22×Q8—33no—
61C22×Q8—36no—
62C22×Q8—33no—
63C22×Q8—63no—
64C22×Q8—62no—
65C22×Q8—61no—
66C22×Q8—63no—
67C22×Q8—63no—
68C22×Q8—63no—
69C22×Q8—63no—
70C22×Q8—66no—
71C22×Q8—63no—
72C22×Q8—63no—
73C22×Q8—612no—
74C22×Q8—612no—
75C22×Q8—66no—
76C22×Q8—66no—
77C22×Q8—62no—
78C22×Q8—66no—
79C22×Q8—66no—
80C22×Q8S413yescomplemented
81C22×D4—31no—
82C22×D4—63no—
83C23×C4—33no—
84C23×C4—33no—
85C23×C4—61no—
86C23×C4—63no—
87C23×C4—62no—
88C23×C4—61no—
89C23×C4—63no—
90C23×C4—62no—
91C2×SD16—63no—
92C2×SD16—612no—
93C2×SD16—63no—
94C2×SD16—63no—
95C2×SD16—63no—
96C22×C8—33no—
97C22×C8—31no—
98#121—63no—
99#121—63no—
100#121—63no—
101#121—63no—
102#120—123no—
103#120—121no—
104C2×C4⋊C4—33no—
105C2×C22⋊C4—63no—
106C2×C22⋊C4—63no—
107#103—63no—
108#103—612no—
109#103—63no—
110#103—63no—
111#103—63no—
112C22⋊C8—63no—
113C22⋊C8—63no—
114C22⋊C8—63no—
115C22⋊C8—63no—
116#95—63no—
117C2×A4—43no—
118C2×A4—81no—
119S4—163no—
120C3×Q8—161no—
121D12—161no—
122C3⋊C8—161no—
123C24—33no—
124C24—33no—
125C24—61no—
126C24—61no—
127C24—61no—
128C24#21511yescharacteristic, complemented
129C2×Q8—33no—
130C2×Q8—312no—
131C2×Q8—33no—
132C2×Q8—33no—
133C2×Q8—33no—
134C2×Q8—63no—
135C2×Q8—66no—
136C2×Q8—66no—
137C2×Q8—63no—
138C2×Q8—63no—
139C2×Q8—63no—
140C2×Q8—63no—
141C2×Q8—66no—
142C2×Q8—63no—
143C2×Q8—66no—
144C2×Q8—63no—
145C2×Q8—66no—
146C2×Q8—612no—
147C2×Q8—612no—
148C2×Q8—612no—
149C2×Q8—66no—
150C2×Q8—63no—
151C2×Q8—62no—
152C2×Q8—61no—
153C2×Q8—612no—
154C2×Q8—66no—
155C2×Q8—612no—
156C2×Q8—66no—
157C2×Q8—66no—
158C2×Q8—66no—
159C2×Q8—61no—
160C2×D4—63no—
161C2×D4—63no—
162C2×D4—123no—
163C2×D4—123no—
164C2×D4—123no—
165C2×D4—123no—
166C2×D4—123no—
167C2×D4—123no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—23no—
171C22×C4—31no—
172C22×C4—33no—
173C22×C4—33no—
174C22×C4—33no—
175C22×C4—33no—
176C22×C4—66no—
177C22×C4—63no—
178C22×C4—63no—
179C22×C4—61no—
180C22×C4—62no—
181C22×C4—63no—
182C22×C4—61no—
183C22×C4—63no—
184C22×C4—63no—
185C22×C4—66no—
186C22×C4—63no—
187C22×C4—66no—
188C22×C4—66no—
189C22×C4—612no—
190C22×C4—63no—
191C22×C4—63no—
192C22×C4C2×S413yes—
193SD16—126no—
194SD16—126no—
195SD16—123no—
196SD16—121no—
197C2×C8—66no—
198C2×C8—63no—
199C2×C8—63no—
200C4⋊C4—63no—
201C4⋊C4—63no—
202C22⋊C4—123no—
203C22⋊C4—123no—
204C22⋊C4—123no—
205C22⋊C4—123no—
206C2×D3—321no—
207A4—43no—
208A4—81no—
209C3×C4—161no—
210C3×C4—321no—
211C23—21no—
212C23—31no—
213C23—33no—
214C23—33no—
215C23—33no—
216C23—33no—
217C23—33no—
218C23—61no—
219C23—61no—
220C23—63no—
221C23—66no—
222C23—63no—
223C23—63no—
224C23—123no—
225C23—121no—
226C23—123no—
227C23GL(2,ℤ/4)13yes—
228Q8—36no—
229Q8—36no—
230Q8—33no—
231Q8—63no—
232Q8—62no—
233Q8—612no—
234Q8—63no—
235Q8—66no—
236Q8—62no—
237Q8—66no—
238Q8—61no—
239Q8#102011yescharacteristic, complemented
240D4—123no—
241D4—121no—
242D4—243no—
243D4—243no—
244D4—243no—
245D4—243no—
246C2×C4—33no—
247C2×C4—33no—
248C2×C4—33no—
249C2×C4—33no—
250C2×C4—63no—
251C2×C4—62no—
252C2×C4—63no—
253C2×C4—66no—
254C2×C4—63no—
255C2×C4—612no—
256C2×C4—63no—
257C2×C4—63no—
258C2×C4—66no—
259C2×C4—66no—
260C2×C4—61no—
261C2×C4—63no—
262C2×C4—61no—
263C2×C4—62no—
264C2×C4—123no—
265C2×C4—123no—
266C2×C4—123no—
267C8—123no—
268C8—121no—
269C2×C3—161no—
270D3—641no—
271C22—21no—
272C22—33no—
273C22—31no—
274C22—33no—
275C22—33no—
276C22—33no—
277C22—33no—
278C22—63no—
279C22—66no—
280C22—63no—
281C22—63no—
282C22—61no—
283C22—61no—
284C22—243no—
285C22—243no—
286C22—241no—
287C22#541713yescomplemented
288C4—21no—
289C4—33no—
290C4—61no—
291C4—63no—
292C4—62no—
293C4—243no—
294C4C2×#102011yescharacteristic
295C3—161noSylow
296C2—33no—
297C2—33no—
298C2—61no—
299C2—61no—
300C2—481no—
301C2#8752411yescharacteristic
subgroupG/H
centerC2#87524
commutator, cosocleC4×C22⋊A4C22
FrattiniC4C2×#1020
FittingC24×Q8D3
socleC25#67
prime radicalC4⋊#1020C2
hypercenterQ8#1020
hypocenterC22⋊A4SD16
LCSG ▷ C4×C22⋊A4 ▷ C2×C22⋊A4 ▷ C22⋊A4
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊A4C2semidirect
3. #85561C2semidirect
9. C24×Q8D3semidirect
47. C22⋊A4SD16semidirect
80. C22×Q8S4semidirect
128. C24#215semidirect
239. Q8#1020semidirect
287. C22#5417semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊A42
C4⋊#10202
#855612
#328793
#54174