Finite Group Databasebeta

← Q8×#1020 #11715804

Subgroups — Q8×#1020 #11715804

Order 768.

1654 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 1654 conjugacy classes of subgroups — too many to draw. Only its 49 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×#1020C213yesmaximal
3#87502C213yesmaximal, complemented
4Q8×#120—31nomaximal, Sylow
5C2×#1020C2211yescharacteristic
6C4×C22⋊A4C2213yes—
7#5936C2213yes—
8Q8×S4—43nomaximal
9C24×Q8D311yescharacteristic, complemented
10C2×D4×Q8—33no—
11#2848—33no—
12#2434—39no—
13#2409—39no—
14Q8×C22⋊C4—33no—
15C4×#120—33no—
16C2×C22⋊A4C2311yescharacteristic
17#1020Q812yesdirect product factor
18Q8×A4—43no—
19Q8×A4—81no—
20C4×S4—49no—
21A4⋊Q8—49no—
22C23×Q8—33no—
23C23×Q8—39no—
24C23×Q8—31no—
25C23×Q8—61no—
26C23×Q8—63no—
27C24×C4C2×D313yes—
28D4×Q8—69no—
29D4×Q8—63no—
30D4×Q8—69no—
31D4×Q8—63no—
32C2×C4⋊Q8—39no—
33C2×C22⋊Q8—39no—
34C2×C22⋊Q8—39no—
35C2×#120—31no—
36C2×C4×Q8—33no—
37C2×C4×D4—39no—
38#395—318no—
39#393—36no—
40#391—318no—
41#386—39no—
42#385—318no—
43#385—39no—
44#380—39no—
45#379—33no—
46C4×C22⋊C4—39no—
47C22⋊A4C2×Q811yescharacteristic, complemented
48C2×S4—43no—
49D3×Q8—161no—
50C4×A4—49no—
51C4×A4—83no—
52SL(2,ℤ/4)—49no—
53C25C22×D311yescharacteristic
54C22×Q8—21no—
55C22×Q8—31no—
56C22×Q8—39no—
57C22×Q8—39no—
58C22×Q8—33no—
59C22×Q8—33no—
60C22×Q8—36no—
61C22×Q8—39no—
62C22×Q8—63no—
63C22×Q8—63no—
64C22×Q8—69no—
65C22×Q8—69no—
66C22×Q8—618no—
67C22×Q8—63no—
68C22×Q8—69no—
69C22×Q8—618no—
70C22×Q8—69no—
71C22×Q8—63no—
72C22×Q8—62no—
73C22×Q8S413yescomplemented
74C22×D4—33no—
75C23×C4—39no—
76C23×C4—39no—
77C23×C4—33no—
78C23×C4—63no—
79C23×C4—63no—
80C4⋊Q8—69no—
81C4⋊Q8—618no—
82C4⋊Q8—69no—
83C22⋊Q8—69no—
84C22⋊Q8—69no—
85C22⋊Q8—69no—
86C22⋊Q8—69no—
87C22⋊Q8—618no—
88C22⋊Q8—69no—
89C22⋊Q8—69no—
90#120—36no—
91#120—32no—
92C4×Q8—69no—
93C4×Q8—63no—
94C4×D4—69no—
95C4×D4—69no—
96C4×D4—69no—
97C4×D4—69no—
98C2×C4⋊C4—39no—
99C2×C4⋊C4—39no—
100C2×C4⋊C4—318no—
101C2×C22⋊C4—33no—
102C2×C22⋊C4—39no—
103C2×C22⋊C4—39no—
104C2×C22⋊C4—39no—
105C2×C42—39no—
106#95—318no—
107#95—318no—
108C2×A4—43no—
109C2×A4—81no—
110S4—46no—
111C3×Q8—161no—
112C4×D3—163no—
113Dic6—163no—
114C24—33no—
115C24—33no—
116C24—31no—
117C24—61no—
118C24—61no—
119C24D3×Q811yescharacteristic, complemented
120C2×Q8—39no—
121C2×Q8—33no—
122C2×Q8—39no—
123C2×Q8—318no—
124C2×Q8—39no—
125C2×Q8—618no—
126C2×Q8—618no—
127C2×Q8—618no—
128C2×Q8—618no—
129C2×Q8—69no—
130C2×Q8—69no—
131C2×Q8—69no—
132C2×Q8—618no—
133C2×Q8—69no—
134C2×Q8—63no—
135C2×Q8—618no—
136C2×Q8—66no—
137C2×Q8—61no—
138C2×Q8—126no—
139C2×Q8—129no—
140C2×Q8—121no—
141C2×D4—36no—
142C2×D4—36no—
143C2×D4—63no—
144C2×D4—63no—
145C2×D4—63no—
146C2×D4—63no—
147C22×C4—23no—
148C22×C4—33no—
149C22×C4—33no—
150C22×C4—39no—
151C22×C4—39no—
152C22×C4—39no—
153C22×C4—39no—
154C22×C4—39no—
155C22×C4—33no—
156C22×C4—618no—
157C22×C4—69no—
158C22×C4—69no—
159C22×C4—69no—
160C22×C4—69no—
161C22×C4—63no—
162C22×C4—69no—
163C22×C4C2×S419yes—
164C4⋊C4—69no—
165C4⋊C4—69no—
166C4⋊C4—69no—
167C4⋊C4—69no—
168C4⋊C4—618no—
169C4⋊C4—618no—
170C22⋊C4—36no—
171C22⋊C4—36no—
172C22⋊C4—69no—
173C22⋊C4—69no—
174C22⋊C4—69no—
175C22⋊C4—69no—
176C22⋊C4—69no—
177C22⋊C4—69no—
178C42—69no—
179C42—69no—
180C2×D3—161no—
181A4—43no—
182A4—81no—
183C3×C4—163no—
184Dic3—163no—
185C23—21no—
186C23—31no—
187C23—33no—
188C23—33no—
189C23—33no—
190C23—33no—
191C23—33no—
192C23—32no—
193C23—61no—
194C23—61no—
195C23—63no—
196C23—66no—
197C23—63no—
198C23—63no—
199C23—63no—
200C23—63no—
201C23C22×S413yes—
202Q8—36no—
203Q8—39no—
204Q8—63no—
205Q8—618no—
206Q8—69no—
207Q8—62no—
208Q8—63no—
209Q8—129no—
210Q8—123no—
211Q8#102011yescharacteristic, direct product factor
212D4—66no—
213D4—66no—
214D4—66no—
215D4—66no—
216C2×C4—39no—
217C2×C4—39no—
218C2×C4—39no—
219C2×C4—36no—
220C2×C4—39no—
221C2×C4—69no—
222C2×C4—618no—
223C2×C4—69no—
224C2×C4—63no—
225C2×C4—63no—
226C2×C4—63no—
227C2×C4—63no—
228C2×C4—69no—
229C2×C4—69no—
230C2×C4—618no—
231C2×C4—129no—
232C2×C4—123no—
233C2×C3—161no—
234D3—162no—
235C22—21no—
236C22—33no—
237C22—31no—
238C22—33no—
239C22—33no—
240C22—33no—
241C22—33no—
242C22—63no—
243C22—66no—
244C22—63no—
245C22—63no—
246C22—66no—
247C22—61no—
248C22—61no—
249C22—123no—
250C22—121no—
251C22Q8×S413yescomplemented
252C4—39no—
253C4—66no—
254C4—63no—
255C4—129no—
256C4—123no—
257C4C2×#102013yes—
258C3—161noSylow
259C2—33no—
260C2—33no—
261C2—61no—
262C2—61no—
263C2—122no—
264C2C22×#102011yescharacteristic
subgroupG/H
center, FrattiniC2C22×#1020
commutator, cosocleC2×C22⋊A4C23
FittingC24×Q8D3
socleC25C22×D3
prime radicalC2×#1020C22
hypercenterQ8#1020
hypocenterC22⋊A4C2×Q8
LCSG ▷ C2×C22⋊A4 ▷ C22⋊A4
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊A4C2semidirect
3. #87502C2semidirect
9. C24×Q8D3semidirect
17. #1020Q8direct (N × H)
47. C22⋊A4C2×Q8semidirect
73. C22×Q8S4semidirect
119. C24D3×Q8semidirect
211. Q8#1020direct (N × H)
251. C22Q8×S4semidirect

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
#875022
Q8×#1203
Q8×S44