Finite Group Databasebeta

← #3503⋊C9 #11919755

Subgroups — #3503⋊C9 #11919755

Order 1152.

477 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 477 conjugacy classes of subgroups — too many to draw. Only its 109 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×Q8⋊C9C212yesmaximal, complemented
2#557⋊C9C212yesmaximal, complemented
3#520⋊C9C212yesmaximal
4C4×#41⋊C9C211yesmaximal, characteristic
5C3×#3503C311yesmaximal, characteristic
6C2×#41⋊C9C2211yescharacteristic
7C9×C4⋊Q8—41nomaximal
8C4×Q8⋊C9C2211yescharacteristic
9C4×Q8⋊C9C2211yescharacteristic
10C4×Q8⋊C9C2214yes—
11C3×Q82C2×C312yes—
12C3×#557C2×C312yes—
13C3×#554—32no—
14C3×D4×Q8—32no—
15C3×#544—32no—
16C3×#533—31no—
17C2×C3×C4⋊Q8—31no—
18C3×#520C2×C312yes—
19C3×C4×#41C2×C311yescharacteristic
20C2×Q8×C9—42no—
21#41⋊C9Q812yescomplemented
22C2×Q8⋊C9C2311yescharacteristic
23C9×C4⋊C4—42no—
24C9×C4⋊C4—42no—
25C42×C9—41no—
26#3503C911yescharacteristic, complemented, Sylow
27C2×C3×#41C22×C311yescharacteristic
28C22×C3×Q8—32no—
29C3×C4⋊Q8—32no—
30C3×C4⋊Q8—32no—
31C3×C4⋊Q8—32no—
32C3×C4⋊Q8—34no—
33C3×C4⋊Q8—31no—
34C3×C4⋊Q8A411yescharacteristic
35C3×#125—34no—
36C3×#125—32no—
37C3×C22⋊Q8—38no—
38C3×C22⋊Q8—32no—
39C3×C22⋊Q8—32no—
40C3×C4×Q8—32no—
41C3×C4×Q8—32no—
42C3×C4×Q8C22×C311yescharacteristic
43C3×C4×Q8C22×C314yes—
44C3×C4×Q8C22×C311yescharacteristic
45C3×C4×D4—31no—
46C3×C4×D4—31no—
47C3×C4×D4—34no—
48C3×#117—32no—
49C3×#117—31no—
50C2×C3×C4⋊C4—32no—
51C2×C3×C4⋊C4—32no—
52C2×C3×C42—31no—
53Q8×C9—84no—
54C2×C4×C9—41no—
55C2×C4×C9—41no—
56C2×C4×C9—41no—
57C2×C4×C9—44no—
58Q8⋊C9C2×Q811yescharacteristic, complemented
59Q82C2×C912yes—
60#557C2×C912yes—
61#554—32no—
62D4×Q8—32no—
63#544—32no—
64#533—31no—
65C2×C4⋊Q8—31no—
66#520C2×C912yes—
67C4×#41C2×C911yescharacteristic
68C3×#41—32no—
69C3×#41C3×Q812yes—
70C2×C3×Q8—34no—
71C2×C3×Q8—32no—
72C2×C3×Q8—124no—
73C2×C3×Q8C2×A412yes—
74C2×C3×Q8C23×C311yescharacteristic
75C2×C3×D4—31no—
76C22×C3×C4—31no—
77C22×C3×C4—34no—
78C22×C3×C4—31no—
79C22×C3×C4—31no—
80C3×C4⋊C4—34no—
81C3×C4⋊C4—31no—
82C3×C4⋊C4—31no—
83C3×C4⋊C4—34no—
84C3×C4⋊C4—32no—
85C3×C4⋊C4—32no—
86C3×C4⋊C4—34no—
87C3×C4⋊C4—34no—
88C3×C4⋊C4C2×A412yes—
89C3×C4⋊C4C2×A412yes—
90C3×C22⋊C4—32no—
91C3×C22⋊C4—34no—
92C3×C42—31no—
93C3×C42—31no—
94C3×C42—31no—
95C3×C42—34no—
96C3×C42C2×A411yescharacteristic
97C22×C9—41no—
98C4×C9—42no—
99C4×C9—42no—
100C4×C9—42no—
101C4×C9—84no—
102C2×#41C22×C911yescharacteristic
103C22×Q8—32no—
104C4⋊Q8—32no—
105C4⋊Q8—32no—
106C4⋊Q8—32no—
107C4⋊Q8—34no—
108C4⋊Q8—31no—
109C4⋊Q8C22⋊C911yescharacteristic
110#125—34no—
111#125—32no—
112C22⋊Q8—38no—
113C22⋊Q8—32no—
114C22⋊Q8—32no—
115C4×Q8—32no—
116C4×Q8—32no—
117C4×Q8C22×C911yescharacteristic
118C4×Q8C22×C911yescharacteristic
119C4×Q8C22×C914yes—
120C4×D4—31no—
121C4×D4—31no—
122C4×D4—34no—
123#117—32no—
124#117—31no—
125C2×C4⋊C4—32no—
126C2×C4⋊C4—32no—
127C2×C42—31no—
128C23×C3—31no—
129C3×Q8—24no—
130C3×Q8—31no—
131C3×Q8—64no—
132C3×Q8—124no—
133C3×Q8C2×C3×Q811yescharacteristic
134C3×D4—32no—
135C3×D4—32no—
136C2×C3×C4—31no—
137C2×C3×C4—32no—
138C2×C3×C4—32no—
139C2×C3×C4—31no—
140C2×C3×C4—31no—
141C2×C3×C4—34no—
142C2×C3×C4—62no—
143C2×C3×C4—62no—
144C2×C3×C4—124no—
145C2×C3×C4C22×A414yes—
146C2×C3×C4C22×A411yescharacteristic
147C2×C3×C4C22×A411yescharacteristic
148C2×C3×C4C22×A411yescharacteristic
149C2×C9—41no—
150C2×C9—41no—
151C2×C9—41no—
152#41—32no—
153#41Q8×C912yescomplemented
154C2×Q8—34no—
155C2×Q8—32no—
156C2×Q8—124no—
157C2×Q8C2×C22⋊C912yes—
158C2×Q8C23×C911yescharacteristic
159C2×D4—31no—
160C22×C4—31no—
161C22×C4—34no—
162C22×C4—31no—
163C22×C4—31no—
164C4⋊C4—34no—
165C4⋊C4—31no—
166C4⋊C4—31no—
167C4⋊C4—34no—
168C4⋊C4—32no—
169C4⋊C4—32no—
170C4⋊C4—34no—
171C4⋊C4—34no—
172C4⋊C4C2×C22⋊C912yes—
173C4⋊C4C2×C22⋊C912yes—
174C22⋊C4—32no—
175C22⋊C4—34no—
176C42—31no—
177C42—31no—
178C42—31no—
179C42—34no—
180C42C2×C22⋊C911yescharacteristic
181C22×C3—32no—
182C22×C3—61no—
183C22×C3—61no—
184C22×C3C23×A411yescharacteristic
185C3×C4—24no—
186C3×C4—31no—
187C3×C4—31no—
188C3×C4—61no—
189C3×C4—64no—
190C3×C4—61no—
191C3×C4Q8×A412yes—
192C3×C4#99412yes—
193C3×C42−1+4⋊C312yes—
194C9—41noSylow
195C23—31no—
196Q8—24no—
197Q8—31no—
198Q8—64no—
199Q8—124no—
200Q8C2×Q8×C911yescharacteristic
201D4—32no—
202D4—32no—
203C2×C4—31no—
204C2×C4—32no—
205C2×C4—32no—
206C2×C4—31no—
207C2×C4—31no—
208C2×C4—34no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—124no—
212C2×C4C22×C22⋊C914yes—
213C2×C4C22×C22⋊C911yescharacteristic
214C2×C4C22×C22⋊C911yescharacteristic
215C2×C4C22×C22⋊C911yescharacteristic
216C2×C3—62no—
217C2×C3C2×Q8×A411yescharacteristic
218C2×C3C2×#99411yescharacteristic
219C2×C3C2×2−1+4⋊C311yescharacteristic
220C22—32no—
221C22—61no—
222C22—61no—
223C22C23×C22⋊C911yescharacteristic
224C4—24no—
225C4—31no—
226C4—31no—
227C4—61no—
228C4—64no—
229C4—61no—
230C4Q8×C22⋊C912yes—
231C4#6370012yes—
232C42−1+4⋊C912yes—
233C3#3503⋊C311yescharacteristic
234C2—62no—
235C2C2×Q8×C22⋊C911yescharacteristic
236C2C2×#6370011yescharacteristic
237C2C2×2−1+4⋊C911yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×A4
commutatorC2×Q8C23×C9
FittingC3×#3503C3
prime radicalC2×C3×#41C22×C3
cosocleC2×C3×Q8C23×C3
hypercenterC3×C4⋊Q8A4
hypocenterQ8C2×Q8×C9
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. Q8×Q8⋊C9C2semidirect
2. #557⋊C9C2semidirect
21. #41⋊C9Q8semidirect
26. #3503C9semidirect
58. Q8⋊C9C2×Q8semidirect
153. #41Q8×C9semidirect

Maximal subgroups

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

subgroupindex
Q8×Q8⋊C92
#557⋊C92
#520⋊C92
C4×#41⋊C92
C3×#35033
C9×C4⋊Q84