Finite Group Databasebeta

← C4⋊Q8×C22⋊C9 #11919713

Subgroups — C4⋊Q8×C22⋊C9 #11919713

Order 1152.

700 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 130 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×Q8×C22⋊C9C212yesmaximal
2C4⋊C4×C22⋊C9C214yesmaximal
3C42×C22⋊C9C211yesmaximal, characteristic
4C22×C3×C4⋊Q8C311yesmaximal, characteristic
5Q8×C22⋊C9—24no—
6C2×C4×C22⋊C9C2214yes—
7C2×C4×C22⋊C9C2211yescharacteristic
8C2×C4×C22⋊C9C2212yes—
9C9×C4⋊Q8—41nomaximal
10C23×C3×Q8C2×C312yes—
11C2×C3×C4⋊Q8—31no—
12C2×C3×C4⋊Q8—31no—
13C2×C3×C4⋊Q8—34no—
14C2×C3×C4⋊Q8—32no—
15C22×C3×C4⋊C4C2×C314yes—
16C22×C3×C42C2×C311yescharacteristic
17C22×C22⋊C9C2311yescharacteristic
18C2×Q8×C9—42no—
19C4×C22⋊C9—24no—
20C4×C22⋊C9D412yes—
21C4×C22⋊C9Q814yescomplemented
22C9×C4⋊C4—44no—
23C42×C9—41no—
24C22×C4⋊Q8C911yescharacteristic, complemented, Sylow
25C22×C3×Q8—24no—
26C22×C3×Q8—32no—
27C22×C3×Q8—34no—
28C22×C3×Q8—32no—
29C22×C3×Q8—64no—
30C23×C3×C4C22×C314yes—
31C23×C3×C4C22×C311yescharacteristic
32C23×C3×C4C22×C312yes—
33C3×C4⋊Q8—38no—
34C3×C4⋊Q8—32no—
35C3×C4⋊Q8—34no—
36C3×C4⋊Q8—32no—
37C3×C4⋊Q8—34no—
38C3×C4⋊Q8—31no—
39C3×C4⋊Q8A411yescharacteristic
40C2×C3×C4⋊C4—34no—
41C2×C3×C4⋊C4—34no—
42C2×C3×C4⋊C4—38no—
43C2×C3×C42—31no—
44C2×C3×C42—32no—
45C2×C3×C42—31no—
46C2×C22⋊C9C2×D411yescharacteristic
47C2×C22⋊C9C2×Q812yes—
48Q8×C9—84no—
49C2×C4×C9—42no—
50C2×C4×C9—41no—
51C2×C4×C9—44no—
52C23×Q8C2×C912yes—
53C2×C4⋊Q8—31no—
54C2×C4⋊Q8—31no—
55C2×C4⋊Q8—34no—
56C2×C4⋊Q8—32no—
57C22×C4⋊C4C2×C914yes—
58C22×C42C2×C911yescharacteristic
59C24×C3C23×C311yescharacteristic
60C2×C3×Q8—34no—
61C2×C3×Q8—34no—
62C2×C3×Q8—32no—
63C2×C3×Q8—64no—
64C2×C3×Q8—64no—
65C2×C3×Q8—64no—
66C2×C3×Q8—68no—
67C2×C3×Q8—64no—
68C2×C3×Q8—68no—
69C2×C3×Q8C2×A412yes—
70C22×C3×C4—24no—
71C22×C3×C4—32no—
72C22×C3×C4—31no—
73C22×C3×C4—32no—
74C22×C3×C4—31no—
75C22×C3×C4—34no—
76C22×C3×C4—32no—
77C22×C3×C4—34no—
78C22×C3×C4—34no—
79C22×C3×C4—64no—
80C22×C3×C4C3×D412yes—
81C22×C3×C4C3×Q814yes—
82C3×C4⋊C4—38no—
83C3×C4⋊C4—38no—
84C3×C4⋊C4—34no—
85C3×C4⋊C4C2×A414yes—
86C3×C42—32no—
87C3×C42—31no—
88C3×C42—32no—
89C3×C42C2×A411yescharacteristic
90C22×C9—41no—
91C22⋊C9C4⋊Q811yescharacteristic, direct product factor
92C4×C9—42no—
93C4×C9—44no—
94C4×C9—84no—
95C22×Q8—24no—
96C22×Q8—32no—
97C22×Q8—34no—
98C22×Q8—32no—
99C22×Q8—64no—
100C23×C4C22×C914yes—
101C23×C4C22×C911yescharacteristic
102C23×C4C22×C912yes—
103C4⋊Q8—38no—
104C4⋊Q8—32no—
105C4⋊Q8—34no—
106C4⋊Q8—32no—
107C4⋊Q8—34no—
108C4⋊Q8—31no—
109C4⋊Q8C22⋊C911yescharacteristic, direct product factor
110C2×C4⋊C4—34no—
111C2×C4⋊C4—34no—
112C2×C4⋊C4—38no—
113C2×C42—31no—
114C2×C42—32no—
115C2×C42—31no—
116C23×C3—31no—
117C23×C3—32no—
118C23×C3—31no—
119C23×C3C2×C3×D411yescharacteristic
120C23×C3C2×C3×Q812yes—
121C3×Q8—24no—
122C3×Q8—68no—
123C3×Q8—68no—
124C3×Q8—64no—
125C2×C3×C4—34no—
126C2×C3×C4—32no—
127C2×C3×C4—32no—
128C2×C3×C4—31no—
129C2×C3×C4—32no—
130C2×C3×C4—32no—
131C2×C3×C4—34no—
132C2×C3×C4—34no—
133C2×C3×C4—32no—
134C2×C3×C4—34no—
135C2×C3×C4—34no—
136C2×C3×C4—64no—
137C2×C3×C4—64no—
138C2×C3×C4—64no—
139C2×C3×C4—64no—
140C2×C3×C4C22×A414yes—
141C2×C3×C4C22×A411yescharacteristic
142C2×C3×C4C22×A412yes—
143C2×C9—41no—
144C2×C9—42no—
145C24C23×C911yescharacteristic
146C2×Q8—34no—
147C2×Q8—34no—
148C2×Q8—32no—
149C2×Q8—64no—
150C2×Q8—64no—
151C2×Q8—64no—
152C2×Q8—68no—
153C2×Q8—64no—
154C2×Q8—68no—
155C2×Q8C2×C22⋊C912yes—
156C22×C4—24no—
157C22×C4—32no—
158C22×C4—31no—
159C22×C4—32no—
160C22×C4—31no—
161C22×C4—32no—
162C22×C4—34no—
163C22×C4—34no—
164C22×C4—34no—
165C22×C4—64no—
166C22×C4D4×C912yes—
167C22×C4Q8×C914yescomplemented
168C4⋊C4—38no—
169C4⋊C4—38no—
170C4⋊C4—34no—
171C4⋊C4C2×C22⋊C914yes—
172C42—32no—
173C42—31no—
174C42—32no—
175C42C2×C22⋊C911yescharacteristic
176C22×C3—31no—
177C22×C3—32no—
178C22×C3—32no—
179C22×C3—31no—
180C22×C3—31no—
181C22×C3—32no—
182C22×C3—32no—
183C22×C3C3×C4⋊Q811yescharacteristic
184C22×C3C23×A411yescharacteristic
185C3×C4—24no—
186C3×C4—32no—
187C3×C4—34no—
188C3×C4—64no—
189C3×C4D4×A412yes—
190C3×C4Q8×A414yes—
191C9—41noSylow
192C23—31no—
193C23—32no—
194C23—31no—
195C23C2×D4×C911yescharacteristic
196C23C2×Q8×C912yes—
197Q8—24no—
198Q8—68no—
199Q8—68no—
200Q8—64no—
201C2×C4—32no—
202C2×C4—34no—
203C2×C4—32no—
204C2×C4—31no—
205C2×C4—32no—
206C2×C4—32no—
207C2×C4—34no—
208C2×C4—34no—
209C2×C4—32no—
210C2×C4—34no—
211C2×C4—34no—
212C2×C4—64no—
213C2×C4—64no—
214C2×C4—64no—
215C2×C4—64no—
216C2×C4C22×C22⋊C914yes—
217C2×C4C22×C22⋊C911yescharacteristic
218C2×C4C22×C22⋊C912yes—
219C2×C3—31no—
220C2×C3—32no—
221C2×C3—31no—
222C2×C3C2×D4×A411yescharacteristic
223C2×C3C2×Q8×A412yes—
224C22—31no—
225C22—32no—
226C22—32no—
227C22—31no—
228C22—31no—
229C22—32no—
230C22—32no—
231C22C9×C4⋊Q811yescharacteristic, complemented
232C22C23×C22⋊C911yescharacteristic
233C4—24no—
234C4—32no—
235C4—34no—
236C4—64no—
237C4D4×C22⋊C912yes—
238C4Q8×C22⋊C914yescomplemented
239C3A4×C4⋊Q811yescharacteristic
240C2—31no—
241C2—31no—
242C2—32no—
243C2C2×D4×C22⋊C911yescharacteristic
244C2C2×Q8×C22⋊C912yes—
subgroupG/H
center, FrattiniC22×C3C23×A4
commutatorC24C23×C9
FittingC22×C3×C4⋊Q8C3
socle, prime radical, cosocleC24×C3C23×C3
hypercenterC3×C4⋊Q8A4
hypocenterC22C9×C4⋊Q8
LCSG ▷ C24 ▷ C22
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
21. C4×C22⋊C9Q8semidirect
24. C22×C4⋊Q8C9semidirect
91. C22⋊C9C4⋊Q8direct (N × H)
109. C4⋊Q8C22⋊C9direct (N × H)
167. C22×C4Q8×C9semidirect
231. C22C9×C4⋊Q8semidirect
238. C4Q8×C22⋊C9semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C22⋊C92
C4⋊C4×C22⋊C92
C42×C22⋊C92
C22×C3×C4⋊Q83
C9×C4⋊Q84