Finite Group Databasebeta

← C22×(C22⋊C9)⋊Q8 #11921387

Subgroups — C22×(C22⋊C9)⋊Q8 #11921387

Order 1152.

1578 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1578 conjugacy classes of subgroups — too many to draw. Only its 186 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
1C22×C4×C22⋊C9C211yesmaximal, characteristic
2C22×(C22⋊C9)⋊C4C212yesmaximal
3C2×(C22⋊C9)⋊Q8C2112yesmaximal, direct product factor
4C22×#924—31nomaximal
5C23×C22⋊C9C2211yescharacteristic
6C22×Dic18—41nomaximal
7C2×C4×C22⋊C9C2216yes—
8C2×(C22⋊C9)⋊C4C22112yes—
9(C22⋊C9)⋊Q8C22116yesdirect product factor
10C24×C3×C4D311yescharacteristic
11C23×Dic6—31no—
12C22×#217—32no—
13C22×C4⋊Dic3—31no—
14C2×#924—312no—
15C2×#924—312no—
16C22×Dic3⋊C4—32no—
17C22×C22⋊C9Q814yescomplemented
18C22×C22⋊C9C2313yes—
19C22×C4×C9—41no—
20C22×Dic9—42no—
21C2×Dic18—412no—
22C4×C22⋊C9C2314yes—
23(C22⋊C9)⋊C4C2318yes—
24C22×C22⋊Q8—91noSylow
25C25×C3C2×D311yescharacteristic
26C23×C3×C4—31no—
27C23×C3×C4—31no—
28C23×C3×C4—36no—
29C23×C3×C4C2×D316yes—
30C23×Dic3—32no—
31C23×Dic3—32no—
32C22×Dic6—312no—
33C22×Dic6—61no—
34C22×Dic6—66no—
35C22×Dic6—61no—
36C2×#217—312no—
37C2×#217—312no—
38C2×C4⋊Dic3—312no—
39#924—348no—
40#924—316no—
41C2×Dic3⋊C4—324no—
42C23×C9—41no—
43C2×C22⋊C9C2×Q816yescomplemented
44C2×C22⋊C9C2411yescharacteristic
45C2×C4×C9—46no—
46C2×Dic9—412no—
47Dic18—416no—
48C23×Q8—91no—
49C24×C4D911yescharacteristic
50C2×C22⋊Q8—912no—
51C2×C22⋊Q8—912no—
52C22×C4⋊C4—92no—
53C22×C4⋊C4—91no—
54C22×C22⋊C4—92no—
55C24×C3—31no—
56C24×C3—34no—
57C24×C3—33no—
58C24×C3Dic614yes—
59C24×C3C22×D313yes—
60C22×C3×C4—31no—
61C22×C3×C4—36no—
62C22×C3×C4—36no—
63C22×C3×C4—36no—
64C22×C3×C4—312no—
65C22×C3×C4—64no—
66C22×C3×C4—63no—
67C22×C3×C4S411yescharacteristic
68C22×C3×C4C22×D314yes—
69C22×Dic3—312no—
70C22×Dic3—312no—
71C22×Dic3—66no—
72C22×Dic3—68no—
73C22×Dic3—62no—
74C2×Dic6—316no—
75C2×Dic6—612no—
76C2×Dic6—624no—
77C2×Dic6—612no—
78#217—324no—
79#217—38no—
80C4⋊Dic3—316no—
81Dic3⋊C4—332no—
82C22×C9—44no—
83C22×C9—43no—
84C22⋊C9C22×Q811yescharacteristic, complemented
85C4×C9—44no—
86Dic9—48no—
87C25C2×D911yescharacteristic
88C22×Q8—912no—
89C22×Q8—181no—
90C22×Q8—186no—
91C22×Q8—181no—
92C23×C4—31no—
93C23×C4—31no—
94C23×C4—36no—
95C23×C4—92no—
96C23×C4—92no—
97C23×C4C2×D916yes—
98C22⋊Q8—948no—
99C22⋊Q8—916no—
100C2×C4⋊C4—924no—
101C2×C4⋊C4—912no—
102C2×C22⋊C4—912no—
103C2×C22⋊C4—912no—
104C23×C3—34no—
105C23×C3—34no—
106C23×C3—33no—
107C23×C3—33no—
108C23×C3—33no—
109C23×C3—312no—
110C23×C3—36no—
111C23×C3—61no—
112C23×C3—66no—
113C23×C3C2×Dic616yes—
114C23×C3C2×S411yescharacteristic
115C23×C3C23×D311yescharacteristic
116C2×C3×C4—36no—
117C2×C3×C4—34no—
118C2×C3×C4—312no—
119C2×C3×C4—34no—
120C2×C3×C4—66no—
121C2×C3×C4C2×S416yes—
122C2×Dic3—38no—
123C2×Dic3—38no—
124C2×Dic3—612no—
125C2×Dic3—624no—
126C2×Dic3—612no—
127Dic6—616no—
128Dic6—616no—
129C2×C9—46no—
130C2×C9—41no—
131C24—31no—
132C24—33no—
133C24—34no—
134C24Dic1814yescomplemented
135C24C22×D913yes—
136C2×Q8—916no—
137C2×Q8—1812no—
138C2×Q8—1824no—
139C2×Q8—1812no—
140C22×C4—31no—
141C22×C4—36no—
142C22×C4—36no—
143C22×C4—36no—
144C22×C4—312no—
145C22×C4—64no—
146C22×C4—63no—
147C22×C4—912no—
148C22×C4—912no—
149C22×C4—186no—
150C22×C4—188no—
151C22×C4—182no—
152C22×C4(C22⋊C9)⋊C211yescharacteristic
153C22×C4C22×D914yes—
154C4⋊C4—932no—
155C4⋊C4—916no—
156C22⋊C4—924no—
157C22⋊C4—98no—
158C22×C3—36no—
159C22×C3—31no—
160C22×C3—36no—
161C22×C3—36no—
162C22×C3—312no—
163C22×C3—31no—
164C22×C3—33no—
165C22×C3—63no—
166C22×C3—64no—
167C22×C3A4⋊Q814yes—
168C22×C3C22×Dic611yescharacteristic
169C22×C3C22×S413yes—
170C3×C4—34no—
171C3×C4C22×S414yes—
172Dic3—68no—
173Dic3—68no—
174C9—41noSylow
175C23—33no—
176C23—33no—
177C23—34no—
178C23—34no—
179C23—33no—
180C23—312no—
181C23—36no—
182C23—61no—
183C23—66no—
184C23C2×Dic1816yescomplemented
185C23C2×(C22⋊C9)⋊C211yescharacteristic
186C23C23×D911yescharacteristic
187Q8—1816no—
188Q8—1816no—
189C2×C4—36no—
190C2×C4—34no—
191C2×C4—312no—
192C2×C4—34no—
193C2×C4—66no—
194C2×C4—98no—
195C2×C4—98no—
196C2×C4—1812no—
197C2×C4—1824no—
198C2×C4—1812no—
199C2×C4C2×(C22⋊C9)⋊C216yes—
200C2×C3—31no—
201C2×C3—31no—
202C2×C3—36no—
203C2×C3C2×A4⋊Q816yes—
204C2×C3C23×S411yescharacteristic
205C22—36no—
206C22—31no—
207C22—31no—
208C22—33no—
209C22—36no—
210C22—36no—
211C22—312no—
212C22—63no—
213C22—64no—
214C22(C22⋊C9)⋊Q814yesdirect product factor
215C22C22×Dic1811yescharacteristic, complemented
216C22C22×(C22⋊C9)⋊C213yes—
217C4—34no—
218C4—188no—
219C4—188no—
220C4C22×(C22⋊C9)⋊C214yes—
221C3C22×A4⋊Q811yescharacteristic
222C2—31no—
223C2—31no—
224C2—36no—
225C2C2×(C22⋊C9)⋊Q816yesdirect product factor
226C2C23×(C22⋊C9)⋊C211yescharacteristic
subgroupG/H
centerC23C2×(C22⋊C9)⋊C2
commutator, cosocleC2×C22⋊C9C24
FrattiniC2×C3C23×S4
FittingC24×C3×C4D3
socle, prime radicalC25×C3C2×D3
hypercenterC22×C4(C22⋊C9)⋊C2
hypocenterC22⋊C9C22×Q8
LCSG ▷ C2×C22⋊C9 ▷ C22⋊C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×(C22⋊C9)⋊Q8C2direct (N × H)
9. (C22⋊C9)⋊Q8C22direct (N × H)
17. C22×C22⋊C9Q8semidirect
43. C2×C22⋊C9C2×Q8semidirect
84. C22⋊C9C22×Q8semidirect
134. C24Dic18semidirect
184. C23C2×Dic18semidirect
214. C22(C22⋊C9)⋊Q8direct (N × H)
215. C22C22×Dic18semidirect
225. C2C2×(C22⋊C9)⋊Q8direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊C92
C22×(C22⋊C9)⋊C42
C2×(C22⋊C9)⋊Q82
C22×#9243
C22×Dic184