Finite Group Databasebeta

← C22×#64265 #11923726

Subgroups — C22×#64265 #11923726

Order 1152.

1776 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1776 conjugacy classes of subgroups — too many to draw. Only its 201 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×C3×#41⋊C3C211yesmaximal, characteristic
2C2×#64265C2112yesmaximal, direct product factor
3C22×C3⋊#226C212yesmaximal, complemented
4C22×#984—33nomaximal
5C22×#951—31nomaximal
6C22×#634—41nomaximal
7C2×C3×#41⋊C3C2216yes—
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9#64265C22116yesdirect product factor
10C2×C3⋊#226C22112yescomplemented
11C22×C3×#41D311yescharacteristic
12C23×Dic6—31no—
13C22×#41⋊C3D313yes—
14C2×#984—336no—
15C22×#226—36no—
16C2×#951—312no—
17C2×#951—312no—
18C22×#216—32no—
19C22×#214—32no—
20C22×#208—31no—
21C22×C32×C4—41no—
22C22×C3⋊Dic3—42no—
23C2×#634—412no—
24C3×#41⋊C3C2314yes—
25C2×C3×SL(2,𝔽3)C2313yes—
26C3⋊#226C2318yescomplemented
27C22×#137—91noSylow
28C2×C3×#41—36no—
29C2×C3×#41C2×D316yes—
30C22×C3×Q8C2×D311yescharacteristic
31C22×C3×D4—31no—
32C23×C3×C4—31no—
33C23×Dic3—61no—
34C22×Dic6—32no—
35C22×Dic6—312no—
36C22×Dic6—66no—
37C22×Dic6—61no—
38C22×Dic6—123no—
39C2×#41⋊C3C2×D3118yes—
40C22×SL(2,𝔽3)C2×D313yes—
41#984—348no—
42C2×#226—336no—
43#951—348no—
44#951—316no—
45C2×#216—312no—
46C2×#216—312no—
47C2×#214—324no—
48C2×#208—312no—
49C22×C3⋊C8—32no—
50C23×C32—41no—
51C2×C32×C4—46no—
52C2×C3⋊Dic3—412no—
53#634—416no—
54C3×SL(2,𝔽3)C2411yescharacteristic
55C22×#41C3⋊D311yescharacteristic
56C23×Q8—91no—
57C2×#137—912no—
58C2×#137—912no—
59C22×Q16—92no—
60C22×SD16—92no—
61C22×M16—91no—
62C24×C3—31no—
63C3×#41—312no—
64C3×#41—64no—
65C3×#41C22×D314yes—
66C2×C3×Q8—33no—
67C2×C3×Q8C22×D313yes—
68C2×C3×D4—36no—
69C2×C3×D4—63no—
70C22×C3×C4—31no—
71C22×C3×C4—36no—
72C22×C3×C4—36no—
73C22×C3×C4—43no—
74C22×C3×C4S411yescharacteristic
75C22×Dic3—612no—
76C22×Dic3—62no—
77C22×Dic3—126no—
78C2×Dic6—312no—
79C2×Dic6—316no—
80C2×Dic6—66no—
81C2×Dic6—624no—
82C2×Dic6—612no—
83C2×Dic6—1236no—
84#41⋊C3C22×D3112yes—
85C2×SL(2,𝔽3)C22×D319yes—
86#226—324no—
87#216—324no—
88#216—38no—
89#214—332no—
90#208—316no—
91C2×C3⋊C8—312no—
92C22×C32—44no—
93C22×C32—43no—
94C32×C4—44no—
95C3⋊Dic3—48no—
96C2×#41—36no—
97C2×#41C2×C3⋊D316yes—
98C22×Q8—92no—
99C22×Q8—912no—
100C22×Q8—186no—
101C22×Q8—181no—
102C22×Q8C2×C3⋊D311yescharacteristic
103C22×D4—31no—
104C23×C4—31no—
105C23×C4—181no—
106#137—948no—
107#137—916no—
108C2×Q16—912no—
109C2×Q16—912no—
110C2×SD16—924no—
111C2×M16—912no—
112C22×C8—92no—
113C23×C3—36no—
114C23×C3—43no—
115C23×C3—64no—
116C23×C3C2×S411yescharacteristic
117C3×Q8—33no—
118C3×Q8—61no—
119C3×Q8C23×D311yescharacteristic
120C3×D4—34no—
121C3×D4—66no—
122C2×C3×C4—33no—
123C2×C3×C4—33no—
124C2×C3×C4—34no—
125C2×C3×C4—312no—
126C2×C3×C4—418no—
127C2×C3×C4C2×S416yes—
128C2×Dic3—616no—
129C2×Dic3—612no—
130C2×Dic3—1236no—
131Dic6—38no—
132Dic6—612no—
133Dic6—616no—
134Dic6—1248no—
135SL(2,𝔽3)C23×D313yes—
136C3⋊C8—38no—
137C2×C32—46no—
138C2×C32—41no—
139C24—31no—
140#41—312no—
141#41—64no—
142#41C22×C3⋊D314yes—
143C2×Q8—33no—
144C2×Q8—912no—
145C2×Q8—916no—
146C2×Q8—186no—
147C2×Q8—1824no—
148C2×Q8—1812no—
149C2×Q8C22×C3⋊D313yes—
150C2×D4—36no—
151C2×D4—63no—
152C22×C4—31no—
153C22×C4—36no—
154C22×C4—36no—
155C22×C4—1812no—
156C22×C4—182no—
157C22×C4C3⋊S411yescharacteristic
158Q16—924no—
159Q16—98no—
160SD16—932no—
161M16—916no—
162C2×C8—912no—
163C22×C3—34no—
164C22×C3—412no—
165C22×C3—49no—
166C22×C3—612no—
167C22×C3#98414yescomplemented
168C22×C3C22×S413yes—
169C3×C4—31no—
170C3×C4—33no—
171C3×C4—412no—
172C3×C4C22×S414yes—
173Dic3—68no—
174Dic3—1224no—
175C32—41noSylow
176C23—36no—
177C23—64no—
178C23C2×C3⋊S411yescharacteristic
179Q8—33no—
180Q8—61no—
181Q8—98no—
182Q8—1812no—
183Q8—1816no—
184Q8C23×C3⋊D311yescharacteristic
185D4—34no—
186D4—66no—
187C2×C4—33no—
188C2×C4—33no—
189C2×C4—34no—
190C2×C4—312no—
191C2×C4—1816no—
192C2×C4—1812no—
193C2×C4C2×C3⋊S416yes—
194C8—98no—
195C2×C3—43no—
196C2×C3—418no—
197C2×C3—64no—
198C2×C3C2×#98416yescomplemented
199C2×C3C23×S411yescharacteristic
200C22—34no—
201C22—612no—
202C22#6426514yesdirect product factor
203C22C22×C3⋊S413yes—
204C4—31no—
205C4—33no—
206C4—188no—
207C4C22×C3⋊S414yes—
208C3—43no—
209C3C22×#98411yescharacteristic, complemented
210C2—64no—
211C2C2×#6426516yesdirect product factor
212C2C23×C3⋊S411yescharacteristic
subgroupG/H
centerC23C2×C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C24
FrattiniC2C23×C3⋊S4
FittingC22×C3×#41D3
socleC23×C3C2×S4
prime radicalC22×C3×#41⋊C3C2
hypercenterC22×C4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#64265C2direct (N × H)
3. C22×C3⋊#226C2semidirect
9. #64265C22direct (N × H)
10. C2×C3⋊#226C22semidirect
26. C3⋊#226C23semidirect
167. C22×C3#984semidirect
198. C2×C3C2×#984semidirect
202. C22#64265direct (N × H)
209. C3C22×#984semidirect
211. C2C2×#64265direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×#642652
C22×C3⋊#2262
C22×#9843
C22×#9513
C22×#6344