Finite Group Databasebeta

← C22×#4623 #11712540

Subgroups — C22×#4623 #11712540

Order 768.

1012 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 1012 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C23×C42⋊C3C211yesmaximal, characteristic
2C2×#4623C216yesmaximal, direct product factor
3C22×#339—31nomaximal, Sylow
4C22×SL(2,ℤ/4)—41nomaximal
5C22×C42⋊C3C414yescomplemented
6C22×C42⋊C3C2213yes—
7#4623C2214yesdirect product factor
8C23×M16—31no—
9C22×#117—31no—
10C23×C42D311yescharacteristic
11C2×#339—36no—
12C2×#339—36no—
13C23×A4—41no—
14C2×SL(2,ℤ/4)—46no—
15C2×C42⋊C3C2×C416yescomplemented
16C2×C42⋊C3C2311yescharacteristic
17C24×C4—31no—
18C22×M16—36no—
19C22×M16—38no—
20C22×M16—63no—
21C22×M16—64no—
22C23×C8—61no—
23C2×#117—36no—
24C2×#117—36no—
25C2×#117—63no—
26C2×#117—63no—
27C22×C4⋊C4—31no—
28C22×C4⋊C4—31no—
29C22×C22⋊C4—61no—
30C22×C42—34no—
31C22×C42—33no—
32C22×C42—61no—
33C22×C42Dic314yescomplemented
34C22×C42C2×D313yes—
35#339—312no—
36#339—34no—
37C22×A4—44no—
38C22×A4—43no—
39C22×Dic3—161no—
40SL(2,ℤ/4)—44no—
41C42⋊C3C22×C411yescharacteristic, complemented
42C25S411yescharacteristic
43C23×C4—31no—
44C23×C4—33no—
45C23×C4—33no—
46C23×C4—31no—
47C23×C4—34no—
48C23×C4—34no—
49C23×C4—34no—
50C23×C4—34no—
51C23×C4—33no—
52C23×C4—33no—
53C23×C4—61no—
54C23×C4—61no—
55C2×M16—324no—
56C2×M16—34no—
57C2×M16—66no—
58C2×M16—624no—
59C2×M16—612no—
60C22×C8—66no—
61C22×C8—68no—
62#117—312no—
63#117—34no—
64#117—66no—
65#117—612no—
66#117—66no—
67C2×C4⋊C4—36no—
68C2×C4⋊C4—36no—
69C2×C4⋊C4—63no—
70C2×C4⋊C4—63no—
71C2×C22⋊C4—66no—
72C2×C22⋊C4—66no—
73C2×C42—33no—
74C2×C42—312no—
75C2×C42—36no—
76C2×C42—66no—
77C2×C42—61no—
78C2×C42—66no—
79C2×C42—66no—
80C2×C42C2×Dic316yescomplemented
81C2×C42C22×D311yescharacteristic
82C23×C3—161no—
83C2×A4—46no—
84C2×A4—41no—
85C2×Dic3—166no—
86C24—31no—
87C24—33no—
88C24—34no—
89C24SL(2,ℤ/4)14yes—
90C24C2×S413yes—
91C22×C4—33no—
92C22×C4—33no—
93C22×C4—33no—
94C22×C4—33no—
95C22×C4—33no—
96C22×C4—33no—
97C22×C4—36no—
98C22×C4—34no—
99C22×C4—36no—
100C22×C4—312no—
101C22×C4—34no—
102C22×C4—34no—
103C22×C4—34no—
104C22×C4—312no—
105C22×C4—312no—
106C22×C4—312no—
107C22×C4—312no—
108C22×C4—36no—
109C22×C4—36no—
110C22×C4—36no—
111C22×C4—312no—
112C22×C4—31no—
113C22×C4—33no—
114C22×C4—66no—
115C22×C4—66no—
116C22×C4—124no—
117C22×C4—123no—
118C22×C4—121no—
119M16—38no—
120M16—624no—
121M16—64no—
122C2×C8—64no—
123C2×C8—624no—
124C4⋊C4—34no—
125C4⋊C4—34no—
126C4⋊C4—66no—
127C4⋊C4—66no—
128C22⋊C4—612no—
129C22⋊C4—64no—
130C42—36no—
131C42—31no—
132C42—63no—
133C42—64no—
134C42—64no—
135C42—612no—
136C42C22×Dic311yescharacteristic, complemented
137C22×C3—163no—
138C22×C3—164no—
139A4—41no—
140Dic3—164no—
141C23—33no—
142C23—33no—
143C23—34no—
144C23—34no—
145C23—33no—
146C23—312no—
147C23—36no—
148C23—61no—
149C23—66no—
150C23#85711yescharacteristic
151C23C2×SL(2,ℤ/4)16yes—
152C23C22×S411yescharacteristic
153C2×C4—31no—
154C2×C4—33no—
155C2×C4—31no—
156C2×C4—33no—
157C2×C4—36no—
158C2×C4—31no—
159C2×C4—36no—
160C2×C4—36no—
161C2×C4—36no—
162C2×C4—36no—
163C2×C4—324no—
164C2×C4—36no—
165C2×C4—31no—
166C2×C4—36no—
167C2×C4—312no—
168C2×C4—36no—
169C2×C4—36no—
170C2×C4—312no—
171C2×C4—31no—
172C2×C4—31no—
173C2×C4—36no—
174C2×C4—64no—
175C2×C4—64no—
176C2×C4—1212no—
177C2×C4—126no—
178C2×C4—126no—
179C8—68no—
180C2×C3—161no—
181C2×C3—166no—
182C22—36no—
183C22—31no—
184C22—31no—
185C22—33no—
186C22—36no—
187C22—36no—
188C22—312no—
189C22—63no—
190C22—64no—
191C22#462314yesdirect product factor
192C22C2×#85713yes—
193C22C22×SL(2,ℤ/4)11yescharacteristic
194C4—31no—
195C4—31no—
196C4—36no—
197C4—31no—
198C4—31no—
199C4—36no—
200C4—124no—
201C4—124no—
202C3—161noSylow
203C2—31no—
204C2—31no—
205C2—36no—
206C2C2×#462316yesdirect product factor
207C2C22×#85711yescharacteristic
subgroupG/H
center, hypercenterC23#857
commutator, hypocenterC42⋊C3C22×C4
FrattiniC23C22×S4
FittingC23×C42D3
socleC25S4
prime radicalC23×C42⋊C3C2
cosocleC2×C42⋊C3C23
LCSG ▷ C42⋊C3
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#4623C2direct (N × H)
5. C22×C42⋊C3C4semidirect
7. #4623C22direct (N × H)
15. C2×C42⋊C3C2×C4semidirect
33. C22×C42Dic3semidirect
41. C42⋊C3C22×C4semidirect
80. C2×C42C2×Dic3semidirect
136. C42C22×Dic3semidirect
191. C22#4623direct (N × H)
206. C2C2×#4623direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C42⋊C32
C2×#46232
C22×#3393
C22×SL(2,ℤ/4)4