Finite Group Databasebeta

← C22×#88691 #13164852

Subgroups — C22×#88691 #13164852

Order 1728.

1148 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 193 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×#88691C216yesmaximal, direct product factor
2C22×#6204C211yesmaximal, characteristic, complemented
3C22×C32×#41C311yesmaximal, characteristic
4C22×#41⋊C9C313yesmaximal, complemented
5C22×C4×3−1+2—41nomaximal
6#88691C2214yesdirect product factor
7C2×#6204C2213yescomplemented
8C2×C32×#41—36no—
9C2×C32×#41C2×C316yes—
10C22×C32×Q8C2×C311yescharacteristic
11C22×C32×D4—31no—
12C23×C32×C4—31no—
13C2×#41⋊C9C2×C3118yescomplemented
14C22×Q8⋊C9C2×C313yescomplemented
15C23×3−1+2—41no—
16C2×C4×3−1+2—46no—
17#6204C2311yescharacteristic, complemented
18C22×C3×#41—31no—
19C22×C3×#41C3211yescharacteristic
20C24×C32—31no—
21C32×#41—38no—
22C32×#41—312no—
23C32×#41C22×C314yes—
24C2×C32×Q8—33no—
25C2×C32×Q8C22×C313yes—
26C2×C32×D4—36no—
27C2×C32×D4—36no—
28C22×C32×C4—31no—
29C22×C32×C4—36no—
30C22×C32×C4—36no—
31C22×C32×C4A411yescharacteristic
32C22×C4×C9—43no—
33#41⋊C9C22×C3112yescomplemented
34C2×Q8⋊C9C22×C319yescomplemented
35C22×3−1+2—44no—
36C22×3−1+2—43no—
37C4×3−1+2—44no—
38C2×C3×#41—318no—
39C2×C3×#41—36no—
40C2×C3×#41—36no—
41C2×C3×#41C2×C3216yes—
42C22×C3×Q8—31no—
43C22×C3×Q8C2×C3211yescharacteristic
44C22×C3×D4—31no—
45C22×C3×D4—33no—
46C23×C3×C4—33no—
47C23×C3×C4—31no—
48C23×C32—36no—
49C23×C32—64no—
50C23×C32C2×A411yescharacteristic
51C32×Q8—33no—
52C32×Q8—32no—
53C32×Q8C23×C311yescharacteristic
54C32×D4—34no—
55C32×D4—312no—
56C2×C32×C4—33no—
57C2×C32×C4—33no—
58C2×C32×C4—312no—
59C2×C32×C4—34no—
60C2×C32×C4C2×A416yes—
61C23×C9—43no—
62C2×C4×C9—418no—
63Q8⋊C9C23×C313yescomplemented
64C22×#413−1+211yescharacteristic, complemented, Sylow
65C2×3−1+2—46no—
66C2×3−1+2—41no—
67C24×C3—33no—
68C24×C3—31no—
69C3×#41—324no—
70C3×#41—38no—
71C3×#41—336no—
72C3×#41—312no—
73C3×#41—34no—
74C3×#41C22×C3214yes—
75C2×C3×Q8—39no—
76C2×C3×Q8—33no—
77C2×C3×Q8—33no—
78C2×C3×Q8C22×C3213yes—
79C2×C3×D4—36no—
80C2×C3×D4—318no—
81C2×C3×D4—318no—
82C2×C3×D4—36no—
83C22×C3×C4—33no—
84C22×C3×C4—318no—
85C22×C3×C4—318no—
86C22×C3×C4—31no—
87C22×C3×C4—36no—
88C22×C3×C4—36no—
89C22×C3×C4—31no—
90C22×C3×C4C3×A411yescharacteristic
91C22×C32—34no—
92C22×C32—612no—
93C22×C32#41⋊C314yes—
94C22×C32C22×A413yes—
95C32×C4—33no—
96C32×C4—31no—
97C32×C4C22×A414yes—
98C22×C9—49no—
99C22×C9—412no—
100C4×C9—412no—
101C2×#41—36no—
102C2×#41C2×3−1+216yescomplemented
103C22×Q8C2×3−1+211yescharacteristic
104C22×D4—31no—
105C23×C4—31no—
1063−1+2—41noSylow
107C23×C3—318no—
108C23×C3—36no—
109C23×C3—31no—
110C23×C3—612no—
111C23×C3—64no—
112C23×C3C2×C3×A411yescharacteristic
113C3×Q8—33no—
114C3×Q8—39no—
115C3×Q8—36no—
116C3×Q8—32no—
117C3×Q8—31no—
118C3×Q8C23×C3211yescharacteristic
119C3×D4—34no—
120C3×D4—312no—
121C3×D4—336no—
122C3×D4—312no—
123C2×C3×C4—39no—
124C2×C3×C4—39no—
125C2×C3×C4—312no—
126C2×C3×C4—336no—
127C2×C3×C4—33no—
128C2×C3×C4—33no—
129C2×C3×C4—312no—
130C2×C3×C4—34no—
131C2×C3×C4—36no—
132C2×C3×C4C2×C3×A416yes—
133C2×C32—64no—
134C2×C32C2×#41⋊C316yes—
135C2×C32C23×A411yescharacteristic
136C2×C9—418no—
137C2×C9—43no—
138C24—31no—
139#41—38no—
140#41—312no—
141#41C22×3−1+214yescomplemented
142C2×Q8—33no—
143C2×Q8C22×3−1+213yes—
144C2×D4—36no—
145C2×D4—36no—
146C22×C4—31no—
147C22×C4—36no—
148C22×C4—36no—
149C22×C4#109411yescharacteristic
150C22×C3—312no—
151C22×C3—34no—
152C22×C3—33no—
153C22×C3—34no—
154C22×C3—636no—
155C22×C3—612no—
156C22×C3C3×#41⋊C314yes—
157C22×C3C22×C3×A413yes—
158C3×C4—33no—
159C3×C4—39no—
160C3×C4—31no—
161C3×C4—33no—
162C3×C4—34no—
163C3×C4C22×C3×A414yes—
164C32C22×#41⋊C311yescharacteristic
165C9—43no—
166C23—36no—
167C23—64no—
168C23C2×#109411yescharacteristic
169Q8—33no—
170Q8—32no—
171Q8C23×3−1+211yescharacteristic
172D4—34no—
173D4—312no—
174C2×C4—33no—
175C2×C4—33no—
176C2×C4—34no—
177C2×C4—312no—
178C2×C4C2×#109416yes—
179C2×C3—31no—
180C2×C3—36no—
181C2×C3—612no—
182C2×C3—64no—
183C2×C3C2×C3×#41⋊C316yes—
184C2×C3C23×C3×A411yescharacteristic
185C22—34no—
186C22—612no—
187C22#8869114yesdirect product factor
188C22C22×#109413yes—
189C4—31no—
190C4—33no—
191C4C22×#109414yes—
192C3—31no—
193C3C22×C3×#41⋊C311yescharacteristic
194C2—64no—
195C2C2×#8869116yesdirect product factor
196C2C23×#109411yescharacteristic
subgroupG/H
centerC22×C3×C4C3×A4
commutator, cosocleC3×Q8C23×C32
FrattiniC2×C3C23×C3×A4
Fitting, prime radicalC22×C32×#41C3
socleC23×C3C2×C3×A4
hypercenterC22×C32×C4A4
hypocenterQ8C23×3−1+2
LCSG ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C22×C3×C4 ◁ C22×C32×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88691C2direct (N × H)
2. C22×#6204C2semidirect
4. C22×#41⋊C9C3semidirect
6. #88691C22direct (N × H)
7. C2×#6204C22semidirect
13. C2×#41⋊C9C2×C3semidirect
14. C22×Q8⋊C9C2×C3semidirect
17. #6204C23semidirect
33. #41⋊C9C22×C3semidirect
34. C2×Q8⋊C9C22×C3semidirect
63. Q8⋊C9C23×C3semidirect
64. C22×#413−1+2semidirect
102. C2×#41C2×3−1+2semidirect
141. #41C22×3−1+2semidirect
187. C22#88691direct (N × H)
195. C2C2×#88691direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886912
C22×#62042
C22×C32×#413
C22×#41⋊C93
C22×C4×3−1+24