Finite Group Databasebeta

← C22×#88611 #13164488

Subgroups — C22×#88611 #13164488

Order 1728.

1120 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 1120 conjugacy classes of subgroups — too many to draw. Only its 94 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×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#88611C216yesmaximal, direct product factor
3C22×C3×GL(2,𝔽3)—33nomaximal
4C22×C3×#215—31nomaximal
5C23×#272—41nomaximal
6C2×Q8⋊H(3,𝔽3)C2213yescomplemented
7#88611C2214yesdirect product factor
8C22×C32×Q8D311yescharacteristic, complemented
9C22×C3×D12—31no—
10C22×C3×SL(2,𝔽3)D313yescomplemented
11C2×C3×GL(2,𝔽3)—318no—
12C2×C3×#215—36no—
13C2×C3×#215—36no—
14C22×C3×C3⋊C8—31no—
15C23×H(3,𝔽3)—41no—
16C22×#272—46no—
17C22×#272—48no—
18Q8⋊H(3,𝔽3)C2311yescharacteristic, complemented
19C22×GL(2,𝔽3)—33no—
20C22×C3×SD16—91no—
21C22×#215—31no—
22C23×C3×D3—61no—
23C23×C3×D3—123no—
24C2×C32×Q8—33no—
25C2×C32×Q8C2×D313yescomplemented
26C22×C32×C4—31no—
27C2×C3×D12—36no—
28C2×C3×D12—63no—
29C2×C3×SL(2,𝔽3)C2×D319yescomplemented
30C3×GL(2,𝔽3)—312no—
31C3×#215—312no—
32C3×#215—34no—
33C2×C3×C3⋊C8—36no—
34C22×H(3,𝔽3)—44no—
35C22×H(3,𝔽3)—43no—
36C2×#272—44no—
37C2×#272—424no—
38C22×C3×Q8—31no—
39C22×C3×Q8C3⋊D311yescharacteristic
40C22×C3×D4—91no—
41C22×D12—31no—
42C22×SL(2,𝔽3)—33no—
43C2×GL(2,𝔽3)—318no—
44C2×C3×SD16—96no—
45C2×C3×SD16—96no—
46C22×C3×C8—91no—
47C2×#215—36no—
48C2×#215—36no—
49C22×C3⋊C8—31no—
50C23×C32—43no—
51C23×C32S411yescharacteristic
52C22×C3×D3—66no—
53C22×C3×D3—1218no—
54C22×C3×D3—1224no—
55C22×C3×D3—124no—
56C32×Q8—33no—
57C32×Q8—61no—
58C32×Q8C22×D311yescharacteristic, complemented
59C2×C32×C4—33no—
60C2×C32×C4—33no—
61C3×D12—34no—
62C3×D12—66no—
63C3×SL(2,𝔽3)C22×D313yescomplemented
64C3×C3⋊C8—34no—
65C22×SD16—91noSylow
66C2×H(3,𝔽3)—46no—
67C2×H(3,𝔽3)—41no—
68#272—48no—
69C24×C3—181no—
70C23×D3—61no—
71C23×D3—123no—
72C2×C3×Q8—33no—
73C2×C3×Q8—33no—
74C2×C3×Q8—33no—
75C2×C3×Q8—63no—
76C2×C3×Q8C2×C3⋊D313yes—
77C2×C3×D4—96no—
78C2×C3×D4—183no—
79C22×C3×C4—31no—
80C22×C3×C4—31no—
81C22×C3×C4—61no—
82C2×D12—36no—
83C2×D12—63no—
84C2×SL(2,𝔽3)—39no—
85GL(2,𝔽3)—312no—
86C3×SD16—912no—
87C3×SD16—94no—
88C2×C3×C8—96no—
89#215—312no—
90#215—34no—
91C2×C3⋊C8—36no—
92C22×C32—49no—
93C22×C32—412no—
94C22×C32GL(2,𝔽3)14yescomplemented
95C22×C32C2×S413yes—
96C2×C3×D3—64no—
97C2×C3×D3—1212no—
98C2×C3×D3—1272no—
99C2×C3×D3—1212no—
100C32×C4—31no—
101C32×C4—33no—
102C22×Q8#27211yescharacteristic, complemented
103C22×D4—91no—
104C2×SD16—96no—
105C2×SD16—96no—
106C22×C8—91no—
107H(3,𝔽3)—41noSylow
108C23×C3—31no—
109C23×C3—123no—
110C23×C3—186no—
111C23×C3—364no—
112C23×C3C3⋊S411yescharacteristic
113C22×D3—66no—
114C22×D3—1218no—
115C22×D3—124no—
116C22×D3—1224no—
117C3×Q8—33no—
118C3×Q8—33no—
119C3×Q8—31no—
120C3×Q8—63no—
121C3×Q8—61no—
122C3×Q8—63no—
123C3×Q8C22×C3⋊D311yescharacteristic
124C3×D4—94no—
125C3×D4—186no—
126C2×C3×C4—33no—
127C2×C3×C4—33no—
128C2×C3×C4—33no—
129C2×C3×C4—33no—
130C2×C3×C4—63no—
131C2×C3×C4—63no—
132D12—34no—
133D12—66no—
134SL(2,𝔽3)—33no—
135C3×C8—94no—
136C3⋊C8—34no—
137C2×C32—418no—
138C2×C32—43no—
139C2×C32C2×GL(2,𝔽3)16yescomplemented
140C2×C32C22×S411yescharacteristic
141C3×D3—1224no—
142C3×D3—124no—
143C24—181no—
144C2×Q8—33no—
145C2×Q8C2×#27213yescomplemented
146C2×D4—96no—
147C2×D4—183no—
148C22×C4—31no—
149SD16—912no—
150SD16—94no—
151C2×C8—96no—
152C22×C3—34no—
153C22×C3—33no—
154C22×C3—1212no—
155C22×C3—129no—
156C22×C3—184no—
157C22×C3—3612no—
158C22×C3C3⋊GL(2,𝔽3)14yes—
159C22×C3C2×C3⋊S413yes—
160C2×D3—64no—
161C2×D3—1212no—
162C2×D3—1212no—
163C2×D3—1272no—
164C3×C4—31no—
165C3×C4—33no—
166C3×C4—31no—
167C3×C4—33no—
168C3×C4—61no—
169C3×C4—63no—
170C32—43no—
171C32C22×GL(2,𝔽3)11yescharacteristic, complemented
172C23—186no—
173C23—364no—
174C23#625811yescharacteristic
175Q8—33no—
176Q8—61no—
177Q8C22×#27211yescharacteristic, complemented
178D4—94no—
179D4—186no—
180C2×C4—33no—
181C2×C4—33no—
182C8—94no—
183C2×C3—31no—
184C2×C3—36no—
185C2×C3—123no—
186C2×C3—1218no—
187C2×C3—364no—
188C2×C3C2×C3⋊GL(2,𝔽3)16yes—
189C2×C3C22×C3⋊S411yescharacteristic
190D3—124no—
191D3—1224no—
192C22—184no—
193C22—3612no—
194C22#8861114yesdirect product factor
195C22C2×#625813yes—
196C4—31no—
197C4—33no—
198C3—31no—
199C3—123no—
200C3C22×C3⋊GL(2,𝔽3)11yescharacteristic
201C2—364no—
202C2C2×#8861116yesdirect product factor
203C2C22×#625811yescharacteristic
subgroupG/H
center, socle, hypercenterC23×C3C3⋊S4
commutator, cosocle, hypocenterQ8⋊H(3,𝔽3)C23
FrattiniC2×C3C22×C3⋊S4
FittingC22×C32×Q8D3
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C22×Q8⋊H(3,𝔽3)C2semidirect
2. C2×#88611C2direct (N × H)
6. C2×Q8⋊H(3,𝔽3)C22semidirect
7. #88611C22direct (N × H)
8. C22×C32×Q8D3semidirect
10. C22×C3×SL(2,𝔽3)D3semidirect
18. Q8⋊H(3,𝔽3)C23semidirect
25. C2×C32×Q8C2×D3semidirect
29. C2×C3×SL(2,𝔽3)C2×D3semidirect
58. C32×Q8C22×D3semidirect
63. C3×SL(2,𝔽3)C22×D3semidirect
94. C22×C32GL(2,𝔽3)semidirect
102. C22×Q8#272semidirect
139. C2×C32C2×GL(2,𝔽3)semidirect
145. C2×Q8C2×#272semidirect
171. C32C22×GL(2,𝔽3)semidirect
177. Q8C22×#272semidirect
194. C22#88611direct (N × H)
202. C2C2×#88611direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8⋊H(3,𝔽3)2
C2×#886112
C22×C3×GL(2,𝔽3)3
C22×C3×#2153
C23×#2724