Finite Group Databasebeta

← C22×#88430 #13150590

Subgroups — C22×#88430 #13150590

Order 1728.

1119 conjugacy classes of subgroups; only the 299 normal subgroups are shown.

normal subgroups only

This group has 1119 conjugacy classes of subgroups — too many to draw. Only its 299 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×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C22×#6180C211yesmaximal, characteristic, complemented
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4C2×#88430C2112yesmaximal, direct product factor
5C22×#3712—31nomaximal
6C22×#3711—31nomaximal
7C23×(C3⋊D3)⋊C3C411yescharacteristic
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3C411yescharacteristic
10C2×C4×(C3⋊D3)⋊C3C416yes—
11C2×C4×(C3⋊D3)⋊C3C2216yes—
12C2×#6180C2216yescomplemented
13C2×(C3⋊(C3⋊C8))⋊C3C2216yescomplemented
14#88430C22116yesdirect product factor
15C22×C4×C3⋊D3D311yescharacteristic
16C22×C3×C4×D3—31no—
17C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
18C22×C3×C3⋊C8—31no—
19C22×C3×C3⋊C8—31no—
20C22×C3×C3⋊C8—31no—
21C22×C3×C3⋊C8—61no—
22C2×#3712—312no—
23C2×#3711—312no—
24C23×H(3,𝔽3)C2×C411yescharacteristic
25C22×(C3⋊D3)⋊C3—24no—
26C22×(C3⋊D3)⋊C3C2×C416yes—
27C2×C4×H(3,𝔽3)C2×C413yes—
28C2×C4×H(3,𝔽3)C2313yes—
29C2×(C3⋊Dic3)⋊C3C2×C416yes—
30C4×(C3⋊D3)⋊C3C2×C4112yes—
31C4×(C3⋊D3)⋊C3C2314yes—
32#6180C2314yescomplemented
33(C3⋊(C3⋊C8))⋊C3C2314yescomplemented
34C22×#208—91no—
35C22×ΓL(1,𝔽25)—91no—
36C22×ΓL(1,𝔽25)—91no—
37C23×C3⋊D3Dic311yescharacteristic
38C23×C3×D3—31no—
39C22×C32×C4—21no—
40C22×C32×C4C2×D311yescharacteristic
41C22×C32×C4C2×D311yescharacteristic
42C22×C3⋊Dic3Dic311yescharacteristic
43C2×C4×C3⋊D3Dic316yes—
44C2×C4×C3⋊D3C2×D316yes—
45C22×C3×Dic3—31no—
46C2×C3×C4×D3—36no—
47C2×C3×C4×D3—36no—
48C2×C3⋊(C3⋊C8)C2×D316yescomplemented
49C2×C3×C3⋊C8—36no—
50C2×C3×C3⋊C8—36no—
51C2×C3×C3⋊C8—36no—
52C2×C3×C3⋊C8—66no—
53#3712—316no—
54#3711—316no—
55C22×H(3,𝔽3)M1614yescomplemented
56C22×H(3,𝔽3)C22×C413yes—
57C2×(C3⋊D3)⋊C3—212no—
58C2×(C3⋊D3)⋊C3C22×C414yes—
59C4×H(3,𝔽3)C22×C413yes—
60C4×H(3,𝔽3)C2411yescharacteristic
61(C3⋊Dic3)⋊C3C22×C414yes—
62C23×C3×C4—91no—
63C22×C4×D3—31no—
64C22×C4×D3—91no—
65C22×C3×C8—91no—
66C22×C3×C8—91no—
67C2×#208—912no—
68C22×C3⋊C8—31no—
69C22×C3⋊C8—31no—
70C22×C3⋊C8—31no—
71C22×C3⋊C8—61no—
72C22×C3⋊C8—91no—
73C2×ΓL(1,𝔽25)—912no—
74C2×ΓL(1,𝔽25)—912no—
75C23×C32—21no—
76C23×C32C4×D311yescharacteristic
77C23×C32C2×Dic311yescharacteristic
78C22×C3⋊D3—24no—
79C22×C3⋊D3C2×Dic316yes—
80C22×C3×D3—36no—
81C22×C3×D3—64no—
82C2×C32×C4—23no—
83C2×C32×C4—23no—
84C2×C32×C4C4×D313yes—
85C2×C32×C4C2×Dic313yes—
86C2×C32×C4C22×D313yes—
87C2×C32×C4C22×D313yes—
88C2×C3⋊Dic3C2×Dic316yes—
89C4×C3⋊D3C2×Dic3112yes—
90C4×C3⋊D3C22×D314yes—
91C2×C3×Dic3—36no—
92C3×C4×D3—312no—
93C3×C4×D3—34no—
94C3⋊(C3⋊C8)C22×D314yescomplemented
95C3×C3⋊C8—34no—
96C3×C3⋊C8—34no—
97C3×C3⋊C8—34no—
98C3×C3⋊C8—64no—
99C22×M16—271noSylow
100C2×H(3,𝔽3)C2×M1616yescomplemented
101C2×H(3,𝔽3)C23×C411yescharacteristic
102(C3⋊D3)⋊C3—24no—
103C24×C3—91no—
104C23×D3—31no—
105C23×D3—91no—
106C22×C3×C4—31no—
107C22×C3×C4—31no—
108C22×C3×C4—61no—
109C22×C3×C4—96no—
110C22×C3×C4—96no—
111C22×C3×C4—91no—
112C22×C3×C4D3211yescharacteristic
113C22×Dic3—31no—
114C22×Dic3—91no—
115C2×C4×D3—36no—
116C2×C4×D3—36no—
117C2×C4×D3—96no—
118C2×C4×D3—96no—
119C2×C3×C8—96no—
120C2×C3×C8—96no—
121#208—916no—
122C2×C3⋊C8—36no—
123C2×C3⋊C8—36no—
124C2×C3⋊C8—36no—
125C2×C3⋊C8—66no—
126C2×C3⋊C8—96no—
127ΓL(1,𝔽25)—916no—
128ΓL(1,𝔽25)—916no—
129C22×C32—24no—
130C22×C32—23no—
131C22×C32ΓL(1,𝔽25)14yescomplemented
132C22×C32#20814yescomplemented
133C22×C32C2×C4×D313yes—
134C22×C32C22×Dic313yes—
135C2×C3⋊D3—212no—
136C2×C3⋊D3C22×Dic314yes—
137C2×C3×D3—34no—
138C2×C3×D3—612no—
139C32×C4—21no—
140C32×C4—23no—
141C32×C4C2×C4×D313yes—
142C32×C4C22×Dic313yes—
143C32×C4C23×D311yescharacteristic
144C32×C4C23×D311yescharacteristic
145C3⋊Dic3C22×Dic314yes—
146C3×Dic3—34no—
147C23×C4—91no—
148C2×M16—2712no—
149C22×C8—91no—
150C22×C8—91no—
151H(3,𝔽3)C22×M1611yescharacteristic, complemented, Sylow
152C23×C3—31no—
153C23×C3—31no—
154C23×C3—61no—
155C23×C3—96no—
156C23×C3—184no—
157C23×C3D3×Dic311yescharacteristic
158C22×D3—36no—
159C22×D3—64no—
160C22×D3—96no—
161C22×D3—184no—
162C2×C3×C4—33no—
163C2×C3×C4—33no—
164C2×C3×C4—33no—
165C2×C3×C4—33no—
166C2×C3×C4—63no—
167C2×C3×C4—63no—
168C2×C3×C4—94no—
169C2×C3×C4—912no—
170C2×C3×C4—96no—
171C2×C3×C4D3×Dic313yes—
172C2×C3×C4C2×D3213yes—
173C2×Dic3—36no—
174C2×Dic3—96no—
175C4×D3—34no—
176C4×D3—312no—
177C4×D3—912no—
178C4×D3—94no—
179C3×C8—94no—
180C3×C8—94no—
181C3⋊C8—34no—
182C3⋊C8—34no—
183C3⋊C8—34no—
184C3⋊C8—64no—
185C3⋊C8—94no—
186C2×C32—21no—
187C2×C32—26no—
188C2×C32C2×ΓL(1,𝔽25)16yescomplemented
189C2×C32C2×#20816yescomplemented
190C2×C32C22×C4×D311yescharacteristic
191C2×C32C23×Dic311yescharacteristic
192C3⋊D3—24no—
193C3×D3—64no—
194C24—91no—
195C22×C4—91no—
196C22×C4—96no—
197C22×C4—96no—
198C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
199M16—2716no—
200C2×C8—96no—
201C2×C8—96no—
202C22×C3—33no—
203C22×C3—33no—
204C22×C3—34no—
205C22×C3—34no—
206C22×C3—63no—
207C22×C3—64no—
208C22×C3—94no—
209C22×C3—1812no—
210C22×C3#371114yes—
211C22×C3C2×D3×Dic313yes—
212C2×D3—34no—
213C2×D3—612no—
214C2×D3—94no—
215C2×D3—1812no—
216C3×C4—31no—
217C3×C4—31no—
218C3×C4—33no—
219C3×C4—33no—
220C3×C4—61no—
221C3×C4—63no—
222C3×C4—94no—
223C3×C4C2×D3×Dic313yes—
224C3×C4C22×D3211yescharacteristic
225Dic3—34no—
226Dic3—94no—
227C32—21no—
228C32C22×ΓL(1,𝔽25)11yescharacteristic, complemented
229C32C22×#20811yescharacteristic, complemented
230C23—96no—
231C23—184no—
232C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
233C2×C4—96no—
234C2×C4—94no—
235C2×C4—912no—
236C2×C4((C3⋊D3)⋊C3)⋊C413yes—
237C2×C4C2×((C3⋊D3)⋊C3)⋊C213yes—
238C8—94no—
239C8—94no—
240C2×C3—31no—
241C2×C3—31no—
242C2×C3—36no—
243C2×C3—36no—
244C2×C3—61no—
245C2×C3—66no—
246C2×C3—184no—
247C2×C3C2×#371116yes—
248C2×C3C22×D3×Dic311yescharacteristic
249D3—64no—
250D3—184no—
251C22—94no—
252C22—1812no—
253C22#8843014yesdirect product factor
254C22C2×((C3⋊D3)⋊C3)⋊C413yes—
255C4—94no—
256C4C2×((C3⋊D3)⋊C3)⋊C413yes—
257C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
258C3—31no—
259C3—31no—
260C3—61no—
261C3C22×#371111yescharacteristic
262C2—184no—
263C2C2×#8843016yesdirect product factor
264C2C22×((C3⋊D3)⋊C3)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC3×C4C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3D3×Dic3
prime radicalC23×(C3⋊D3)⋊C3C4
cosocleC4×H(3,𝔽3)C24
hypocenterH(3,𝔽3)C22×M16
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×#6180C2semidirect
3. C22×(C3⋊(C3⋊C8))⋊C3C2semidirect
4. C2×#88430C2direct (N × H)
12. C2×#6180C22semidirect
13. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
14. #88430C22direct (N × H)
17. C22×C3⋊(C3⋊C8)D3semidirect
32. #6180C23semidirect
33. (C3⋊(C3⋊C8))⋊C3C23semidirect
48. C2×C3⋊(C3⋊C8)C2×D3semidirect
55. C22×H(3,𝔽3)M16semidirect
94. C3⋊(C3⋊C8)C22×D3semidirect
100. C2×H(3,𝔽3)C2×M16semidirect
131. C22×C32ΓL(1,𝔽25)semidirect
132. C22×C32#208semidirect
151. H(3,𝔽3)C22×M16semidirect
188. C2×C32C2×ΓL(1,𝔽25)semidirect
189. C2×C32C2×#208semidirect
228. C32C22×ΓL(1,𝔽25)semidirect
229. C32C22×#208semidirect
253. C22#88430direct (N × H)
263. C2C2×#88430direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×(C3⋊D3)⋊C32
C22×#61802
C22×(C3⋊(C3⋊C8))⋊C32
C2×#884302
C22×#37123
C22×#37113