Finite Group Databasebeta

← C2×C8×C22⋊H(3,𝔽3) #13152213

Subgroups — C2×C8×C22⋊H(3,𝔽3) #13152213

Order 1728.

536 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 110 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×C4×C22⋊H(3,𝔽3)C211yesmaximal, characteristic
2C8×C22⋊H(3,𝔽3)C212yesmaximal, direct product factor
3C23×C32×C8C311yesmaximal, characteristic, complemented
4C2×C3×C8×A4C313yesmaximal, complemented
5C22×C22⋊H(3,𝔽3)C411yescharacteristic
6C4×C22⋊H(3,𝔽3)C411yescharacteristic
7C4×C22⋊H(3,𝔽3)C2211yescharacteristic
8C2×C8×H(3,𝔽3)—41nomaximal
9C23×C32×C4C2×C311yescharacteristic
10C2×C3×C4×A4C2×C313yes—
11C22×C32×C8—31no—
12C22×C32×C8—31no—
13C22×C32×C8—32no—
14C22×C32×C8C2×C312yescomplemented
15C3×C8×A4C2×C316yescomplemented
16C2×C22⋊H(3,𝔽3)C812yesdirect product factor
17C2×C22⋊H(3,𝔽3)C2×C411yescharacteristic
18C2×C4×H(3,𝔽3)—41no—
19C8×H(3,𝔽3)—42no—
20C23×C3×C8—31no—
21C23×C3×C8C3211yescharacteristic
22C2×C8×A4—33no—
23C24×C32C3×C411yescharacteristic
24C22×C3×A4C3×C413yes—
25C22×C32×C4—31no—
26C22×C32×C4—31no—
27C22×C32×C4—31no—
28C22×C32×C4—31no—
29C22×C32×C4C3×C411yescharacteristic
30C22×C32×C4C22×C311yescharacteristic
31C3×C4×A4C3×C413yes—
32C3×C4×A4C22×C313yes—
33C2×C32×C8—32no—
34C2×C32×C8—31no—
35C2×C32×C8—32no—
36C2×C32×C8—32no—
37C2×C32×C8—32no—
38C2×C32×C8—43no—
39C2×C32×C8A411yescharacteristic, complemented
40C22×H(3,𝔽3)—41no—
41C22⋊H(3,𝔽3)C2×C811yescharacteristic, direct product factor
42C4×H(3,𝔽3)—41no—
43C4×H(3,𝔽3)—41no—
44C23×C3×C4—31no—
45C23×C3×C4C2×C3211yescharacteristic
46C2×C4×A4—33no—
47C22×C3×C8—33no—
48C22×C3×C8—33no—
49C22×C3×C8—36no—
50C22×C3×C8—31no—
51C22×C3×C8—31no—
52C22×C3×C8—32no—
53C22×C3×C8—32no—
54C22×C3×C8C2×C3212yes—
55C8×A4—36no—
56C23×C32—32no—
57C23×C32—31no—
58C23×C32—31no—
59C23×C32C3×C812yescomplemented
60C23×C32C2×C3×C411yescharacteristic
61C2×C3×A4C3×C816yescomplemented
62C2×C3×A4C2×C3×C413yes—
63C2×C32×C4—32no—
64C2×C32×C4—31no—
65C2×C32×C4—31no—
66C2×C32×C4—31no—
67C2×C32×C4—31no—
68C2×C32×C4—31no—
69C2×C32×C4—31no—
70C2×C32×C4—31no—
71C2×C32×C4—43no—
72C2×C32×C4C2×A411yescharacteristic
73C32×C8—32no—
74C32×C8—46no—
75C32×C8C2×A412yescomplemented
76C23×C8H(3,𝔽3)11yescharacteristic, complemented, Sylow
77C2×H(3,𝔽3)—42no—
78C2×H(3,𝔽3)—41no—
79C24×C3—31no—
80C24×C3C32×C411yescharacteristic
81C22×A4—33no—
82C22×C3×C4—33no—
83C22×C3×C4—33no—
84C22×C3×C4—33no—
85C22×C3×C4—33no—
86C22×C3×C4—31no—
87C22×C3×C4—31no—
88C22×C3×C4—31no—
89C22×C3×C4—31no—
90C22×C3×C4—31no—
91C22×C3×C4—31no—
92C22×C3×C4C32×C411yescharacteristic
93C22×C3×C4C22×C3211yescharacteristic
94C4×A4—33no—
95C4×A4—33no—
96C2×C3×C8—36no—
97C2×C3×C8—32no—
98C2×C3×C8—33no—
99C2×C3×C8—36no—
100C2×C3×C8—36no—
101C2×C3×C8—36no—
102C2×C3×C8—31no—
103C2×C3×C8—32no—
104C2×C3×C8—32no—
105C2×C3×C8—32no—
106C2×C3×C8—31no—
107C2×C3×C8—123no—
108C2×C3×C8C3×A411yescharacteristic
109C22×C32—32no—
110C22×C32—31no—
111C22×C32—32no—
112C22×C32—32no—
113C22×C32—32no—
114C22×C32—31no—
115C22×C32—31no—
116C22×C32—43no—
117C22×C32C2×C3×C811yescharacteristic, complemented
118C22×C32C4×A411yescharacteristic
119C3×A4C2×C3×C813yescomplemented
120C32×C4—31no—
121C32×C4—31no—
122C32×C4—43no—
123C32×C4—43no—
124C32×C4C4×A411yescharacteristic
125C32×C4C22×A411yescharacteristic
126C23×C4C2×H(3,𝔽3)11yescharacteristic
127C22×C8—31no—
128C22×C8—31no—
129C22×C8—32no—
130C22×C8C2×H(3,𝔽3)12yescomplemented
131H(3,𝔽3)—41noSylow
132C23×C3—31no—
133C23×C3—32no—
134C23×C3—33no—
135C23×C3—33no—
136C23×C3—36no—
137C23×C3—31no—
138C23×C3—32no—
139C23×C3—31no—
140C23×C3C32×C812yes—
141C23×C3C2×C32×C411yescharacteristic
142C2×A4—36no—
143C2×A4—33no—
144C2×C3×C4—33no—
145C2×C3×C4—33no—
146C2×C3×C4—36no—
147C2×C3×C4—32no—
148C2×C3×C4—33no—
149C2×C3×C4—33no—
150C2×C3×C4—33no—
151C2×C3×C4—33no—
152C2×C3×C4—33no—
153C2×C3×C4—31no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—31no—
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—123no—
162C2×C3×C4C2×C3×A411yescharacteristic
163C3×C8—36no—
164C3×C8—32no—
165C3×C8—32no—
166C3×C8—126no—
167C3×C8C2×C3×A412yes—
168C2×C32—32no—
169C2×C32—31no—
170C2×C32—31no—
171C2×C32—43no—
172C2×C32—46no—
173C2×C32C8×A412yescomplemented
174C2×C32C2×C4×A411yescharacteristic
175C24C4×H(3,𝔽3)11yescharacteristic
176C22×C4—31no—
177C22×C4—31no—
178C22×C4—31no—
179C22×C4—31no—
180C22×C4C4×H(3,𝔽3)11yescharacteristic
181C22×C4C22×H(3,𝔽3)11yescharacteristic
182C2×C8—32no—
183C2×C8—31no—
184C2×C8—32no—
185C2×C8—32no—
186C2×C8—32no—
187C2×C8C22⋊H(3,𝔽3)11yescharacteristic, direct product factor
188C22×C3—32no—
189C22×C3—31no—
190C22×C3—33no—
191C22×C3—33no—
192C22×C3—36no—
193C22×C3—33no—
194C22×C3—36no—
195C22×C3—32no—
196C22×C3—36no—
197C22×C3—36no—
198C22×C3—31no—
199C22×C3—31no—
200C22×C3—32no—
201C22×C3—32no—
202C22×C3—31no—
203C22×C3—31no—
204C22×C3—123no—
205C22×C3C2×C32×C811yescharacteristic
206C22×C3C3×C4×A411yescharacteristic
207A4—33no—
208C3×C4—33no—
209C3×C4—33no—
210C3×C4—31no—
211C3×C4—31no—
212C3×C4—31no—
213C3×C4—31no—
214C3×C4—123no—
215C3×C4—123no—
216C3×C4C3×C4×A411yescharacteristic
217C3×C4C22×C3×A411yescharacteristic
218C32—43no—
219C32C2×C8×A411yescharacteristic, complemented
220C23—31no—
221C23—32no—
222C23—31no—
223C23C8×H(3,𝔽3)12yescomplemented
224C23C2×C4×H(3,𝔽3)11yescharacteristic
225C2×C4—32no—
226C2×C4—31no—
227C2×C4—31no—
228C2×C4—31no—
229C2×C4—31no—
230C2×C4—31no—
231C2×C4—31no—
232C2×C4—31no—
233C2×C4C2×C22⋊H(3,𝔽3)11yescharacteristic
234C8—32no—
235C8C2×C22⋊H(3,𝔽3)12yesdirect product factor
236C2×C3—33no—
237C2×C3—36no—
238C2×C3—33no—
239C2×C3—31no—
240C2×C3—32no—
241C2×C3—31no—
242C2×C3—31no—
243C2×C3—32no—
244C2×C3—123no—
245C2×C3—126no—
246C2×C3C3×C8×A412yes—
247C2×C3C2×C3×C4×A411yescharacteristic
248C22—32no—
249C22—31no—
250C22—32no—
251C22—31no—
252C22—31no—
253C22—32no—
254C22—32no—
255C22C2×C8×H(3,𝔽3)11yescharacteristic, complemented
256C22C4×C22⋊H(3,𝔽3)11yescharacteristic
257C4—31no—
258C4—31no—
259C4C4×C22⋊H(3,𝔽3)11yescharacteristic
260C4C22×C22⋊H(3,𝔽3)11yescharacteristic
261C3—31no—
262C3—123no—
263C3C2×C3×C8×A411yescharacteristic
264C2—31no—
265C2—32no—
266C2—31no—
267C2C8×C22⋊H(3,𝔽3)12yesdirect product factor
268C2C2×C4×C22⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC2×C3×C8C3×A4
commutatorC22×C3C2×C32×C8
FrattiniC3×C4C22×C3×A4
FittingC23×C32×C8C3
socleC24×C3C32×C4
prime radicalC22×C22⋊H(3,𝔽3)C4
cosocleC22×C3×C4C22×C32
hypercenterC2×C32×C8A4
hypocenterC22C2×C8×H(3,𝔽3)
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C2×C3×C8 ◁ C2×C32×C8

Complements

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

NH ≅ G/Ntype
2. C8×C22⋊H(3,𝔽3)C2direct (N × H)
3. C23×C32×C8C3semidirect
4. C2×C3×C8×A4C3semidirect
14. C22×C32×C8C2×C3semidirect
15. C3×C8×A4C2×C3semidirect
16. C2×C22⋊H(3,𝔽3)C8direct (N × H)
39. C2×C32×C8A4semidirect
41. C22⋊H(3,𝔽3)C2×C8direct (N × H)
59. C23×C32C3×C8semidirect
61. C2×C3×A4C3×C8semidirect
75. C32×C8C2×A4semidirect
76. C23×C8H(3,𝔽3)semidirect
117. C22×C32C2×C3×C8semidirect
119. C3×A4C2×C3×C8semidirect
130. C22×C8C2×H(3,𝔽3)semidirect
173. C2×C32C8×A4semidirect
187. C2×C8C22⋊H(3,𝔽3)direct (N × H)
219. C32C2×C8×A4semidirect
223. C23C8×H(3,𝔽3)semidirect
235. C8C2×C22⋊H(3,𝔽3)direct (N × H)
255. C22C2×C8×H(3,𝔽3)semidirect
267. C2C8×C22⋊H(3,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊H(3,𝔽3)2
C8×C22⋊H(3,𝔽3)2
C23×C32×C83
C2×C3×C8×A43
C2×C8×H(3,𝔽3)4