Finite Group Databasebeta

← C22⋊C4×D27⋊C3 #72276208

Subgroups — C22⋊C4×D27⋊C3 #72276208

Order 2592.

562 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 131 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×D27⋊C3C211yesmaximal, characteristic
2C2×C4×D27⋊C3C212yesmaximal, complemented
3C22⋊C4×C27⋊C3C211yesmaximal, characteristic, complemented
4#88372⋊C3C211yesmaximal, characteristic, complemented
5(C2×D27⋊C3)⋊C4C212yesmaximal, complemented
6C3×C22⋊C4×D9—31nomaximal
7C22⋊C4×D27C311yesmaximal, characteristic, complemented
8C23×C27⋊C3C2211yescharacteristic
9C22×D27⋊C3—24no—
10C22×D27⋊C3C414yescomplemented
11C22×D27⋊C3C2211yescharacteristic
12C22×D27⋊C3C2211yescharacteristic
13C2×C4×C27⋊C3C2212yescomplemented
14C2×(C27⋊C3)⋊C4C2212yescomplemented
15C4×D27⋊C3—24no—
16C23×C3×D9—31no—
17C2×C3×C4×D9—32no—
18C23×D27C2×C311yescharacteristic
19C3×C9×C22⋊C4D311yescharacteristic
20C3×#3676—31no—
21C3×(C2×D9)⋊C4—32no—
22C2×C4×D27C2×C312yescomplemented
23C22⋊C4×C27—21no—
24C22⋊C4×C27C2×C311yescharacteristic, complemented
25#88372C2×C311yescharacteristic, complemented
26(C2×D27)⋊C4C2×C312yescomplemented
27C22×C27⋊C3—22no—
28C22×C27⋊C3C2×C412yescomplemented
29C22×C27⋊C3C2311yescharacteristic
30C2×D27⋊C3—22no—
31C2×D27⋊C3—28no—
32C2×D27⋊C3C2×C412yes—
33C2×D27⋊C3C2×C412yescomplemented
34C2×D27⋊C3D414yes—
35C4×C27⋊C3—22no—
36(C27⋊C3)⋊C4—22no—
37C3×D3×C22⋊C4—91no—
38C22⋊C4×D9—31no—
39C23×C3×C9C2×D311yescharacteristic
40C22×C3×D9—31no—
41C22×C3×D9—34no—
42C22×C3×D9—31no—
43C22×C3×D9—64no—
44C2×C3×C4×C9C2×D312yes—
45C2×C3×Dic9—32no—
46C3×C4×D9—64no—
47C23×C27—21no—
48C23×C27C22×C311yescharacteristic
49C22×D27—24no—
50C22×D27C3×C414yescomplemented
51C22×D27C22×C311yescharacteristic
52C22×D27C22×C311yescharacteristic
53C2×C4×C27—22no—
54C2×C4×C27C22×C312yescomplemented
55C2×Dic27C22×C312yescomplemented
56C4×D27—24no—
57C2×C27⋊C3—22no—
58C2×C27⋊C3C22×C411yescharacteristic
59C2×C27⋊C3C2×D412yes—
60D27⋊C3—22no—
61D27⋊C3C22⋊C414yesdirect product factor
62C23×C3×D3—91no—
63C2×C3×C4×D3—92no—
64C23×D9—31no—
65C32×C22⋊C4D911yescharacteristic
66C3×#217—91no—
67C3×(C2×D3)⋊C4—92no—
68C2×C4×D9—32no—
69C9×C22⋊C4—21no—
70C9×C22⋊C4C3×D311yescharacteristic
71#3676—31no—
72(C2×D9)⋊C4—32no—
73C22×C3×C9—22no—
74C22×C3×C9C4×D312yes—
75C22×C3×C9C22×D311yescharacteristic
76C2×C3×D9—32no—
77C2×C3×D9—34no—
78C2×C3×D9—32no—
79C2×C3×D9—68no—
80C2×C3×D9—62no—
81C3×C4×C9—22no—
82C3×Dic9—62no—
83C22×C27—21no—
84C22×C27—22no—
85C22×C27—22no—
86C22×C27—42no—
87C22×C27C2×C3×C412yescomplemented
88C22×C27C23×C311yescharacteristic
89C2×D27—22no—
90C2×D27—28no—
91C2×D27C2×C3×C412yes—
92C2×D27C2×C3×C412yescomplemented
93C2×D27C3×D414yes—
94C4×C27—22no—
95C4×C27—42no—
96Dic27—22no—
97C2×C3×C22⋊C4—271no—
98D3×C22⋊C4—91no—
99C27⋊C3C2×C22⋊C411yescharacteristic, complemented, Sylow
100C23×C32C2×D911yescharacteristic
101C22×C3×D3—91no—
102C22×C3×D3—94no—
103C22×C3×D3—91no—
104C22×C3×D3—184no—
105C2×C32×C4C2×D912yes—
106C2×C3×Dic3—92no—
107C3×C4×D3—184no—
108C23×C9—21no—
109C23×C9C2×C3×D311yescharacteristic
110C22×D9—31no—
111C22×D9—34no—
112C22×D9—31no—
113C22×D9—64no—
114C2×C4×C9—22no—
115C2×C4×C9C2×C3×D312yes—
116C2×Dic9—32no—
117C4×D9—64no—
118C2×C3×C9—22no—
119C2×C3×C9C2×C4×D311yescharacteristic
120C2×C3×C9D3×D412yes—
121C3×D9—34no—
122C3×D9—62no—
123C2×C27—22no—
124C2×C27—21no—
125C2×C27—22no—
126C2×C27—42no—
127C2×C27C22×C3×C411yescharacteristic
128C2×C27C2×C3×D412yes—
129D27—22no—
130D27C3×C22⋊C414yescomplemented
131C24×C3—271no—
132C23×D3—91no—
133C22×C3×C4—272no—
134C2×C4×D3—92no—
135C3×C22⋊C4—31no—
136C3×C22⋊C4—272no—
137C3×C22⋊C4—271no—
138C3×C22⋊C4C3×D911yescharacteristic
139#217—91no—
140(C2×D3)⋊C4—92no—
141C22×C32—22no—
142C22×C32C4×D912yes—
143C22×C32C22×D911yescharacteristic
144C2×C3×D3—92no—
145C2×C3×D3—94no—
146C2×C3×D3—92no—
147C2×C3×D3—188no—
148C2×C3×D3—182no—
149C32×C4—22no—
150C3×Dic3—182no—
151C22×C9—22no—
152C22×C9—22no—
153C22×C9—21no—
154C22×C9—42no—
155C22×C9C3×C4×D312yes—
156C22×C9C22×C3×D311yescharacteristic
157C2×D9—32no—
158C2×D9—34no—
159C2×D9—32no—
160C2×D9—68no—
161C2×D9—62no—
162C4×C9—22no—
163C4×C9—42no—
164Dic9—62no—
165C2×C22⋊C4—271noSylow
166C3×C9D3×C22⋊C411yescharacteristic
167C27—21no—
168C27C2×C3×C22⋊C411yescharacteristic, complemented
169C23×C3—31no—
170C23×C3—271no—
171C23×C3—271no—
172C23×C3—274no—
173C23×C3—544no—
174C23×C3C2×C3×D911yescharacteristic
175C22×D3—91no—
176C22×D3—94no—
177C22×D3—91no—
178C22×D3—184no—
179C2×C3×C4—32no—
180C2×C3×C4—272no—
181C2×C3×C4—544no—
182C2×C3×C4C2×C3×D912yes—
183C2×Dic3—92no—
184C4×D3—184no—
185C2×C32—22no—
186C2×C32C2×C4×D911yescharacteristic
187C2×C32D4×D912yes—
188C3×D3—94no—
189C3×D3—182no—
190C2×C9—22no—
191C2×C9—21no—
192C2×C9—22no—
193C2×C9—42no—
194C2×C9C2×C3×C4×D311yescharacteristic
195C2×C9C3×D3×D412yes—
196D9—34no—
197D9—62no—
198C24—271no—
199C22×C4—272no—
200C22⋊C4—271no—
201C22⋊C4—272no—
202C22⋊C4D27⋊C311yescharacteristic, direct product factor
203C22×C3—22no—
204C22×C3—31no—
205C22×C3—32no—
206C22×C3—62no—
207C22×C3—274no—
208C22×C3—272no—
209C22×C3—272no—
210C22×C3—542no—
211C22×C3—548no—
212C22×C3C3×C4×D912yes—
213C22×C3C22×C3×D911yescharacteristic
214C2×D3—92no—
215C2×D3—94no—
216C2×D3—92no—
217C2×D3—188no—
218C2×D3—182no—
219C3×C4—22no—
220C3×C4—62no—
221C3×C4—542no—
222Dic3—182no—
223C32C22⋊C4×D911yescharacteristic
224C9—21no—
225C9C3×D3×C22⋊C411yescharacteristic
226C23—271no—
227C23—271no—
228C23—274no—
229C23—544no—
230C23C2×D27⋊C311yescharacteristic
231C2×C4—272no—
232C2×C4—544no—
233C2×C4C2×D27⋊C312yescomplemented
234C2×C3—22no—
235C2×C3—31no—
236C2×C3—32no—
237C2×C3—62no—
238C2×C3—274no—
239C2×C3—542no—
240C2×C3C2×C3×C4×D911yescharacteristic
241C2×C3C3×D4×D912yes—
242D3—94no—
243D3—182no—
244C22—22no—
245C22—272no—
246C22—274no—
247C22—272no—
248C22—548no—
249C22—542no—
250C22C4×D27⋊C312yescomplemented
251C22C22×D27⋊C311yescharacteristic
252C4—22no—
253C4—542no—
254C3—31no—
255C3C3×C22⋊C4×D911yescharacteristic
256C2—22no—
257C2—274no—
258C2—542no—
259C2C2×C4×D27⋊C311yescharacteristic
260C2D4×D27⋊C312yes—
subgroupG/H
centerC22C22×D27⋊C3
commutatorC2×C27C22×C3×C4
FrattiniC22×C9C22×C3×D3
FittingC22⋊C4×C27⋊C3C2
socleC22×C3C22×C3×D9
prime radicalC23×D27⋊C3C2
cosocleC22×C27C23×C3
hypercenterC22⋊C4D27⋊C3
hypocenterC27C2×C3×C22⋊C4
LCSG ▷ C2×C27 ▷ C27
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×D27⋊C3C2semidirect
3. C22⋊C4×C27⋊C3C2semidirect
4. #88372⋊C3C2semidirect
5. (C2×D27⋊C3)⋊C4C2semidirect
7. C22⋊C4×D27C3semidirect
10. C22×D27⋊C3C4semidirect
13. C2×C4×C27⋊C3C22semidirect
14. C2×(C27⋊C3)⋊C4C22semidirect
22. C2×C4×D27C2×C3semidirect
24. C22⋊C4×C27C2×C3semidirect
25. #88372C2×C3semidirect
26. (C2×D27)⋊C4C2×C3semidirect
28. C22×C27⋊C3C2×C4semidirect
33. C2×D27⋊C3C2×C4semidirect
50. C22×D27C3×C4semidirect
54. C2×C4×C27C22×C3semidirect
55. C2×Dic27C22×C3semidirect
61. D27⋊C3C22⋊C4direct (N × H)
87. C22×C27C2×C3×C4semidirect
92. C2×D27C2×C3×C4semidirect
99. C27⋊C3C2×C22⋊C4semidirect
130. D27C3×C22⋊C4semidirect
168. C27C2×C3×C22⋊C4semidirect
202. C22⋊C4D27⋊C3direct (N × H)
233. C2×C4C2×D27⋊C3semidirect
250. C22C4×D27⋊C3semidirect

Maximal subgroups

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

subgroupindex
C23×D27⋊C32
C2×C4×D27⋊C32
C22⋊C4×C27⋊C32
#88372⋊C32
(C2×D27⋊C3)⋊C42
C3×C22⋊C4×D93
C22⋊C4×D273