Finite Group Databasebeta

← C2×#88939 #11724345

Subgroups — C2×#88939 #11724345

Order 864.

500 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 67 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×#1113C211yesmaximal, characteristic
2C22×C3×#157C211yesmaximal, characteristic, complemented
3C2×#6294C211yesmaximal, characteristic, complemented
4#88939C214yesmaximal, direct product factor
5C2×D32⋊C4—31nomaximal
6C22×C3×C3⋊D3—21no—
7C22×C3×C3⋊D3C2211yescharacteristic
8C2×#1113—21no—
9C2×#1113—22no—
10C2×#1113C414yescomplemented
11C2×#1113C2212yes—
12C2×C3×#157—22no—
13C2×C3×#157C2212yescomplemented
14C2×C3×C3⋊Dic3—21no—
15#6294—21no—
16#6294C2212yescomplemented
17C22×D32—31no—
18C22×D32—61no—
19C22×#157D311yescharacteristic, complemented
20C2×#624—31no—
21C2×D3×Dic3—61no—
22D32⋊C4—34no—
23C22×C33D411yescharacteristic
24C2×C3×C3⋊D3—24no—
25C2×C3×C3⋊D3—22no—
26C2×C3×C3⋊D3C2×C412yes—
27C2×C3×C3⋊D3D411yescharacteristic
28C2×C3×C3⋊D3D412yes—
29C2×C3×C3⋊D3C2311yescharacteristic
30#1113—24no—
31#1113—22no—
32#1113C2×C414yescomplemented
33C3×#157—22no—
34C3×C3⋊Dic3—22no—
35C2×(C2×D3)⋊C4—91nomaximal
36C22×C3⋊D3—21no—
37C22×C3⋊D3C2×D311yescharacteristic
38C22×C3×D3—61no—
39C22×C3×D3—61no—
40C22×C3×D3—61no—
41C22×C3×D3—61no—
42C22×C3×D3—121no—
43C2×D32—34no—
44C2×D32—32no—
45C2×D32—64no—
46C2×D32—62no—
47C2×D32—62no—
48C2×D32—64no—
49C2×D32—61no—
50C2×D32—62no—
51C2×#157—22no—
52C2×#157C2×D312yescomplemented
53C2×C3⋊Dic3—21no—
54C2×C3×Dic3—61no—
55C2×C3×Dic3—61no—
56C2×C3×Dic3—121no—
57#624—32no—
58#624—61no—
59D3×Dic3—62no—
60D3×Dic3—62no—
61C2×C33C22⋊C412yescomplemented
62C2×C33C2×D411yescharacteristic
63C3×C3⋊D3—24no—
64C3×C3⋊D3C22⋊C412yescomplemented
65C3×C3⋊D3C22×C411yescharacteristic
66C3×C3⋊D3C2×D411yescharacteristic
67C23×D3—91no—
68C23×D3—181no—
69C22×C3×C4—91no—
70C22×Dic3—91no—
71C2×C4×D3—181no—
72(C2×D3)⋊C4—94no—
73C22×C32—21no—
74C22×C32—21no—
75C22×C32—41no—
76C22×C32—41no—
77C22×C32D1211yescharacteristic
78C2×C3⋊D3—22no—
79C2×C3⋊D3—24no—
80C2×C3⋊D3C4×D312yes—
81C2×C3⋊D3D1211yescharacteristic
82C2×C3⋊D3#6712yes—
83C2×C3⋊D3C22×D311yescharacteristic
84C2×C3×D3—62no—
85C2×C3×D3—62no—
86C2×C3×D3—61no—
87C2×C3×D3—61no—
88C2×C3×D3—62no—
89C2×C3×D3—62no—
90C2×C3×D3—61no—
91C2×C3×D3—61no—
92C2×C3×D3—62no—
93C2×C3×D3—64no—
94C2×C3×D3—64no—
95C2×C3×D3—62no—
96C2×C3×D3—124no—
97C2×C3×D3—122no—
98D32—34no—
99D32—64no—
100D32—68no—
101D32—64no—
102D32—64no—
103D32—62no—
104#157—22no—
105C3⋊Dic3—22no—
106C3×Dic3—62no—
107C3×Dic3—62no—
108C3×Dic3—122no—
109C2×C22⋊C4—271noSylow
110C33C2×C22⋊C411yescharacteristic, complemented, Sylow
111C23×C3—91no—
112C23×C3—181no—
113C22×D3—61no—
114C22×D3—61no—
115C22×D3—61no—
116C22×D3—61no—
117C22×D3—92no—
118C22×D3—94no—
119C22×D3—121no—
120C22×D3—182no—
121C22×D3—184no—
122C22×D3—181no—
123C22×D3—182no—
124C22×D3—182no—
125C22×D3—184no—
126C2×C3×C4—92no—
127C2×C3×C4—181no—
128C2×C3×C4—182no—
129C2×Dic3—61no—
130C2×Dic3—61no—
131C2×Dic3—92no—
132C2×Dic3—121no—
133C2×Dic3—181no—
134C4×D3—182no—
135C4×D3—182no—
136C2×C32—21no—
137C2×C32—21no—
138C2×C32—22no—
139C2×C32—22no—
140C2×C32—42no—
141C2×C32—42no—
142C2×C32—41no—
143C2×C32—41no—
144C2×C32(C2×D3)⋊C412yescomplemented
145C2×C32C2×D1211yescharacteristic
146C3⋊D3—24no—
147C3⋊D3(C2×D3)⋊C412yescomplemented
148C3⋊D3C2×C4×D311yescharacteristic
149C3⋊D3C2×#6711yescharacteristic
150C3×D3—62no—
151C3×D3—62no—
152C3×D3—61no—
153C3×D3—61no—
154C3×D3—61no—
155C3×D3—61no—
156C3×D3—64no—
157C3×D3—64no—
158C3×D3—124no—
159C24—271no—
160C22×C4—91no—
161C22×C4—271no—
162C22⋊C4—274no—
163C22×C3—21no—
164C22×C3—21no—
165C22×C3—41no—
166C22×C3—41no—
167C22×C3—91no—
168C22×C3—91no—
169C22×C3—92no—
170C22×C3—92no—
171C22×C3—182no—
172C22×C3—184no—
173C22×C3GO+(4,𝔽2)11yescharacteristic
174C2×D3—61no—
175C2×D3—61no—
176C2×D3—62no—
177C2×D3—61no—
178C2×D3—62no—
179C2×D3—61no—
180C2×D3—62no—
181C2×D3—62no—
182C2×D3—62no—
183C2×D3—64no—
184C2×D3—62no—
185C2×D3—64no—
186C2×D3—94no—
187C2×D3—122no—
188C2×D3—124no—
189C2×D3—184no—
190C2×D3—182no—
191C2×D3—184no—
192C2×D3—188no—
193C2×D3—184no—
194C3×C4—182no—
195C3×C4—182no—
196Dic3—62no—
197Dic3—62no—
198Dic3—122no—
199C32—21no—
200C32—21no—
201C32—41no—
202C32—41no—
203C32C2×(C2×D3)⋊C411yescharacteristic, complemented
204C23—91no—
205C23—181no—
206C23—272no—
207C23—274no—
208C23—541no—
209C23—542no—
210C2×C4—92no—
211C2×C4—182no—
212C2×C4—181no—
213C2×C4—272no—
214C2×C4—541no—
215C2×C3—21no—
216C2×C3—21no—
217C2×C3—22no—
218C2×C3—22no—
219C2×C3—41no—
220C2×C3—41no—
221C2×C3—42no—
222C2×C3—42no—
223C2×C3—91no—
224C2×C3—91no—
225C2×C3—92no—
226C2×C3—184no—
227C2×C3D32⋊C412yescomplemented
228C2×C3C2×GO+(4,𝔽2)11yescharacteristic
229D3—62no—
230D3—61no—
231D3—61no—
232D3—62no—
233D3—61no—
234D3—61no—
235D3—64no—
236D3—64no—
237D3—124no—
238C22—91no—
239C22—91no—
240C22—92no—
241C22—92no—
242C22—182no—
243C22—184no—
244C22—274no—
245C22—542no—
246C22—544no—
247C22#157⋊D311yescharacteristic
248C4—182no—
249C4—182no—
250C3—21no—
251C3—21no—
252C3—41no—
253C3—41no—
254C3C2×D32⋊C411yescharacteristic, complemented
255C2—91no—
256C2—91no—
257C2—92no—
258C2—184no—
259C2#8893912yesdirect product factor
260C2C2×#157⋊D311yescharacteristic
subgroupG/H
center, hypercenterC22#157⋊D3
commutatorC3×C3⋊D3C22×C4
FrattiniC2C2×#157⋊D3
Fitting, socleC22×C33D4
prime radicalC22×#1113C2
cosocleC2×C3×C3⋊D3C23
hypocenterC33C2×C22⋊C4
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C22×C3×#157C2semidirect
3. C2×#6294C2semidirect
4. #88939C2direct (N × H)
10. C2×#1113C4semidirect
13. C2×C3×#157C22semidirect
16. #6294C22semidirect
19. C22×#157D3semidirect
32. #1113C2×C4semidirect
52. C2×#157C2×D3semidirect
61. C2×C33C22⋊C4semidirect
64. C3×C3⋊D3C22⋊C4semidirect
110. C33C2×C22⋊C4semidirect
144. C2×C32(C2×D3)⋊C4semidirect
147. C3⋊D3(C2×D3)⋊C4semidirect
203. C32C2×(C2×D3)⋊C4semidirect
227. C2×C3D32⋊C4semidirect
254. C3C2×D32⋊C4semidirect
259. C2#88939direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#11132
C22×C3×#1572
C2×#62942
#889392
C2×D32⋊C43
C2×(C2×D3)⋊C49