Finite Group Databasebeta

← #68952

Subgroups — #68952

Order 384.

412 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 107 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×#67⋊C4C211yesmaximal, characteristic
2C2×#806C212yesmaximal
3Dic3×C22⋊C4C212yesmaximal, complemented
4C3×#342C211yesmaximal, characteristic, complemented
5#4526C211yesmaximal, characteristic, complemented
6C4×#99—31nomaximal, Sylow
7C22×#67C2211yescharacteristic
8C23×Dic3C2211yescharacteristic
9C22×C4×D3C411yescharacteristic, complemented
10C2×C3×C22⋊C4C2211yescharacteristic
11C2×C3×C22⋊C4C2212yes—
12C2×#217C2212yes—
13C2×(C2×D3)⋊C4C411yescharacteristic, complemented
14C2×Dic3⋊C4C411yescharacteristic, complemented
15C2×C4×Dic3—22no—
16C2×C4×Dic3C411yescharacteristic, complemented
17#67⋊C4—22no—
18#67⋊C4—22no—
19C3⋊#95—22no—
20#806C418yescomplemented
21C2×C4×D4—31no—
22C2×#99—32no—
23C4×C22⋊C4—32no—
24#342—31no—
25#342D311yescharacteristic, complemented
26C24×C3C2311yescharacteristic
27C23×D3C2×C411yescharacteristic
28C22×C3×C4—22no—
29C22×C3×C4C2×C411yescharacteristic
30C22×C3×C4C2×C411yescharacteristic
31C2×#67—22no—
32C2×#67—21no—
33C2×#67—21no—
34C2×#67C2×C414yes—
35C22×Dic3—22no—
36C22×Dic3—42no—
37C22×Dic3C2×C411yescharacteristic
38C22×Dic3C2×C411yescharacteristic
39C22×Dic3C2×C411yescharacteristic
40C22×Dic3D412yes—
41C22×Dic3D412yes—
42C2×C4×D3—21no—
43C2×C4×D3—21no—
44C2×C4×D3—42no—
45C3×C22⋊C4—22no—
46C3×C22⋊C4—22no—
47C3×C22⋊C4C2×C414yescomplemented
48#217—22no—
49#217C2×C414yescomplemented
50(C2×D3)⋊C4—22no—
51Dic3⋊C4—22no—
52C4×Dic3—21no—
53C4×Dic3—21no—
54C4×Dic3—44no—
55C22×D4—31no—
56C23×C4—31no—
57C23×C4—31no—
58C4×D4—62no—
59C4×D4—62no—
60C2×C4⋊C4—31no—
61C2×C22⋊C4—32no—
62C2×C22⋊C4—31no—
63C2×C22⋊C4C2×D311yescharacteristic
64C2×C22⋊C4C2×D312yes—
65C2×C42—31no—
66C2×C42—62no—
67#99—38no—
68#95—62no—
69C23×C3—41no—
70C23×C3—41no—
71C23×C3C22×C411yescharacteristic
72C23×C3C22×C412yes—
73C23×C3C2×D412yes—
74C23×C3#4111yescharacteristic
75C23×C3#4111yescharacteristic
76C22×D3—21no—
77C22×D3—21no—
78C22×D3—41no—
79C22×D3—41no—
80C22×D3C4212yescomplemented
81C2×C3×C4—21no—
82C2×C3×C4—21no—
83C2×C3×C4—24no—
84C2×C3×C4—42no—
85C2×C3×C4—42no—
86C2×C3×C4—41no—
87C2×C3×C4—41no—
88#67—24no—
89#67—24no—
90C2×Dic3—21no—
91C2×Dic3—21no—
92C2×Dic3—24no—
93C2×Dic3—22no—
94C2×Dic3—24no—
95C2×Dic3—41no—
96C2×Dic3—42no—
97C2×Dic3—42no—
98C2×Dic3—42no—
99C2×Dic3C4212yescomplemented
100C2×Dic3C22⋊C411yescharacteristic
101C2×Dic3C22⋊C412yes—
102C2×Dic3C22⋊C411yescharacteristic
103C4×D3—42no—
104C4×D3—42no—
105C24—31no—
106C24C22×D311yescharacteristic
107C2×D4—34no—
108C2×D4—62no—
109C2×D4—61no—
110C2×D4—61no—
111C22×C4—22no—
112C22×C4—31no—
113C22×C4—31no—
114C22×C4—32no—
115C22×C4—31no—
116C22×C4—32no—
117C22×C4—61no—
118C22×C4—61no—
119C22×C4—62no—
120C22×C4—122no—
121C22×C4—122no—
122C22×C4C4×D311yescharacteristic
123C22×C4C4×D311yescharacteristic
124C4⋊C4—62no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—34no—
128C22⋊C4—62no—
129C22⋊C4—62no—
130C22⋊C4C4×D314yescomplemented
131C42—61no—
132C42—61no—
133C42—124no—
134C22×C3—22no—
135C22×C3—21no—
136C22×C3—22no—
137C22×C3—21no—
138C22×C3—42no—
139C22×C3—41no—
140C22×C3—41no—
141C22×C3C2×C4211yescharacteristic
142C22×C3C2×C22⋊C411yescharacteristic
143C22×C3#11711yescharacteristic
144C22×C3C4×D412yes—
145C22×C3C4×D412yes—
146C2×D3—22no—
147C2×D3—42no—
148C2×D3—41no—
149C2×D3—41no—
150C2×D3—42no—
151C3×C4—44no—
152C3×C4—41no—
153C3×C4—41no—
154Dic3—22no—
155Dic3—21no—
156Dic3—22no—
157Dic3—44no—
158Dic3#9912yescomplemented
159C23—32no—
160C23—41no—
161C23—41no—
162C23—61no—
163C23—61no—
164C23—121no—
165C23—121no—
166C23C2×C4×D311yescharacteristic
167C23C2×C4×D312yes—
168C23D3×D412yes—
169C23#23711yescharacteristic
170C23#23911yescharacteristic
171D4—64no—
172D4—64no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—24no—
176C2×C4—31no—
177C2×C4—32no—
178C2×C4—31no—
179C2×C4—32no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—64no—
185C2×C4—62no—
186C2×C4—61no—
187C2×C4—61no—
188C2×C4—64no—
189C2×C4—122no—
190C2×C4—122no—
191C2×C4—122no—
192C2×C4—121no—
193C2×C4—122no—
194C2×C4—122no—
195C2×C3—22no—
196C2×C3—22no—
197C2×C3—21no—
198C2×C3—21no—
199C2×C3C4×C22⋊C411yescharacteristic
200C2×C3C2×#9911yescharacteristic
201C2×C3#41011yescharacteristic
202D3—42no—
203C22—22no—
204C22—21no—
205C22—22no—
206C22—21no—
207C22—42no—
208C22—41no—
209C22—41no—
210C22—62no—
211C22—122no—
212C22—122no—
213C22—121no—
214C22—121no—
215C22C42×D311yescharacteristic
216C22D3×C22⋊C411yescharacteristic
217C22#67⋊C412yes—
218C22#89211yescharacteristic
219C22#89312yes—
220C4—32no—
221C4—44no—
222C4—41no—
223C4—41no—
224C4—62no—
225C4—61no—
226C4—62no—
227C4—124no—
228C3C4×#9911yescharacteristic, complemented, Sylow
229C2—22no—
230C2—22no—
231C2—21no—
232C2—21no—
233C2—122no—
234C2#466611yescharacteristic
235C2#474011yescharacteristic
236C2D3×#9911yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC22×C3C2×C42
FrattiniC24C22×D3
FittingC3×#342C2
socleC22×C3C2×C22⋊C4
prime radicalC22×#67C22
cosocleC24×C3C23
hypercenter#342D3
hypocenterC3C4×#99
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
3. Dic3×C22⋊C4C2semidirect
4. C3×#342C2semidirect
5. #4526C2semidirect
9. C22×C4×D3C4semidirect
13. C2×(C2×D3)⋊C4C4semidirect
14. C2×Dic3⋊C4C4semidirect
16. C2×C4×Dic3C4semidirect
20. #806C4semidirect
25. #342D3semidirect
47. C3×C22⋊C4C2×C4semidirect
49. #217C2×C4semidirect
80. C22×D3C42semidirect
99. C2×Dic3C42semidirect
130. C22⋊C4C4×D3semidirect
158. Dic3#99semidirect
228. C3C4×#99semidirect

Maximal subgroups

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

subgroupindex
C2×#67⋊C42
C2×#8062
Dic3×C22⋊C42
C3×#3422
#45262
C4×#993