Finite Group Databasebeta

← #70028

Subgroups — #70028

Order 384.

380 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 95 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×D4×Dic3C211yesmaximal, characteristic
2C2×C3×#99C211yesmaximal, characteristic
3C2×#834C211yesmaximal, characteristic
4#4942C212yesmaximal, complemented
5#4527C212yesmaximal, complemented
6#1799—31nomaximal, Sylow
7C22×C3×D4C2211yescharacteristic
8C23×Dic3C411yescharacteristic, complemented
9C23×Dic3C2211yescharacteristic
10C2×C3×C22⋊C4C2212yes—
11C2×#217C411yescharacteristic, complemented
12C2×#217C2212yes—
13C2×#217C2211yescharacteristic
14D4×Dic3—22no—
15D4×Dic3—22no—
16C2×C4⋊Dic3C411yescharacteristic, complemented
17C2×Dic3⋊C4—22no—
18C2×C4×Dic3C411yescharacteristic, complemented
19C3×#99—22no—
20#834—22no—
21C3⋊#95—22no—
22C2×C4×D4—31no—
23C2×#99—31no—
24C2×#99D311yescharacteristic
25#380—32no—
26#342—32no—
27C24×C3C2×C411yescharacteristic
28C24×C3C2311yescharacteristic
29C2×C3×D4—22no—
30C2×C3×D4—22no—
31C2×C3×D4D412yes—
32C2×C3×D4Q812yes—
33C22×C3×C4—22no—
34C22×C3×C4C2×C411yescharacteristic
35C22×Dic3—21no—
36C22×Dic3—21no—
37C22×Dic3—22no—
38C22×Dic3—42no—
39C22×Dic3—42no—
40C22×Dic3C2×C411yescharacteristic
41C22×Dic3C2×C411yescharacteristic
42C22×Dic3C2×C411yescharacteristic
43C22×Dic3C2×C411yescharacteristic
44C22×Dic3D412yes—
45C22×Dic3D412yes—
46C3×C22⋊C4—22no—
47C3×C22⋊C4—22no—
48#217—22no—
49#217—22no—
50#217—22no—
51#217—22no—
52C4⋊Dic3—22no—
53Dic3⋊C4—44no—
54C4×Dic3—21no—
55C4×Dic3—21no—
56C22×D4C2×D311yescharacteristic
57C23×C4—31no—
58C23×C4—31no—
59C4×D4—62no—
60C4×D4—62no—
61C2×C4⋊C4—31no—
62C2×C4⋊C4—62no—
63C2×C22⋊C4—31no—
64C2×C22⋊C4—31no—
65C2×C22⋊C4—32no—
66C2×C22⋊C4C2×D312yes—
67C2×C42—31no—
68#99—22no—
69#99—62no—
70#95—62no—
71C23×C3—21no—
72C23×C3—21no—
73C23×C3—41no—
74C23×C3—41no—
75C23×C3—41no—
76C23×C3—41no—
77C23×C3C4⋊C412yescomplemented
78C23×C3C22×C411yescharacteristic
79C23×C3C2×D412yes—
80C23×C3C2×D411yescharacteristic
81C23×C3C2×Q811yescharacteristic
82C23×C3#4112yes—
83C3×D4—24no—
84C3×D4—24no—
85C2×C3×C4—21no—
86C2×C3×C4—21no—
87C2×C3×C4—42no—
88C2×C3×C4—42no—
89C2×C3×C4—42no—
90C2×C3×C4C4⋊C412yescomplemented
91C2×Dic3—24no—
92C2×Dic3—22no—
93C2×Dic3—21no—
94C2×Dic3—21no—
95C2×Dic3—41no—
96C2×Dic3—42no—
97C2×Dic3—42no—
98C2×Dic3—42no—
99C2×Dic3—41no—
100C2×Dic3—41no—
101C2×Dic3—42no—
102C2×Dic3—42no—
103C2×Dic3—42no—
104C2×Dic3C22⋊C411yescharacteristic
105C2×Dic3C22⋊C412yes—
106C2×Dic3C22⋊C411yescharacteristic
107C24C4×D311yescharacteristic
108C24C22×D311yescharacteristic
109C2×D4—22no—
110C2×D4—22no—
111C2×D4Dic612yes—
112C2×D4#6712yes—
113C22×C4—22no—
114C22×C4—31no—
115C22×C4—32no—
116C22×C4—31no—
117C22×C4—32no—
118C22×C4—31no—
119C22×C4—31no—
120C22×C4—61no—
121C22×C4—61no—
122C22×C4—62no—
123C22×C4—122no—
124C22×C4—122no—
125C22×C4C4×D311yescharacteristic
126C4⋊C4—62no—
127C4⋊C4—124no—
128C22⋊C4—22no—
129C22⋊C4—22no—
130C22⋊C4—62no—
131C22⋊C4—62no—
132C22⋊C4—62no—
133C22⋊C4—62no—
134C42—61no—
135C42—61no—
136C22×C3—22no—
137C22×C3—21no—
138C22×C3—22no—
139C22×C3—21no—
140C22×C3—22no—
141C22×C3—41no—
142C22×C3—41no—
143C22×C3—42no—
144C22×C3—42no—
145C22×C3—42no—
146C22×C3—41no—
147C22×C3—41no—
148C22×C3C2×C22⋊C411yescharacteristic
149C22×C3C2×C4⋊C411yescharacteristic
150C22×C3#11711yescharacteristic
151C22×C3#12112yes—
152C22×C3C22⋊Q812yes—
153C3×C4—22no—
154C3×C4—82no—
155Dic3—21no—
156Dic3—22no—
157Dic3—41no—
158Dic3—41no—
159Dic3—82no—
160Dic3#9912yescomplemented
161C23—21no—
162C23—21no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23Dic3⋊C412yescomplemented
168C23C2×Dic611yescharacteristic
169C23C2×C4×D311yescharacteristic
170C23D3×D412yes—
171C23#23712yes—
172C23C2×#6711yescharacteristic
173D4—24no—
174D4—24no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—31no—
178C2×C4—31no—
179C2×C4—32no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—64no—
184C2×C4—62no—
185C2×C4—61no—
186C2×C4—61no—
187C2×C4—122no—
188C2×C4—122no—
189C2×C4—121no—
190C2×C4—122no—
191C2×C4—121no—
192C2×C4—121no—
193C2×C4—122no—
194C2×C4—122no—
195C2×C4—122no—
196C2×C4Dic3⋊C412yescomplemented
197C2×C3—22no—
198C2×C3—21no—
199C2×C3—21no—
200C2×C3—22no—
201C2×C3—42no—
202C2×C3#38011yescharacteristic
203C2×C3C2×#9911yescharacteristic
204C2×C3#41011yescharacteristic
205C22—22no—
206C22—21no—
207C22—22no—
208C22—21no—
209C22—22no—
210C22—41no—
211C22—42no—
212C22—41no—
213C22—42no—
214C22—42no—
215C22—41no—
216C22—41no—
217C22#87711yescharacteristic
218C22#87812yes—
219C22D3×C22⋊C411yescharacteristic
220C22C2×Dic3⋊C411yescharacteristic
221C22#93912yes—
222C4—22no—
223C4—32no—
224C4—61no—
225C4—62no—
226C4—82no—
227C4—121no—
228C4—121no—
229C4—242no—
230C3#179911yescharacteristic, complemented, Sylow
231C2—22no—
232C2—21no—
233C2—21no—
234C2—22no—
235C2—42no—
236C2#474011yescharacteristic
237C2D3×#9911yescharacteristic
238C2#494211yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC23×C3C22×C4
FrattiniC24C22×D3
FittingC2×C3×#99C2
socleC22×C3C2×C22⋊C4
prime radicalC22×C3×D4C22
cosocleC24×C3C23
hypercenterC2×#99D3
hypocenterC3#1799
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
4. #4942C2semidirect
5. #4527C2semidirect
8. C23×Dic3C4semidirect
11. C2×#217C4semidirect
16. C2×C4⋊Dic3C4semidirect
18. C2×C4×Dic3C4semidirect
77. C23×C3C4⋊C4semidirect
90. C2×C3×C4C4⋊C4semidirect
160. Dic3#99semidirect
167. C23Dic3⋊C4semidirect
196. C2×C4Dic3⋊C4semidirect
230. C3#1799semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic32
C2×C3×#992
C2×#8342
#49422
#45272
#17993