Finite Group Databasebeta

← #74256

Subgroups — #74256

Order 384.

696 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 169 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×#897C211yesmaximal, characteristic, complemented
2C2×#895C211yesmaximal, characteristic
3C2×#875C211yesmaximal, characteristic, complemented
4C2×C42×D3C211yesmaximal, characteristic, complemented
5C3×#382C211yesmaximal, characteristic, complemented
6#4988C211yesmaximal, characteristic, complemented
7#4973C211yesmaximal, characteristic, complemented
8#4670C214yesmaximal, complemented
9#4665C214yesmaximal, complemented
10#2850—31nomaximal, Sylow
11C22×D12C2212yes—
12C22×D12C2211yescharacteristic
13C22×C4×D3C2212yes—
14C22×C4×D3C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C42C2211yescharacteristic
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2214yes—
21C2×(C2×D3)⋊C4C2214yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×C4⋊Dic3C2212yes—
24C2×C4×Dic3C2212yes—
25C2×C4×Dic3C2211yescharacteristic
26#897—24no—
27#895—24no—
28#875—24no—
29C42×D3—24no—
30C3×#95C2214yes—
31C3⋊#95C2214yes—
32C2×#124—31no—
33C2×C22⋊Q8—31no—
34C2×#121—31no—
35C22×C42—31no—
36#391—31no—
37#390—31no—
38#388—34no—
39#386—34no—
40#382D311yescharacteristic, complemented
41C23×D3C2312yes—
42C23×D3C2311yescharacteristic
43C22×C3×C4C2312yes—
44C22×C3×C4C2312yes—
45C22×C3×C4C2312yes—
46C22×C3×C4C2311yescharacteristic
47C22×Dic3C2312yes—
48C22×Dic3C2311yescharacteristic
49C22×Dic3C2312yes—
50C2×D12—24no—
51C2×D12—28no—
52C2×D12—22no—
53C2×D12—22no—
54C2×D12—22no—
55C2×C4×D3—24no—
56C2×C4×D3—24no—
57C2×C4×D3—22no—
58C2×C4×D3—22no—
59C2×C4×D3—24no—
60C2×C4×D3D414yes—
61C2×Dic6—22no—
62C2×Dic6—22no—
63C2×Dic6—22no—
64C3×C4⋊C4—22no—
65C3×C4⋊C4—22no—
66C3×C42D414yescomplemented
67(C2×D3)⋊C4—24no—
68(C2×D3)⋊C4—28no—
69(C2×D3)⋊C4—24no—
70(C2×D3)⋊C4—28no—
71C4⋊Dic3—24no—
72C4×Dic3—24no—
73C4×Dic3—22no—
74C22×Q8—31no—
75C22×D4—31no—
76C22×D4—32no—
77C23×C4—31no—
78C23×C4—32no—
79#124—64no—
80C22⋊Q8—64no—
81#121—64no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4C2×D311yescharacteristic
84C2×C4⋊C4C2×D311yescharacteristic
85C2×C22⋊C4—32no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4—34no—
88C2×C22⋊C4—34no—
89C2×C42—32no—
90C2×C42—31no—
91C2×C42—64no—
92C2×C42C2×D311yescharacteristic
93#95—34no—
94#95C2×D314yes—
95C23×C3C2411yescharacteristic
96C22×D3—21no—
97C22×D3—21no—
98C22×D3—22no—
99C22×D3—22no—
100C22×D3—22no—
101C22×D3—24no—
102C22×D3—22no—
103C22×D3—24no—
104C22×D3C2×D412yes—
105C22×D3#4114yes—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—24no—
111C2×C3×C4—24no—
112C2×C3×C4C2×D414yes—
113C2×C3×C4C2×D412yes—
114C2×C3×C4C2×D412yes—
115C2×C3×C4#4112yes—
116C2×Dic3—21no—
117C2×Dic3—22no—
118C2×Dic3—21no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3C2×D412yes—
124C2×Dic3#4114yes—
125D12—48no—
126D12—44no—
127C4×D3—216no—
128C4×D3—28no—
129Dic6—44no—
130C24—31no—
131C24—32no—
132C2×Q8—62no—
133C2×Q8—62no—
134C2×Q8—62no—
135C2×D4—64no—
136C2×D4—68no—
137C2×D4—62no—
138C2×D4—62no—
139C2×D4—62no—
140C22×C4—32no—
141C22×C4—31no—
142C22×C4—34no—
143C22×C4—32no—
144C22×C4—64no—
145C22×C4—64no—
146C22×C4—64no—
147C22×C4—62no—
148C22×C4—62no—
149C22×C4C22×D312yes—
150C22×C4C22×D312yes—
151C22×C4C22×D312yes—
152C22×C4C22×D311yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—64no—
156C22⋊C4—64no—
157C22⋊C4—68no—
158C22⋊C4—64no—
159C22⋊C4—68no—
160C42—64no—
161C42—62no—
162C42D1214yescomplemented
163C22×C3C22×D411yescharacteristic
164C22×C3C22×D411yescharacteristic
165C22×C3C2×#4111yescharacteristic
166C22×C3C2×#4112yes—
167C22×C3C2×#4112yes—
168C2×D3—22no—
169C2×D3—24no—
170C2×D3—24no—
171C2×D3—22no—
172C2×D3—42no—
173C2×D3—44no—
174C2×D3—42no—
175C2×D3—44no—
176C2×D3—42no—
177C2×D3#12414yescomplemented
178C3×C4—24no—
179C3×C4—42no—
180C3×C4—42no—
181C3×C4#12114yes—
182Dic3—22no—
183Dic3—24no—
184Dic3—42no—
185C23—32no—
186C23—34no—
187C23—61no—
188C23—61no—
189C23—62no—
190C23—62no—
191C23—62no—
192C23—62no—
193C23—64no—
194C23—64no—
195C23C23×D311yescharacteristic
196Q8—124no—
197D4—128no—
198D4—124no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—32no—
206C2×C4—34no—
207C2×C4—616no—
208C2×C4—61no—
209C2×C4—62no—
210C2×C4—61no—
211C2×C4—62no—
212C2×C4—68no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4C2×D1214yes—
217C2×C4C2×D1212yes—
218C2×C4D3×D412yes—
219C2×C4#23912yes—
220C2×C3C2×#12111yescharacteristic
221C2×C3#52512yes—
222C2×C3C2×#12411yescharacteristic
223C2×C3#52912yes—
224C2×C3#53211yescharacteristic
225D3—24no—
226D3—82no—
227C22—34no—
228C22—62no—
229C22—64no—
230C22—62no—
231C22—64no—
232C22—122no—
233C22—122no—
234C22—124no—
235C22—122no—
236C22—124no—
237C22C22×D1211yescharacteristic
238C22C2×D3×D411yescharacteristic
239C22C2×#23911yescharacteristic
240C22D3×#4112yes—
241C22D3×#4112yes—
242C4—24no—
243C4—42no—
244C4—42no—
245C4—62no—
246C4—64no—
247C4—122no—
248C4#89514yes—
249C3#285011yescharacteristic, complemented, Sylow
250C2—64no—
251C2—242no—
252C2C2×#89511yescharacteristic
253C2#552412yes—
254C2#566811yescharacteristic
255C2D3×#12411yescharacteristic
256C2#569112yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#382C2
prime radicalC2×#895C2
hypercenter#382D3
hypocenterC3#2850
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. C2×#897C2semidirect
3. C2×#875C2semidirect
4. C2×C42×D3C2semidirect
5. C3×#382C2semidirect
6. #4988C2semidirect
7. #4973C2semidirect
8. #4670C2semidirect
9. #4665C2semidirect
40. #382D3semidirect
66. C3×C42D4semidirect
162. C42D12semidirect
177. C2×D3#124semidirect
249. C3#2850semidirect

Maximal subgroups

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

subgroupindex
C2×#8972
C2×#8952
C2×#8752
C2×C42×D32
C3×#3822
#49882
#49732
#46702
#46652
#28503