Finite Group Databasebeta

← #76533

Subgroups — #76533

Order 384.

704 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 157 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×#947C211yesmaximal, characteristic, complemented
2C2×#947C211yesmaximal, characteristic, complemented
3C2×#893C211yesmaximal, characteristic, complemented
4C2×#874C211yesmaximal, characteristic
5C3×#391C211yesmaximal, characteristic, complemented
6#4988C211yesmaximal, characteristic, complemented
7C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
8#4672C214yesmaximal, complemented
9#4665C214yesmaximal, complemented
10#2544—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×D12C2211yescharacteristic
13C22×D12C2214yes—
14C22×D12C2211yescharacteristic
15C22×C4×D3C2212yes—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C42C2211yescharacteristic
18#947—24no—
19#947—24no—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2214yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2214yes—
25C2×Dic3⋊C4C2212yes—
26C2×C4×Dic3C2211yescharacteristic
27C2×C4×Dic3C2211yescharacteristic
28#893—24no—
29#874—24no—
30C3×#95C2214yes—
31C3⋊#95C2212yes—
32C2×#127—31no—
33C2×#124—31no—
34C2×#124—31no—
35C2×C4×D4—31no—
36#394—34no—
37#391D311yescharacteristic, complemented
38#390—31no—
39#386—34no—
40C4×C4⋊C4—31no—
41C23×D3C2312yes—
42C23×D3C2312yes—
43C2×C3×Q8—22no—
44C2×C3×Q8—22no—
45C2×C3×Q8—22no—
46C22×C3×C4C2312yes—
47C22×C3×C4C2312yes—
48C22×C3×C4C2312yes—
49C22×C3×C4C2311yescharacteristic
50C22×Dic3C2312yes—
51C22×Dic3C2311yescharacteristic
52C22×Dic3C2311yescharacteristic
53C2×D12—22no—
54C2×D12—22no—
55C2×D12—28no—
56C2×D12—22no—
57C2×D12—22no—
58C2×D12—22no—
59C2×D12—28no—
60C2×D12—44no—
61C2×D12D414yes—
62C2×C4×D3—24no—
63C2×C4×D3—44no—
64C3×C4⋊C4—22no—
65C3×C42—22no—
66(C2×D3)⋊C4—28no—
67(C2×D3)⋊C4—24no—
68(C2×D3)⋊C4—24no—
69(C2×D3)⋊C4—24no—
70(C2×D3)⋊C4—44no—
71Dic3⋊C4—24no—
72C4×Dic3—22no—
73C4×Dic3D414yescomplemented
74C22×Q8C2×D311yescharacteristic
75C22×D4—31no—
76C22×D4—31no—
77C22×D4—34no—
78C23×C4—32no—
79#127—64no—
80#124—64no—
81#124—64no—
82C4×D4—64no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4C2×D311yescharacteristic
85C2×C22⋊C4—32no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4—34no—
88C2×C22⋊C4—32no—
89C2×C22⋊C4—34no—
90C2×C42—31no—
91C2×C42—31no—
92C2×C42C2×D311yescharacteristic
93#95—32no—
94#95C2×D314yes—
95C23×C3C2411yescharacteristic
96C22×D3—22no—
97C22×D3—22no—
98C22×D3—24no—
99C22×D3—24no—
100C22×D3—22no—
101C22×D3—22no—
102C22×D3—22no—
103C22×D3—24no—
104C22×D3—24no—
105C22×D3C2×D414yes—
106C3×Q8—44no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—24no—
114C2×C3×C4C2×D412yes—
115C2×C3×C4C2×D412yes—
116C2×C3×C4#4112yes—
117C2×C3×C4#4114yes—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3C2×D412yes—
124C2×Dic3C2×D412yes—
125D12—28no—
126D12—416no—
127D12—44no—
128C4×D3—48no—
129C24—32no—
130C24—32no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×D4—34no—
135C2×D4—68no—
136C2×D4—62no—
137C2×D4—62no—
138C2×D4—62no—
139C2×D4—62no—
140C2×D4—62no—
141C2×D4—68no—
142C2×D4—124no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—32no—
146C22×C4—64no—
147C22×C4—124no—
148C22×C4C22×D312yes—
149C22×C4C22×D312yes—
150C22×C4C22×D312yes—
151C22×C4C22×D311yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—64no—
154C22⋊C4—64no—
155C22⋊C4—64no—
156C22⋊C4—64no—
157C22⋊C4—68no—
158C22⋊C4—124no—
159C42—22no—
160C42—34no—
161C42—62no—
162C22×C3C22×D411yescharacteristic
163C22×C3C22×D411yescharacteristic
164C22×C3C2×#4111yescharacteristic
165C22×C3C2×#4112yes—
166C22×C3H(5,𝔽2)12yes—
167C2×D3—24no—
168C2×D3—28no—
169C2×D3—42no—
170C2×D3—42no—
171C2×D3—44no—
172C2×D3—42no—
173C2×D3—42no—
174C2×D3—42no—
175C2×D3—44no—
176C2×D3—44no—
177C3×C4—24no—
178C3×C4—42no—
179C3×C4—42no—
180C3×C4#12114yes—
181Dic3—22no—
182Dic3—22no—
183Dic3—42no—
184C23—34no—
185C23—62no—
186C23—62no—
187C23—64no—
188C23—64no—
189C23—62no—
190C23—62no—
191C23—62no—
192C23—64no—
193C23—64no—
194C23C23×D311yescharacteristic
195Q8—44no—
196D4—68no—
197D4—1216no—
198D4—124no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—32no—
207C2×C4—32no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—128no—
214C2×C4#23512yes—
215C2×C4D3×D412yes—
216C2×C4D3×D412yes—
217C2×C4#23914yes—
218C2×C3C2×#12111yescharacteristic
219C2×C3#53211yescharacteristic
220C2×C3#53511yescharacteristic
221C2×C3#54612yes—
222C2×C3#55912yes—
223D3—44no—
224D3—82no—
225C22—64no—
226C22—68no—
227C22—122no—
228C22—122no—
229C22—124no—
230C22—122no—
231C22—122no—
232C22—122no—
233C22—124no—
234C22—124no—
235C22C2×#23511yescharacteristic
236C22C2×D3×D411yescharacteristic
237C22C2×D3×D411yescharacteristic
238C22C2×#23912yes—
239C22#100912yes—
240C4—24no—
241C4—42no—
242C4—42no—
243C4—62no—
244C4—62no—
245C4—122no—
246C4#88414yes—
247C3#254411yescharacteristic, complemented, Sylow
248C2—124no—
249C2—242no—
250C2C2×#88411yescharacteristic
251C2#558512yes—
252C2#563012yes—
253C2#567411yescharacteristic
254C2#572511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#391C2
prime radicalC2×#874C2
hypercenter#391D3
hypocenterC3#2544
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#947C2semidirect
2. C2×#947C2semidirect
3. C2×#893C2semidirect
5. C3×#391C2semidirect
6. #4988C2semidirect
7. C4×Dic3⋊C4C2semidirect
8. #4672C2semidirect
9. #4665C2semidirect
37. #391D3semidirect
73. C4×Dic3D4semidirect
247. C3#2544semidirect

Maximal subgroups

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

subgroupindex
C2×#9472
C2×#8932
C2×#8742
C3×#3912
#49882
C4×Dic3⋊C42
#46722
#46652
#25443