Finite Group Databasebeta

← #76350

Subgroups — #76350

Order 384.

704 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 153 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×#940C211yesmaximal, characteristic
2C2×#936C211yesmaximal, characteristic, complemented
3C2×#893C211yesmaximal, characteristic, complemented
4C2×#875C211yesmaximal, characteristic, complemented
5C3×#390C211yesmaximal, characteristic, complemented
6#4973C211yesmaximal, characteristic, complemented
7#4959C214yesmaximal, complemented
8#4956C214yesmaximal, complemented
9C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
10#2544—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C22×#67C2214yes—
13C22×D12C2211yescharacteristic
14C22×C4×D3C2212yes—
15C22×Dic6C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2214yes—
18C2×C3×C42C2211yescharacteristic
19C2×#217C2212yes—
20C2×#217C2212yes—
21#940—24no—
22#936—24no—
23C2×(C2×D3)⋊C4C2214yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×C4×Dic3C2211yescharacteristic
27C2×C4×Dic3C2211yescharacteristic
28#893—24no—
29#875—24no—
30C3⋊#95C2212yes—
31C3⋊#95C2214yes—
32C2×#127—31no—
33C2×#124—31no—
34C2×#124—31no—
35C2×C4×D4—31no—
36#394—34no—
37#391—31no—
38#390D311yescharacteristic, complemented
39#386—34no—
40C4×C4⋊C4—31no—
41C24×C3C2312yes—
42C23×D3C2312yes—
43C2×C3×D4—22no—
44C2×C3×D4—22no—
45C2×C3×D4—22no—
46C22×C3×C4C2312yes—
47C22×C3×C4C2312yes—
48C22×C3×C4C2311yescharacteristic
49C2×#67—28no—
50C2×#67—28no—
51C2×#67—44no—
52C22×Dic3C2312yes—
53C22×Dic3C2311yescharacteristic
54C22×Dic3C2311yescharacteristic
55C22×Dic3C2312yes—
56C2×D12—22no—
57C2×D12—22no—
58C2×D12D414yes—
59C2×C4×D3—24no—
60C2×C4×D3—44no—
61C2×Dic6—22no—
62C2×Dic6—22no—
63C2×Dic6—22no—
64C3×C4⋊C4—22no—
65C3×C22⋊C4—44no—
66C3×C42—22no—
67#217—24no—
68#217—24no—
69(C2×D3)⋊C4—28no—
70(C2×D3)⋊C4—24no—
71Dic3⋊C4—24no—
72C4×Dic3—22no—
73C4×Dic3D414yescomplemented
74C22×Q8—31no—
75C22×D4—34no—
76C22×D4—31no—
77C22×D4C2×D311yescharacteristic
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—32no—
88C2×C22⋊C4—34no—
89C2×C22⋊C4C2×D314yes—
90C2×C42—31no—
91C2×C42—31no—
92C2×C42C2×D311yescharacteristic
93#95—32no—
94#95—34no—
95C23×C3—22no—
96C23×C3—22no—
97C23×C3—24no—
98C23×C3—22no—
99C23×C3—24no—
100C23×C3C2411yescharacteristic
101C22×D3—22no—
102C22×D3—22no—
103C22×D3—24no—
104C22×D3—24no—
105C22×D3C2×D414yes—
106C3×D4—44no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—24no—
112C2×C3×C4C2×D412yes—
113C2×C3×C4C2×D412yes—
114C2×C3×C4#4112yes—
115#67—416no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3C2×D412yes—
124C2×Dic3C2×D412yes—
125C2×Dic3#4114yes—
126D12—28no—
127C4×D3—48no—
128Dic6—44no—
129C24—32no—
130C24C22×D312yes—
131C2×Q8—62no—
132C2×Q8—62no—
133C2×Q8—62no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—34no—
138C2×D4—68no—
139C2×D4—68no—
140C2×D4—62no—
141C2×D4—62no—
142C2×D4—124no—
143C22×C4—32no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—32no—
147C22×C4—64no—
148C22×C4—124no—
149C22×C4C22×D312yes—
150C22×C4C22×D312yes—
151C22×C4C22×D311yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—64no—
154C22⋊C4—44no—
155C22⋊C4—64no—
156C22⋊C4—68no—
157C22⋊C4—64no—
158C22⋊C4—64no—
159C42—22no—
160C42—34no—
161C42—62no—
162C22×C3—42no—
163C22×C3—42no—
164C22×C3—42no—
165C22×C3—44no—
166C22×C3—44no—
167C22×C3C22×D411yescharacteristic
168C22×C3C22×D411yescharacteristic
169C22×C3C2×#4111yescharacteristic
170C22×C3C2×#4112yes—
171C22×C3H(5,𝔽2)12yes—
172C2×D3—24no—
173C2×D3—28no—
174C2×D3—42no—
175C2×D3—42no—
176C2×D3—44no—
177C3×C4—42no—
178C3×C4—42no—
179C3×C4#12114yes—
180Dic3—22no—
181Dic3—22no—
182Dic3—24no—
183Dic3—42no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—24no—
188C23—24no—
189C23—34no—
190C23—62no—
191C23—62no—
192C23—64no—
193C23—64no—
194C23C23×D311yescharacteristic
195Q8—124no—
196D4—44no—
197D4—68no—
198D4—1216no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—32no—
205C2×C4—32no—
206C2×C4—34no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—128no—
215C2×C4#23512yes—
216C2×C4D3×D412yes—
217C2×C4D3×D412yes—
218C2×C3—82no—
219C2×C3C2×#12111yescharacteristic
220C2×C3#53211yescharacteristic
221C2×C3#53511yescharacteristic
222C2×C3#54612yes—
223C2×C3#55912yes—
224D3—44no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—44no—
229C22—44no—
230C22—64no—
231C22—68no—
232C22—122no—
233C22—122no—
234C22—124no—
235C22C2×#23511yescharacteristic
236C22C2×D3×D411yescharacteristic
237C22C2×D3×D411yescharacteristic
238C22#100412yes—
239C22D3×#4112yes—
240C4—42no—
241C4—42no—
242C4—62no—
243C4—62no—
244C4—64no—
245C4—122no—
246C4#88414yes—
247C3#254411yescharacteristic, complemented, Sylow
248C2—82no—
249C2—124no—
250C2C2×#88411yescharacteristic
251C2#555912yes—
252C2#561212yes—
253C2#566811yescharacteristic
254C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
prime radicalC2×#940C2
hypercenter#390D3
hypocenterC3#2544
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#936C2semidirect
3. C2×#893C2semidirect
4. C2×#875C2semidirect
5. C3×#390C2semidirect
6. #4973C2semidirect
7. #4959C2semidirect
8. #4956C2semidirect
9. C4×Dic3⋊C4C2semidirect
38. #390D3semidirect
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×#9402
C2×#9362
C2×#8932
C2×#8752
C3×#3902
#49732
#49592
#49562
C4×Dic3⋊C42
#25443