Finite Group Databasebeta

← #74032

Subgroups — #74032

Order 384.

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

normal subgroups only

This group has 632 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×#930C211yesmaximal, characteristic
2C22×Dic3⋊C4C211yesmaximal, characteristic, complemented
3C2×#886C212yesmaximal, complemented
4C3×#380C211yesmaximal, characteristic, complemented
5#4994C214yesmaximal, complemented
6#4958C212yesmaximal, complemented
7#4672C212yesmaximal, complemented
8#4661C212yesmaximal, complemented
9#2618—31nomaximal, Sylow
10C23×C3×C4C2211yescharacteristic
11C22×#67C2212yes—
12C23×Dic3C2211yescharacteristic
13C23×Dic3C2211yescharacteristic
14C22×D12C2211yescharacteristic
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C22⋊C4C2212yes—
17#930—24no—
18C2×(C2×D3)⋊C4C2214yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2214yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×C4⋊Dic3C2214yes—
23C2×Dic3⋊C4—24no—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26#886—28no—
27C3×#95C2212yes—
28C3⋊#95C2214yes—
29C2×#123—32no—
30C2×#121—31no—
31C22×C4⋊C4—31no—
32#398—32no—
33#396—34no—
34#394—32no—
35#386—32no—
36#380D311yescharacteristic, complemented
37C24×C3C2311yescharacteristic
38C23×D3C2312yes—
39C22×C3×C4—42no—
40C22×C3×C4D414yes—
41C22×C3×C4C2314yes—
42C22×C3×C4C2311yescharacteristic
43C22×C3×C4C2311yescharacteristic
44C2×#67—24no—
45C2×#67—24no—
46C2×#67—24no—
47C22×Dic3—22no—
48C22×Dic3—22no—
49C22×Dic3—22no—
50C22×Dic3—42no—
51C22×Dic3D414yescomplemented
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C2×D12—22no—
56C2×D12—42no—
57C3×C4⋊C4—42no—
58C3×C22⋊C4—24no—
59(C2×D3)⋊C4—24no—
60(C2×D3)⋊C4—28no—
61(C2×D3)⋊C4—44no—
62C4⋊Dic3—22no—
63C4⋊Dic3—28no—
64Dic3⋊C4—24no—
65Dic3⋊C4—24no—
66C22×D4—32no—
67C22×D4—31no—
68C23×C4—31no—
69C23×C4—31no—
70C23×C4C2×D311yescharacteristic
71#123—68no—
72#121—64no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4—31no—
76C2×C4⋊C4—34no—
77C2×C4⋊C4—64no—
78C2×C4⋊C4C2×D312yes—
79C2×C22⋊C4—32no—
80C2×C22⋊C4—34no—
81C2×C22⋊C4—34no—
82C2×C22⋊C4C2×D312yes—
83#95—34no—
84#95C2×D312yes—
85C23×C3—22no—
86C23×C3—22no—
87C23×C3C2×D412yes—
88C23×C3C2×D412yes—
89C23×C3#4112yes—
90C23×C3C2411yescharacteristic
91C22×D3—22no—
92C22×D3—22no—
93C22×D3—22no—
94C22×D3—24no—
95C22×D3—24no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—24no—
99C2×C3×C4—24no—
100C2×C3×C4—24no—
101C2×C3×C4—44no—
102C2×C3×C4C2×D412yes—
103C2×C3×C4C2×D412yes—
104#67—48no—
105C2×Dic3—28no—
106C2×Dic3—22no—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—24no—
113C2×Dic3—44no—
114C2×Dic3C2×D414yes—
115D12—44no—
116C24—32no—
117C24C22×D311yescharacteristic
118C2×D4—64no—
119C2×D4—64no—
120C2×D4—64no—
121C2×D4—62no—
122C2×D4—122no—
123C22×C4—34no—
124C22×C4—32no—
125C22×C4—32no—
126C22×C4—32no—
127C22×C4—42no—
128C22×C4—62no—
129C22×C4—62no—
130C22×C4—62no—
131C22×C4—122no—
132C22×C4D1214yes—
133C22×C4C22×D314yes—
134C22×C4C22×D311yescharacteristic
135C22×C4C22×D311yescharacteristic
136C4⋊C4—42no—
137C4⋊C4—68no—
138C4⋊C4—64no—
139C4⋊C4—64no—
140C4⋊C4—62no—
141C22⋊C4—24no—
142C22⋊C4—68no—
143C22⋊C4—64no—
144C22⋊C4—124no—
145C22×C3—22no—
146C22×C3—22no—
147C22×C3—24no—
148C22×C3—24no—
149C22×C3#12114yes—
150C22×C3C22×D411yescharacteristic
151C22×C3C22×D411yescharacteristic
152C22×C3C2×#4111yescharacteristic
153C22×C3H(5,𝔽2)12yes—
154C22×C32−1+412yes—
155C2×D3—42no—
156C2×D3—42no—
157C2×D3—42no—
158C2×D3—44no—
159C2×D3—44no—
160C3×C4—22no—
161C3×C4—22no—
162C3×C4—44no—
163Dic3—24no—
164Dic3—42no—
165Dic3—42no—
166C23—22no—
167C23—22no—
168C23—62no—
169C23—62no—
170C23—64no—
171C23—62no—
172C23—64no—
173C23C2×D1212yes—
174C23D3×D412yes—
175C23#23912yes—
176C23C23×D311yescharacteristic
177D4—124no—
178D4—128no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—34no—
185C2×C4—44no—
186C2×C4—68no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—64no—
193C2×C4—62no—
194C2×C4—124no—
195C2×C4C2×D1212yes—
196C2×C4C2×D1212yes—
197C2×C3—24no—
198C2×C3C2×#12111yescharacteristic
199C2×C3#53411yescharacteristic
200C2×C3#53611yescharacteristic
201C2×C3#53712yes—
202C2×C3#53912yes—
203D3—82no—
204C22—24no—
205C22—22no—
206C22—22no—
207C22—24no—
208C22—124no—
209C22—122no—
210C22—122no—
211C22—122no—
212C22—124no—
213C22#89514yes—
214C22C22×D1211yescharacteristic
215C22C2×D3×D411yescharacteristic
216C22#100412yes—
217C22C2×#23911yescharacteristic
218C22#101012yes—
219C4—22no—
220C4—22no—
221C4—44no—
222C4—64no—
223C4—122no—
224C4—122no—
225C3#261811yescharacteristic, complemented, Sylow
226C2—24no—
227C2—242no—
228C2#549311yescharacteristic
229C2C2×#89511yescharacteristic
230C2#552611yescharacteristic
231C2#559712yes—
232C2#564512yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#380C2
prime radicalC2×#930C2
hypercenter#380D3
hypocenterC3#2618
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C22×Dic3⋊C4C2semidirect
3. C2×#886C2semidirect
4. C3×#380C2semidirect
5. #4994C2semidirect
6. #4958C2semidirect
7. #4672C2semidirect
8. #4661C2semidirect
36. #380D3semidirect
51. C22×Dic3D4semidirect
225. C3#2618semidirect

Maximal subgroups

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

subgroupindex
C2×#9302
C22×Dic3⋊C42
C2×#8862
C3×#3802
#49942
#49582
#46722
#46612
#26183