Finite Group Databasebeta

← #76399

Subgroups — #76399

Order 384.

476 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 133 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×#879C212yesmaximal, complemented
2C3×#390C211yesmaximal, characteristic
3#5222C212yesmaximal
4#4976C212yesmaximal, complemented
5#4951C212yesmaximal, complemented
6#4949C212yesmaximal, complemented
7#4948C212yesmaximal, complemented
8#4928C211yesmaximal, characteristic, complemented
9#4928C211yesmaximal, characteristic, complemented
10#2727—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C22⋊C4C2212yes—
14C2×C3×C22⋊C4C2212yes—
15C2×C3×C42C2211yescharacteristic
16C2×#217C2212yes—
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×C4⋊Dic3C2211yescharacteristic
21C2×C4⋊Dic3C2211yescharacteristic
22C2×C4⋊Dic3C2211yescharacteristic
23C2×C4⋊Dic3C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×C4×Dic3C2212yes—
29#879—28no—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C2×#126—32no—
34#399—32no—
35#396—32no—
36#394—32no—
37#390D311yescharacteristic
38#389—32no—
39#389—31no—
40#389—31no—
41#388—32no—
42C24×C3C2312yes—
43C2×C3×D4—22no—
44C2×C3×D4—41no—
45C2×C3×D4—41no—
46C22×C3×C4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2311yescharacteristic
49C22×C3×C4C2311yescharacteristic
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C3×C4⋊C4—41no—
55C3×C22⋊C4—24no—
56C3×C22⋊C4—24no—
57C3×C42—41no—
58#217—24no—
59#217—24no—
60#217—42no—
61#217—42no—
62C4⋊Dic3—22no—
63C4⋊Dic3—41no—
64C4⋊Dic3—41no—
65C4⋊Dic3D414yescomplemented
66Dic3⋊C4—24no—
67Dic3⋊C4—24no—
68Dic3⋊C4—24no—
69Dic3⋊C4—42no—
70C4×Dic3—24no—
71C22×D4C2×D311yescharacteristic
72#126—68no—
73C2×C4⋊C4—31no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4—32no—
76C2×C4⋊C4—32no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4—31no—
79C2×C4⋊C4—31no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4C2×D311yescharacteristic
82C2×C22⋊C4—32no—
83C2×C22⋊C4—32no—
84C2×C22⋊C4—32no—
85C2×C22⋊C4—32no—
86C2×C22⋊C4C2×D312yes—
87C2×C22⋊C4C2×D312yes—
88C2×C42—32no—
89C2×C42C2×D311yescharacteristic
90#95—32no—
91#95—32no—
92#95—32no—
93C23×C3—22no—
94C23×C3—22no—
95C23×C3—22no—
96C23×C3—22no—
97C23×C3—22no—
98C23×C3—22no—
99C23×C3—22no—
100C23×C3C2411yescharacteristic
101C3×D4—44no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—21no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4C2×D412yes—
112C2×C3×C4#4112yes—
113C2×C3×C4#4112yes—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3C2×D414yes—
126C24C22×D312yes—
127C2×D4—22no—
128C2×D4—41no—
129C2×D4—41no—
130C22×C4—32no—
131C22×C4—32no—
132C22×C4—32no—
133C22×C4—32no—
134C22×C4C22×D312yes—
135C22×C4C22×D311yescharacteristic
136C22×C4C22×D311yescharacteristic
137C22×C4C22×D311yescharacteristic
138C4⋊C4—34no—
139C4⋊C4—41no—
140C4⋊C4—64no—
141C4⋊C4—64no—
142C4⋊C4—64no—
143C4⋊C4—62no—
144C4⋊C4—122no—
145C4⋊C4—121no—
146C4⋊C4—121no—
147C22⋊C4—24no—
148C22⋊C4—24no—
149C22⋊C4—64no—
150C22⋊C4—64no—
151C22⋊C4—122no—
152C22⋊C4—122no—
153C42—41no—
154C42—64no—
155C22×C3—42no—
156C22×C3—42no—
157C22×C3—42no—
158C22×C3—42no—
159C22×C3—42no—
160C22×C3—42no—
161C22×C3—42no—
162C22×C3C22×D411yescharacteristic
163C22×C3C2×#4111yescharacteristic
164C22×C3C2×#4111yescharacteristic
165C22×C3H(5,𝔽2)11yescharacteristic
166C22×C3H(5,𝔽2)11yescharacteristic
167C22×C32−1+411yescharacteristic
168C22×C32−1+411yescharacteristic
169C3×C4—22no—
170C3×C4—22no—
171C3×C4—22no—
172C3×C4—42no—
173Dic3—24no—
174Dic3—42no—
175Dic3—42no—
176Dic3—42no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23C23×D311yescharacteristic
185D4—44no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—34no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—62no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4#23512yes—
208C2×C4D3×D412yes—
209C2×C4#23712yes—
210C2×C3—82no—
211C2×C3#53711yescharacteristic
212C2×C3#53811yescharacteristic
213C2×C3#53911yescharacteristic
214C2×C3#54711yescharacteristic
215C2×C3#55011yescharacteristic
216C2×C3#55611yescharacteristic
217C2×C3#56311yescharacteristic
218C22—42no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22C2×#23511yescharacteristic
226C22C2×D3×D411yescharacteristic
227C22C2×#23711yescharacteristic
228C22#100411yescharacteristic
229C22#100411yescharacteristic
230C22#101011yescharacteristic
231C22#101011yescharacteristic
232C4—22no—
233C4—22no—
234C4—22no—
235C4—42no—
236C4—64no—
237C4—122no—
238C4—122no—
239C4—122no—
240C3#272711yescharacteristic, complemented, Sylow
241C2—82no—
242C2#549411yescharacteristic
243C2#554211yescharacteristic
244C2#555111yescharacteristic
245C2#559711yescharacteristic
246C2#560211yescharacteristic
247C2#567211yescharacteristic
248C2#571811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
prime radicalC22×C3×D4C22
hypercenter#390D3
hypocenterC3#2727
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#879C2semidirect
4. #4976C2semidirect
5. #4951C2semidirect
6. #4949C2semidirect
7. #4948C2semidirect
8. #4928C2semidirect
9. #4928C2semidirect
65. C4⋊Dic3D4semidirect
240. C3#2727semidirect

Maximal subgroups

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

subgroupindex
C2×#8792
C3×#3902
#52222
#49762
#49512
#49492
#49482
#49282
#27273