Finite Group Databasebeta

← #76401

Subgroups — #76401

Order 384.

528 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 149 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×#878C212yesmaximal, complemented
2C3×#390C211yesmaximal, characteristic
3#5220C212yesmaximal
4#4982C212yesmaximal, complemented
5#4950C212yesmaximal, complemented
6#4949C212yesmaximal, complemented
7#4948C212yesmaximal, complemented
8#4929C211yesmaximal, characteristic, complemented
9#4928C211yesmaximal, characteristic, complemented
10#2732—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C23×Dic3C2212yes—
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2212yes—
17C2×C3×C42C2211yescharacteristic
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×C4⋊Dic3C2212yes—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×C4⋊Dic3C2211yescharacteristic
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28#878—28no—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C2×C22⋊Q8—32no—
34#399—32no—
35#398—32no—
36#396—32no—
37#391—31no—
38#390D311yescharacteristic
39#389—31no—
40#386—32no—
41#385—32no—
42C24×C3C2312yes—
43C2×C3×D4—41no—
44C2×C3×D4—41no—
45C2×C3×D4Q814yescomplemented
46C22×C3×C4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2311yescharacteristic
49C22×C3×C4C2311yescharacteristic
50C22×Dic3—24no—
51C22×Dic3—42no—
52C22×Dic3—42no—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C22×Dic3C2312yes—
57C2×Dic6—41no—
58C2×Dic6—41no—
59C2×Dic6D414yescomplemented
60C3×C4⋊C4—41no—
61C3×C22⋊C4—24no—
62C3×C22⋊C4—24no—
63C3×C42—41no—
64#217—24no—
65#217—42no—
66#217—42no—
67C4⋊Dic3—22no—
68C4⋊Dic3—24no—
69C4⋊Dic3—41no—
70C4⋊Dic3—41no—
71Dic3⋊C4—24no—
72Dic3⋊C4—24no—
73Dic3⋊C4—42no—
74C22×Q8—31no—
75C22×D4C2×D311yescharacteristic
76C23×C4—32no—
77C22⋊Q8—68no—
78C2×C4⋊C4—32no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4C2×D311yescharacteristic
86C2×C22⋊C4—32no—
87C2×C22⋊C4—32no—
88C2×C22⋊C4—32no—
89C2×C22⋊C4C2×D312yes—
90C2×C22⋊C4C2×D312yes—
91C2×C42C2×D311yescharacteristic
92#95—32no—
93#95—32no—
94#95—32no—
95#95—32no—
96C23×C3—22no—
97C23×C3—22no—
98C23×C3—22no—
99C23×C3—22no—
100C23×C3—22no—
101C23×C3—22no—
102C23×C3C2×Q814yes—
103C23×C3C2411yescharacteristic
104C3×D4—44no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—21no—
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4—21no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4C2×D412yes—
115C2×C3×C4C2×Q812yes—
116C2×C3×C4#4112yes—
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×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—48no—
129C2×Dic3C2×D414yes—
130Dic6—44no—
131C24C22×D312yes—
132C2×Q8—34no—
133C2×Q8—121no—
134C2×Q8—121no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4Dic614yescomplemented
138C22×C4—32no—
139C22×C4—32no—
140C22×C4—32no—
141C22×C4—32no—
142C22×C4—64no—
143C22×C4—122no—
144C22×C4—122no—
145C22×C4C22×D312yes—
146C22×C4C22×D311yescharacteristic
147C22×C4C22×D311yescharacteristic
148C22×C4C22×D311yescharacteristic
149C4⋊C4—41no—
150C4⋊C4—64no—
151C4⋊C4—64no—
152C4⋊C4—64no—
153C4⋊C4—62no—
154C4⋊C4—122no—
155C4⋊C4—121no—
156C4⋊C4—121no—
157C22⋊C4—24no—
158C22⋊C4—24no—
159C22⋊C4—64no—
160C22⋊C4—122no—
161C22⋊C4—122no—
162C42—41no—
163C22×C3—24no—
164C22×C3—24no—
165C22×C3—24no—
166C22×C3—42no—
167C22×C3—42no—
168C22×C3—42no—
169C22×C3—42no—
170C22×C3C22×D411yescharacteristic
171C22×C3C22×Q811yescharacteristic
172C22×C3C2×#4111yescharacteristic
173C22×C3H(5,𝔽2)11yescharacteristic
174C22×C3H(5,𝔽2)11yescharacteristic
175C22×C3H(5,𝔽2)11yescharacteristic
176C22×C32−1+411yescharacteristic
177C3×C4—22no—
178C3×C4—22no—
179C3×C4—22no—
180C3×C4—42no—
181Dic3—24no—
182Dic3—42no—
183Dic3—42no—
184Dic3—42no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23—22no—
191C23C2×Dic614yes—
192C23C23×D311yescharacteristic
193Q8—124no—
194D4—44no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—34no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—128no—
217C2×C4C2×Dic612yes—
218C2×C4D3×D412yes—
219C2×C4#23712yes—
220C2×C3—44no—
221C2×C3#53411yescharacteristic
222C2×C3#53911yescharacteristic
223C2×C3#54311yescharacteristic
224C2×C3D4×Q811yescharacteristic
225C2×C3#55011yescharacteristic
226C2×C3#55411yescharacteristic
227C2×C3#56211yescharacteristic
228C22—24no—
229C22—24no—
230C22—24no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22C22×Dic611yescharacteristic
236C22C2×D3×D411yescharacteristic
237C22C2×#23711yescharacteristic
238C22#100411yescharacteristic
239C22#100411yescharacteristic
240C22#100911yescharacteristic
241C22#101011yescharacteristic
242C4—22no—
243C4—22no—
244C4—22no—
245C4—42no—
246C4—64no—
247C4—122no—
248C4—122no—
249C4—122no—
250C3#273211yescharacteristic, complemented, Sylow
251C2—44no—
252C2#548311yescharacteristic
253C2D4×Dic611yescharacteristic
254C2#554311yescharacteristic
255C2#560211yescharacteristic
256C2#560511yescharacteristic
257C2#568411yescharacteristic
258C2#571811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
prime radicalC22×C3×D4C22
hypercenter#390D3
hypocenterC3#2732
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#878C2semidirect
4. #4982C2semidirect
5. #4950C2semidirect
6. #4949C2semidirect
7. #4948C2semidirect
8. #4929C2semidirect
9. #4928C2semidirect
45. C2×C3×D4Q8semidirect
59. C2×Dic6D4semidirect
137. C2×D4Dic6semidirect
250. C3#2732semidirect

Maximal subgroups

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

subgroupindex
C2×#8782
C3×#3902
#52202
#49822
#49502
#49492
#49482
#49292
#49282
#27323