Finite Group Databasebeta

← #74143

Subgroups — #74143

Order 384.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C3×#381C211yesmaximal, characteristic, complemented
2#5213C212yesmaximal, complemented
3#4960C211yesmaximal, characteristic
4#4956C212yesmaximal, complemented
5#4947C211yesmaximal, characteristic, complemented
6#4673C212yesmaximal, complemented
7#4665C212yesmaximal, complemented
8#4664C212yesmaximal, complemented
9C3⋊#401C212yesmaximal, complemented
10#2760—31nomaximal, Sylow
11C23×C3×C4C2211yescharacteristic
12C22×#67C2212yes—
13C22×D12C2211yescharacteristic
14C22×C4×D3C2212yes—
15C22×Dic6C2211yescharacteristic
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×#217C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C3×#95C2212yes—
24C3×#95C2212yes—
25C3⋊#95C2211yescharacteristic
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2211yescharacteristic
32#401—32no—
33#393—31no—
34#393—32no—
35#392—31no—
36#386—32no—
37#386—32no—
38#386—32no—
39#381—32no—
40#381D311yescharacteristic, complemented
41C24×C3C2311yescharacteristic
42C23×D3C2312yes—
43C22×C3×C4—22no—
44C22×C3×C4—41no—
45C22×C3×C4—41no—
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2312yes—
48C22×C3×C4C2312yes—
49C22×C3×C4C2311yescharacteristic
50C2×#67—24no—
51C2×#67—24no—
52C2×#67—42no—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C2×D12—24no—
57C2×D12—41no—
58C2×C4×D3—24no—
59C2×C4×D3—42no—
60C2×C4×D3—42no—
61C2×Dic6—41no—
62C2×Dic6—42no—
63C3×C22⋊C4—41no—
64C3×C22⋊C4—41no—
65#217—42no—
66(C2×D3)⋊C4—42no—
67(C2×D3)⋊C4—42no—
68(C2×D3)⋊C4—42no—
69(C2×D3)⋊C4—42no—
70C22×Q8—31no—
71C22×D4—32no—
72C22×D4—31no—
73C23×C4—32no—
74C23×C4C2×D311yescharacteristic
75C2×C22⋊C4—32no—
76C2×C22⋊C4—32no—
77C2×C22⋊C4—32no—
78C2×C22⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C22⋊C4C2×D311yescharacteristic
81C2×C22⋊C4C2×D311yescharacteristic
82#95—31no—
83#95—32no—
84#95—32no—
85#95—32no—
86#95—32no—
87#95—32no—
88#95—31no—
89#95C2×D312yes—
90#95C2×D312yes—
91C23×C3—22no—
92C23×C3—22no—
93C23×C3—21no—
94C23×C3—21no—
95C23×C3#4112yes—
96C23×C3C2411yescharacteristic
97C22×D3—22no—
98C22×D3—22no—
99C22×D3—22no—
100C22×D3—22no—
101C22×D3—22no—
102C22×D3—22no—
103C22×D3#4114yes—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4—21no—
114C2×C3×C4—44no—
115#67—48no—
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—
125D12—44no—
126C4×D3—48no—
127Dic6—44no—
128C24—32no—
129C24C22×D311yescharacteristic
130C2×Q8—121no—
131C2×Q8—122no—
132C2×D4—64no—
133C2×D4—64no—
134C2×D4—64no—
135C2×D4—121no—
136C2×D4—122no—
137C22×C4—22no—
138C22×C4—32no—
139C22×C4—32no—
140C22×C4—32no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—64no—
144C22×C4—122no—
145C22×C4—122no—
146C22×C4C22×D311yescharacteristic
147C22×C4C22×D312yes—
148C22×C4C22×D312yes—
149C22×C4C22×D311yescharacteristic
150C22⋊C4—41no—
151C22⋊C4—41no—
152C22⋊C4—122no—
153C22⋊C4—122no—
154C22⋊C4—122no—
155C22⋊C4—122no—
156C22⋊C4—122no—
157C22×C3—22no—
158C22×C3—24no—
159C22×C3—42no—
160C22×C3—41no—
161C22×C3—41no—
162C22×C3C2×#4111yescharacteristic
163C22×C3C2×#4112yes—
164C22×C3H(5,𝔽2)11yescharacteristic
165C22×C3H(5,𝔽2)11yescharacteristic
166C22×C3H(5,𝔽2)12yes—
167C2×D3—24no—
168C2×D3—24no—
169C2×D3—24no—
170C2×D3—42no—
171C2×D3—42no—
172C2×D3—42no—
173C2×D3—42no—
174C3×C4—42no—
175C3×C4—42no—
176C3×C4—41no—
177C3×C4—41no—
178Dic3—42no—
179Dic3—42no—
180Dic3—42no—
181C23—21no—
182C23—21no—
183C23—22no—
184C23—22no—
185C23—34no—
186C23—62no—
187C23—62no—
188C23—62no—
189C23—62no—
190C23—62no—
191C23—62no—
192C23#23512yes—
193C23C23×D311yescharacteristic
194Q8—124no—
195D4—124no—
196D4—128no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—44no—
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—62no—
217C2×C4—128no—
218C2×C3—42no—
219C2×C3#53812yes—
220C2×C3#53811yescharacteristic
221C2×C3#55111yescharacteristic
222C2×C3#55112yes—
223C2×C3#56011yescharacteristic
224D3—44no—
225C22—22no—
226C22—24no—
227C22—42no—
228C22—41no—
229C22—41no—
230C22—64no—
231C22—64no—
232C22—64no—
233C22—122no—
234C22—122no—
235C22—122no—
236C22—122no—
237C22C2×#23511yescharacteristic
238C22#100411yescharacteristic
239C22#100411yescharacteristic
240C22D3×#4112yes—
241C22#100912yes—
242C4—42no—
243C4—41no—
244C4—41no—
245C4—42no—
246C4—122no—
247C4—122no—
248C4—122no—
249C3#276011yescharacteristic, complemented, Sylow
250C2—42no—
251C2—124no—
252C2#549411yescharacteristic
253C2#552712yes—
254C2#565712yes—
255C2#565911yescharacteristic
256C2#566611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#381C2
prime radical#4960C2
hypercenter#381D3
hypocenterC3#2760
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C3×#381C2semidirect
2. #5213C2semidirect
4. #4956C2semidirect
5. #4947C2semidirect
6. #4673C2semidirect
7. #4665C2semidirect
8. #4664C2semidirect
9. C3⋊#401C2semidirect
40. #381D3semidirect
249. C3#2760semidirect

Maximal subgroups

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

subgroupindex
C3×#3812
#52132
#49602
#49562
#49472
#46732
#46652
#46642
C3⋊#4012
#27603