Finite Group Databasebeta

← #77147

Subgroups — #77147

Order 384.

612 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 129 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×#884C212yesmaximal, complemented
2C3×#394C211yesmaximal, characteristic, complemented
3#5223C212yesmaximal, complemented
4#4988C211yesmaximal, characteristic, complemented
5#4981C211yesmaximal, characteristic, complemented
6#4961C212yesmaximal, complemented
7#4956C212yesmaximal, complemented
8#4945C212yesmaximal, complemented
9#4664C212yesmaximal, complemented
10#2699—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C22×#67C2212yes—
13C22×#67C2212yes—
14C22×D12C2211yescharacteristic
15C22×C4×D3C2212yes—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2212yes—
18C2×C3×C22⋊C4C2212yes—
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×C4⋊Dic3C2211yescharacteristic
25C2×Dic3⋊C4C2212yes—
26C2×C4×Dic3C2211yescharacteristic
27#884—28no—
28C3×#95C2211yescharacteristic
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2211yescharacteristic
33C2×#121—32no—
34#398—31no—
35#394—32no—
36#394D311yescharacteristic, complemented
37#392—32no—
38#390—31no—
39#388—32no—
40#386—32no—
41#381—32no—
42C24×C3C2312yes—
43C23×D3C2312yes—
44C2×C3×D4—42no—
45C2×C3×D4—41no—
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2312yes—
48C22×C3×C4C2311yescharacteristic
49C22×C3×C4C2311yescharacteristic
50C2×#67—24no—
51C2×#67—24no—
52C2×#67—24no—
53C2×#67—42no—
54C2×#67—42no—
55C2×#67—42no—
56C22×Dic3C2312yes—
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C2×D12—41no—
60C2×D12—41no—
61C2×D12D414yescomplemented
62C2×C4×D3—24no—
63C2×C4×D3—24no—
64C2×C4×D3—42no—
65C3×C4⋊C4—41no—
66C3×C22⋊C4—24no—
67C3×C22⋊C4—42no—
68#217—24no—
69#217—42no—
70(C2×D3)⋊C4—24no—
71(C2×D3)⋊C4—42no—
72(C2×D3)⋊C4—42no—
73C4⋊Dic3—41no—
74Dic3⋊C4—24no—
75C4×Dic3—41no—
76C22×D4—32no—
77C22×D4—32no—
78C22×D4—31no—
79C22×D4C2×D311yescharacteristic
80C23×C4—32no—
81#121—68no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4C2×D311yescharacteristic
85C2×C22⋊C4—32no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4—32no—
88C2×C22⋊C4—32no—
89C2×C22⋊C4—32no—
90C2×C22⋊C4C2×D312yes—
91C2×C22⋊C4C2×D312yes—
92C2×C42—31no—
93#95—32no—
94#95—32no—
95#95—32no—
96#95—31no—
97#95C2×D311yescharacteristic
98C23×C3—22no—
99C23×C3—22no—
100C23×C3—22no—
101C23×C3—22no—
102C23×C3—22no—
103C23×C3—22no—
104C23×C3—22no—
105C23×C3C2411yescharacteristic
106C22×D3—22no—
107C22×D3—22no—
108C22×D3—22no—
109C22×D3—22no—
110C22×D3—22no—
111C22×D3—22no—
112C22×D3C2×D414yes—
113C3×D4—44no—
114C2×C3×C4—21no—
115C2×C3×C4—21no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—21no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—21no—
123C2×C3×C4C2×D412yes—
124#67—48no—
125#67—48no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3#4114yes—
135D12—44no—
136C4×D3—48no—
137C24—32no—
138C24C22×D312yes—
139C2×D4—34no—
140C2×D4—42no—
141C2×D4—41no—
142C2×D4—64no—
143C2×D4—64no—
144C2×D4—64no—
145C2×D4—121no—
146C2×D4—121no—
147C2×D4—122no—
148C2×D4—122no—
149C2×D4—122no—
150C22×C4—32no—
151C22×C4—32no—
152C22×C4—32no—
153C22×C4—64no—
154C22×C4—64no—
155C22×C4—122no—
156C22×C4C22×D311yescharacteristic
157C22×C4C22×D312yes—
158C22×C4C22×D311yescharacteristic
159C22×C4C22×D311yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—64no—
162C4⋊C4—121no—
163C22⋊C4—24no—
164C22⋊C4—42no—
165C22⋊C4—64no—
166C22⋊C4—64no—
167C22⋊C4—122no—
168C22⋊C4—122no—
169C22⋊C4—122no—
170C42—121no—
171C22×C3—42no—
172C22×C3—42no—
173C22×C3—42no—
174C22×C3—42no—
175C22×C3—42no—
176C22×C3—42no—
177C22×C3—42no—
178C22×C3C22×D411yescharacteristic
179C22×C3C2×#4112yes—
180C22×C3H(5,𝔽2)11yescharacteristic
181C22×C3H(5,𝔽2)12yes—
182C22×C3H(5,𝔽2)11yescharacteristic
183C2×D3—24no—
184C2×D3—24no—
185C2×D3—24no—
186C2×D3—42no—
187C2×D3—42no—
188C2×D3—42no—
189C2×D3—42no—
190C3×C4—22no—
191C3×C4—42no—
192C3×C4—41no—
193C3×C4—41no—
194Dic3—24no—
195Dic3—42no—
196Dic3—42no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—34no—
205C23—62no—
206C23—62no—
207C23—62no—
208C23—62no—
209C23—62no—
210C23—62no—
211C23C23×D311yescharacteristic
212D4—44no—
213D4—124no—
214D4—128no—
215D4—128no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—34no—
226C2×C4—62no—
227C2×C4—62no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—128no—
235C2×C4D3×D412yes—
236C2×C3—82no—
237C2×C3#53411yescharacteristic
238C2×C3#54012yes—
239C2×C3#54612yes—
240C2×C3#55511yescharacteristic
241C2×C3#56011yescharacteristic
242D3—44no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—64no—
251C22—64no—
252C22—64no—
253C22—122no—
254C22—122no—
255C22—122no—
256C22—122no—
257C22C2×D3×D411yescharacteristic
258C22#100412yes—
259C22#100411yescharacteristic
260C22D3×#4112yes—
261C22#100911yescharacteristic
262C4—22no—
263C4—41no—
264C4—41no—
265C4—42no—
266C4—64no—
267C4—122no—
268C4—122no—
269C3#269911yescharacteristic, complemented, Sylow
270C2—82no—
271C2—124no—
272C2#559511yescharacteristic
273C2#560511yescharacteristic
274C2#561212yes—
275C2#566312yes—
276C2#567011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
hypercenter#394D3
hypocenterC3#2699
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#884C2semidirect
2. C3×#394C2semidirect
3. #5223C2semidirect
4. #4988C2semidirect
5. #4981C2semidirect
6. #4961C2semidirect
7. #4956C2semidirect
8. #4945C2semidirect
9. #4664C2semidirect
36. #394D3semidirect
61. C2×D12D4semidirect
269. C3#2699semidirect

Maximal subgroups

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

subgroupindex
C2×#8842
C3×#3942
#52232
#49882
#49812
#49612
#49562
#49452
#46642
#26993