Finite Group Databasebeta

← #78024

Subgroups — #78024

Order 384.

660 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 145 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×#894C212yesmaximal, complemented
2C2×#67⋊C4C212yesmaximal, complemented
3C3×#399C211yesmaximal, characteristic, complemented
4#4989C212yesmaximal, complemented
5#4972C212yesmaximal, complemented
6#4961C212yesmaximal, complemented
7#4960C211yesmaximal, characteristic
8#4946C211yesmaximal, characteristic, complemented
9#4670C212yesmaximal, complemented
10#2455—31nomaximal, Sylow
11C22×#67C2212yes—
12C23×Dic3C2211yescharacteristic
13C22×D12C2211yescharacteristic
14C22×C4×D3C2212yes—
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2212yes—
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×#217C2211yescharacteristic
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×C4⋊Dic3C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×C4×Dic3C2212yes—
28#894—28no—
29#67⋊C4—28no—
30C3×#95C2211yescharacteristic
31C3⋊#95C2212yes—
32C3⋊#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C2×#123—32no—
35C2×C4×D4—32no—
36#399D311yescharacteristic, complemented
37#394—32no—
38#392—31no—
39#389—32no—
40#388—32no—
41#385—32no—
42#385—31no—
43C24×C3C2311yescharacteristic
44C23×D3C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2312yes—
48C22×C3×C4C2312yes—
49C2×#67—24no—
50C2×#67—24no—
51C2×#67—24no—
52C22×Dic3—24no—
53C22×Dic3—41no—
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C22×Dic3C2312yes—
57C2×D12—24no—
58C2×D12—41no—
59C2×C4×D3—24no—
60C2×C4×D3—24no—
61C2×C4×D3—42no—
62C3×C4⋊C4—24no—
63C3×C4⋊C4—41no—
64C3×C22⋊C4—24no—
65C3×C22⋊C4—41no—
66#217—22no—
67(C2×D3)⋊C4—24no—
68(C2×D3)⋊C4—24no—
69(C2×D3)⋊C4—24no—
70(C2×D3)⋊C4D418yescomplemented
71C4⋊Dic3—24no—
72Dic3⋊C4—24no—
73Dic3⋊C4—24no—
74C4×Dic3—24no—
75C22×D4—32no—
76C22×D4—31no—
77C23×C4—31no—
78C23×C4—32no—
79#123—68no—
80C4×D4—68no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4C2×D312yes—
85C2×C4⋊C4C2×D311yescharacteristic
86C2×C22⋊C4—32no—
87C2×C22⋊C4—31no—
88C2×C22⋊C4—32no—
89C2×C22⋊C4—32no—
90C2×C22⋊C4—32no—
91C2×C22⋊C4C2×D312yes—
92C2×C22⋊C4C2×D311yescharacteristic
93C2×C42—32no—
94#95—32no—
95#95—32no—
96#95—31no—
97#95C2×D311yescharacteristic
98C23×C3—22no—
99C23×C3—22no—
100C23×C3—21no—
101C23×C3—21no—
102C23×C3#4112yes—
103C23×C3C2411yescharacteristic
104C22×D3—22no—
105C22×D3—22no—
106C22×D3—22no—
107C22×D3—22no—
108C22×D3—22no—
109C22×D3—22no—
110C22×D3C2×D414yes—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—21no—
118C2×C3×C4—21no—
119C2×C3×C4—22no—
120C2×C3×C4C2×D414yes—
121#67—216no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—44no—
130C2×Dic3C2×D414yes—
131C2×Dic3#4114yes—
132D12—44no—
133C4×D3—48no—
134C24—32no—
135C24C22×D311yescharacteristic
136C2×D4—64no—
137C2×D4—64no—
138C2×D4—64no—
139C2×D4—64no—
140C2×D4—121no—
141C22×C4—32no—
142C22×C4—32no—
143C22×C4—32no—
144C22×C4—64no—
145C22×C4—64no—
146C22×C4—64no—
147C22×C4—122no—
148C22×C4—121no—
149C22×C4C22×D311yescharacteristic
150C22×C4C22×D312yes—
151C22×C4C22×D311yescharacteristic
152C22×C4C22×D312yes—
153C4⋊C4—24no—
154C4⋊C4—41no—
155C4⋊C4—64no—
156C4⋊C4—64no—
157C4⋊C4—64no—
158C22⋊C4—24no—
159C22⋊C4—38no—
160C22⋊C4—41no—
161C22⋊C4—64no—
162C22⋊C4—64no—
163C22⋊C4—62no—
164C22⋊C4—64no—
165C42—64no—
166C22×C3—24no—
167C22×C3—22no—
168C22×C3—41no—
169C22×C3—42no—
170C22×C3—41no—
171C22×C3C22×D412yes—
172C22×C3C2×#4111yescharacteristic
173C22×C3C2×#4112yes—
174C22×C3H(5,𝔽2)11yescharacteristic
175C22×C32−1+411yescharacteristic
176C2×D3—24no—
177C2×D3—24no—
178C2×D3—24no—
179C2×D3—42no—
180C2×D3—42no—
181C2×D3—42no—
182C2×D3—42no—
183C3×C4—24no—
184C3×C4—42no—
185C3×C4—41no—
186C3×C4—41no—
187Dic3—24no—
188Dic3—24no—
189Dic3—42no—
190C23—22no—
191C23—21no—
192C23—22no—
193C23—21no—
194C23—34no—
195C23—62no—
196C23—62no—
197C23—62no—
198C23—62no—
199C23—62no—
200C23—62no—
201C23#23912yes—
202C23C23×D311yescharacteristic
203D4—616no—
204D4—124no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—34no—
215C2×C4—34no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—128no—
224C2×C4—124no—
225C2×C4D3×D414yes—
226C2×C3—42no—
227C2×C3#52512yes—
228C2×C3#53911yescharacteristic
229C2×C3D4211yescharacteristic
230C2×C3#54712yes—
231C2×C3#55911yescharacteristic
232D3—44no—
233C22—22no—
234C22—24no—
235C22—42no—
236C22—41no—
237C22—41no—
238C22—64no—
239C22—64no—
240C22—64no—
241C22—122no—
242C22—122no—
243C22—122no—
244C22—122no—
245C22C2×D3×D412yes—
246C22#100411yescharacteristic
247C22C2×#23911yescharacteristic
248C22D3×#4112yes—
249C22#101011yescharacteristic
250C4—24no—
251C4—41no—
252C4—42no—
253C4—41no—
254C4—64no—
255C4—64no—
256C4—122no—
257C3#245511yescharacteristic, complemented, Sylow
258C2—42no—
259C2—124no—
260C2#562712yes—
261C2#564511yescharacteristic
262C2#565512yes—
263C2#566411yescharacteristic
264C2#571711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radical#4960C2
hypercenter#399D3
hypocenterC3#2455
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#894C2semidirect
2. C2×#67⋊C4C2semidirect
3. C3×#399C2semidirect
4. #4989C2semidirect
5. #4972C2semidirect
6. #4961C2semidirect
8. #4946C2semidirect
9. #4670C2semidirect
36. #399D3semidirect
70. (C2×D3)⋊C4D4semidirect
257. C3#2455semidirect

Maximal subgroups

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

subgroupindex
C2×#8942
C2×#67⋊C42
C3×#3992
#49892
#49722
#49612
#49602
#49462
#46702
#24553