Finite Group Databasebeta

← #78051

Subgroups — #78051

Order 384.

512 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 131 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×#890C211yesmaximal, characteristic, complemented
2C2×#67⋊C4C211yesmaximal, characteristic
3C3×#399C211yesmaximal, characteristic, complemented
4#4993C211yesmaximal, characteristic, complemented
5#4989C212yesmaximal, complemented
6#4982C212yesmaximal, complemented
7#4961C212yesmaximal
8#4953C211yesmaximal, characteristic, complemented
9#4946C212yesmaximal, complemented
10#4674C211yesmaximal, characteristic, complemented
11#4651C211yesmaximal, characteristic, complemented
12#2633—31nomaximal, Sylow
13C22×#67C2211yescharacteristic
14C23×Dic3C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2212yes—
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2212yes—
24C2×C4⋊Dic3C2211yescharacteristic
25C2×C4⋊Dic3C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2212yes—
29C2×C4×Dic3C2211yescharacteristic
30#890—24no—
31#67⋊C4—24no—
32C3×#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#125—31no—
39C2×C4×D4—31no—
40#399—31no—
41#399D311yescharacteristic, complemented
42#398—32no—
43#397—31no—
44#397—31no—
45#394—32no—
46#385—32no—
47#385—32no—
48#383—31no—
49C24×C3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C2×#67—22no—
56C2×#67—22no—
57C2×#67—22no—
58C22×Dic3—22no—
59C22×Dic3—42no—
60C22×Dic3C2312yes—
61C22×Dic3C2311yescharacteristic
62C22×Dic3C2312yes—
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2311yescharacteristic
65C2×C4×D3—22no—
66C2×C4×D3—42no—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—42no—
69C3×C22⋊C4—22no—
70C3×C22⋊C4—42no—
71#217—24no—
72(C2×D3)⋊C4—24no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—42no—
75C4⋊Dic3—22no—
76C4⋊Dic3—24no—
77Dic3⋊C4—22no—
78Dic3⋊C4—24no—
79Dic3⋊C4—42no—
80C4×Dic3D414yes—
81C22×D4—31no—
82C23×C4—31no—
83C23×C4—31no—
84#125—64no—
85C4×D4—64no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4C2×D311yescharacteristic
92C2×C4⋊C4C2×D312yes—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—31no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4C2×D311yescharacteristic
98C2×C22⋊C4C2×D312yes—
99C2×C42—31no—
100#95—32no—
101#95—32no—
102#95—31no—
103#95—32no—
104#95—32no—
105#95C2×D311yescharacteristic
106C23×C3—21no—
107C23×C3—21no—
108C23×C3—22no—
109C23×C3—22no—
110C23×C3#4112yes—
111C23×C3C2411yescharacteristic
112C22×D3—21no—
113C22×D3—21no—
114C22×D3—22no—
115C22×D3—22no—
116C22×D3#4112yes—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—21no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4C2×D412yes—
127#67—28no—
128C2×Dic3—21no—
129C2×Dic3—21no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—21no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—44no—
140C2×Dic3C2×D412yes—
141C2×Dic3C2×D412yes—
142C4×D3—44no—
143C24—31no—
144C24C22×D311yescharacteristic
145C2×D4—62no—
146C2×D4—62no—
147C2×D4—62no—
148C22×C4—31no—
149C22×C4—31no—
150C22×C4—32no—
151C22×C4—32no—
152C22×C4—31no—
153C22×C4—62no—
154C22×C4—62no—
155C22×C4—122no—
156C22×C4—122no—
157C22×C4C22×D312yes—
158C22×C4C22×D311yescharacteristic
159C22×C4C22×D312yes—
160C22×C4C22×D311yescharacteristic
161C4⋊C4—22no—
162C4⋊C4—42no—
163C4⋊C4—62no—
164C4⋊C4—64no—
165C4⋊C4—64no—
166C4⋊C4—62no—
167C4⋊C4—122no—
168C22⋊C4—22no—
169C22⋊C4—42no—
170C22⋊C4—64no—
171C22⋊C4—64no—
172C22⋊C4—62no—
173C22⋊C4—122no—
174C42—34no—
175C22×C3—22no—
176C22×C3—24no—
177C22×C3—41no—
178C22×C3—41no—
179C22×C3—42no—
180C22×C3C22×D411yescharacteristic
181C22×C3C2×#4111yescharacteristic
182C22×C3C2×#4111yescharacteristic
183C22×C3H(5,𝔽2)12yes—
184C22×C32−1+412yes—
185C2×D3—22no—
186C2×D3—24no—
187C2×D3—41no—
188C2×D3—41no—
189C2×D3—42no—
190C3×C4—22no—
191C3×C4—41no—
192C3×C4—42no—
193C3×C4—42no—
194Dic3—22no—
195Dic3—22no—
196Dic3—42no—
197Dic3—41no—
198Dic3—42no—
199C23—21no—
200C23—22no—
201C23—21no—
202C23—22no—
203C23—32no—
204C23—61no—
205C23—62no—
206C23—61no—
207C23—62no—
208C23#23912yes—
209C23C23×D311yescharacteristic
210D4—68no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—32no—
221C2×C4—32no—
222C2×C4—62no—
223C2×C4—61no—
224C2×C4—61no—
225C2×C4—61no—
226C2×C4—62no—
227C2×C4—62no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4—124no—
234C2×C4—124no—
235C2×C4D3×D412yes—
236C2×C3—42no—
237C2×C3#52511yescharacteristic
238C2×C3#53511yescharacteristic
239C2×C3#53611yescharacteristic
240C2×C3#53912yes—
241C2×C3#53912yes—
242D3—42no—
243C22—24no—
244C22—22no—
245C22—41no—
246C22—42no—
247C22—41no—
248C22—64no—
249C22—62no—
250C22—122no—
251C22—121no—
252C22—121no—
253C22C2×D3×D411yescharacteristic
254C22#100412yes—
255C22C2×#23911yescharacteristic
256C22D3×#4111yescharacteristic
257C22#101012yes—
258C4—22no—
259C4—42no—
260C4—41no—
261C4—42no—
262C4—62no—
263C4—62no—
264C4—122no—
265C4—121no—
266C4—122no—
267C3#263311yescharacteristic, complemented, Sylow
268C2—42no—
269C2—122no—
270C2#562711yescharacteristic
271C2#564512yes—
272C2#564811yescharacteristic
273C2#566012yes—
274C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC22×#67C22
hypercenter#399D3
hypocenterC3#2633
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#890C2semidirect
3. C3×#399C2semidirect
4. #4993C2semidirect
5. #4989C2semidirect
6. #4982C2semidirect
8. #4953C2semidirect
9. #4946C2semidirect
10. #4674C2semidirect
11. #4651C2semidirect
41. #399D3semidirect
267. C3#2633semidirect

Maximal subgroups

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

subgroupindex
C2×#8902
C2×#67⋊C42
C3×#3992
#49932
#49892
#49822
#49612
#49532
#49462
#46742
#46512
#26333