Finite Group Databasebeta

← #78146

Subgroups — #78146

Order 384.

584 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 151 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×#893C211yesmaximal, characteristic
2C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
3C3×#400C211yesmaximal, characteristic, complemented
4#4993C211yesmaximal, characteristic, complemented
5#4986C211yesmaximal, characteristic, complemented
6#4677C211yesmaximal, characteristic, complemented
7#4668C212yesmaximal, complemented
8#4667C212yesmaximal, complemented
9#4667C211yesmaximal, characteristic, complemented
10#4665C212yesmaximal
11#4647C212yesmaximal, complemented
12#2487—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C22×C4×D3C2212yes—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C4C2211yescharacteristic
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×C4⋊Dic3C2211yescharacteristic
25C2×C4⋊Dic3C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×Dic3⋊C4C2212yes—
29C2×C4×Dic3C2211yescharacteristic
30#893—24no—
31D3×C4⋊C4—24no—
32C3×#95C2212yes—
33C3×#95C2212yes—
34C3×#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C2×C4×D4—31no—
39C22×C4⋊C4—31no—
40#400—31no—
41#400D311yescharacteristic, complemented
42#399—31no—
43#399—31no—
44#389—32no—
45#386—32no—
46#385—32no—
47#380—32no—
48#380—31no—
49C23×D3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2312yes—
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×Dic3C2312yes—
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C2×D12—22no—
61C2×D12—22no—
62C2×D12Q814yes—
63C2×C4×D3—24no—
64C2×C4×D3—24no—
65C2×C4×D3—22no—
66C2×C4×D3—24no—
67C2×C4×D3—42no—
68C2×C4×D3—42no—
69C2×C4×D3D414yes—
70C3×C4⋊C4—22no—
71C3×C4⋊C4—22no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4—24no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—42no—
76(C2×D3)⋊C4—42no—
77C4⋊Dic3—22no—
78C4⋊Dic3—42no—
79Dic3⋊C4—24no—
80Dic3⋊C4—24no—
81Dic3⋊C4—42no—
82C4×Dic3—22no—
83C22×D4—31no—
84C23×C4—31no—
85C23×C4—31no—
86C23×C4—32no—
87C4×D4—64no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—64no—
94C2×C4⋊C4C2×D311yescharacteristic
95C2×C4⋊C4C2×D311yescharacteristic
96C2×C22⋊C4—31no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C42—31no—
102#95—32no—
103#95—31no—
104#95—32no—
105#95C2×D312yes—
106#95C2×D311yescharacteristic
107#95C2×D312yes—
108C23×C3C2411yescharacteristic
109C22×D3—22no—
110C22×D3—22no—
111C22×D3—21no—
112C22×D3—22no—
113C22×D3—21no—
114C22×D3—22no—
115C22×D3C2×D412yes—
116C22×D3C2×Q812yes—
117C22×D3C2×Q812yes—
118C22×D3#4112yes—
119C2×C3×C4—21no—
120C2×C3×C4—22no—
121C2×C3×C4—21no—
122C2×C3×C4—22no—
123C2×C3×C4—21no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4C2×D412yes—
131C2×C3×C4C2×Q812yes—
132C2×Dic3—21no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3C2×D412yes—
141C2×Dic3#4114yes—
142D12—28no—
143C4×D3—28no—
144C4×D3—48no—
145C4×D3—44no—
146C24—31no—
147C24—31no—
148C2×D4—34no—
149C2×D4—62no—
150C2×D4—62no—
151C22×C4—31no—
152C22×C4—34no—
153C22×C4—32no—
154C22×C4—32no—
155C22×C4—31no—
156C22×C4—64no—
157C22×C4—62no—
158C22×C4—64no—
159C22×C4—64no—
160C22×C4—122no—
161C22×C4—122no—
162C22×C4C22×D311yescharacteristic
163C22×C4C22×D312yes—
164C22×C4C22×D312yes—
165C22×C4C22×D311yescharacteristic
166C22×C4C22×D311yescharacteristic
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—64no—
170C4⋊C4—64no—
171C4⋊C4—62no—
172C4⋊C4—122no—
173C4⋊C4—122no—
174C22⋊C4—62no—
175C22⋊C4—64no—
176C22⋊C4—62no—
177C22⋊C4—122no—
178C22⋊C4—122no—
179C42—62no—
180C22×C3C22×D411yescharacteristic
181C22×C3C22×Q811yescharacteristic
182C22×C3C2×#4111yescharacteristic
183C22×C3C2×#4112yes—
184C22×C3H(5,𝔽2)12yes—
185C2×D3—22no—
186C2×D3—22no—
187C2×D3—24no—
188C2×D3—24no—
189C2×D3—22no—
190C2×D3—24no—
191C2×D3—41no—
192C2×D3—41no—
193C2×D3—42no—
194C2×D3C22⋊Q814yes—
195C3×C4—22no—
196C3×C4—22no—
197C3×C4—41no—
198C3×C4—42no—
199C3×C4—42no—
200Dic3—42no—
201Dic3—41no—
202Dic3—42no—
203Dic3#12314yes—
204C23—32no—
205C23—32no—
206C23—32no—
207C23—32no—
208C23—62no—
209C23—62no—
210C23—61no—
211C23—62no—
212C23—61no—
213C23—62no—
214C23C23×D311yescharacteristic
215D4—68no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—32no—
228C2×C4—34no—
229C2×C4—68no—
230C2×C4—61no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—124no—
239C2×C4—128no—
240C2×C4D3×D412yes—
241C2×C4D3×Q812yes—
242C2×C3C2×C22⋊Q811yescharacteristic
243C2×C3C2×#12311yescharacteristic
244C2×C3#53511yescharacteristic
245C2×C3#55312yes—
246C2×C3#55412yes—
247D3—24no—
248D3—42no—
249C22—34no—
250C22—62no—
251C22—62no—
252C22—64no—
253C22—64no—
254C22—64no—
255C22—62no—
256C22—121no—
257C22—121no—
258C22—122no—
259C22C2×D3×D411yescharacteristic
260C22C2×D3×Q811yescharacteristic
261C22D3×#4112yes—
262C22D3×#4111yescharacteristic
263C22#100912yes—
264C4—22no—
265C4—22no—
266C4—34no—
267C4—41no—
268C4—42no—
269C4—42no—
270C4—122no—
271C4—121no—
272C4—122no—
273C3#248711yescharacteristic, complemented, Sylow
274C2—64no—
275C2—122no—
276C2D3×C22⋊Q811yescharacteristic
277C2D3×#12311yescharacteristic
278C2#567411yescharacteristic
279C2#569012yes—
280C2#570912yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#400C2
prime radicalC22×D12C22
hypercenter#400D3
hypocenterC3#2487
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×D3×C4⋊C4C2semidirect
3. C3×#400C2semidirect
4. #4993C2semidirect
5. #4986C2semidirect
6. #4677C2semidirect
7. #4668C2semidirect
8. #4667C2semidirect
9. #4667C2semidirect
11. #4647C2semidirect
41. #400D3semidirect
273. C3#2487semidirect

Maximal subgroups

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

subgroupindex
C2×#8932
C2×D3×C4⋊C42
C3×#4002
#49932
#49862
#46772
#46682
#46672
#46652
#46472
#24873