Finite Group Databasebeta

← #78356

Subgroups — #78356

Order 384.

584 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 147 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×#897C211yesmaximal, characteristic, complemented
2C2×#893C211yesmaximal, characteristic
3C3×#400C211yesmaximal, characteristic, complemented
4#4993C211yesmaximal, characteristic, complemented
5#4979C211yesmaximal, characteristic, complemented
6#4673C211yesmaximal, characteristic, complemented
7#4668C212yesmaximal, complemented
8#4667C212yesmaximal, complemented
9#4665C212yesmaximal
10D3×#95C211yesmaximal, characteristic, complemented
11#4655C212yesmaximal, complemented
12#2564—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×C4×D3C2212yes—
15C22×C4×D3C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C22×Dic6C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2212yes—
25C2×C4⋊Dic3C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×C4×Dic3C2211yescharacteristic
29#897—24no—
30#893—24no—
31C3×#95C2212yes—
32C3×#95C2211yescharacteristic
33C3×#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C2×C22⋊Q8—31no—
39C2×C4×D4—31no—
40#400D311yescharacteristic, complemented
41#399—31no—
42#395—31no—
43#393—31no—
44#387—32no—
45#386—32no—
46#385—32no—
47#380—32no—
48C2×#95—31no—
49C23×D3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2312yes—
60C2×D12—22no—
61C2×D12—22no—
62C2×D12Q814yescomplemented
63C2×C4×D3—24no—
64C2×C4×D3—22no—
65C2×C4×D3—24no—
66C2×C4×D3—42no—
67C2×C4×D3—42no—
68C2×C4×D3—42no—
69C2×C4×D3D414yes—
70C2×Dic6—22no—
71C2×Dic6—42no—
72C3×C4⋊C4—22no—
73C3×C4⋊C4—22no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—24no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—24no—
78(C2×D3)⋊C4—42no—
79C4⋊Dic3—24no—
80Dic3⋊C4—24no—
81Dic3⋊C4—42no—
82C4×Dic3—22no—
83C22×Q8—31no—
84C22×D4—31no—
85C23×C4—31no—
86C23×C4—32no—
87C23×C4—31no—
88C22⋊Q8—64no—
89C4×D4—64no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4C2×D311yescharacteristic
94C2×C4⋊C4C2×D311yescharacteristic
95C2×C22⋊C4—31no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C42—31no—
101#95—32no—
102#95—31no—
103#95—32no—
104#95—32no—
105#95C2×D311yescharacteristic
106#95C2×D312yes—
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—21no—
121C2×C3×C4—22no—
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—22no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—21no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3C2×D412yes—
141C2×Dic3#4114yes—
142D12—28no—
143C4×D3—44no—
144C4×D3—48no—
145C4×D3—44no—
146Dic6—44no—
147C24—31no—
148C24—31no—
149C2×Q8—62no—
150C2×Q8—122no—
151C2×D4—34no—
152C2×D4—62no—
153C2×D4—62no—
154C22×C4—32no—
155C22×C4—31no—
156C22×C4—34no—
157C22×C4—31no—
158C22×C4—32no—
159C22×C4—62no—
160C22×C4—64no—
161C22×C4—64no—
162C22×C4—122no—
163C22×C4—122no—
164C22×C4—122no—
165C22×C4C22×D312yes—
166C22×C4C22×D312yes—
167C22×C4C22×D311yescharacteristic
168C22×C4C22×D311yescharacteristic
169C22×C4C22×D311yescharacteristic
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—64no—
173C4⋊C4—64no—
174C4⋊C4—122no—
175C22⋊C4—64no—
176C22⋊C4—62no—
177C22⋊C4—64no—
178C22⋊C4—62no—
179C22⋊C4—122no—
180C42—62no—
181C22×C3C22×D411yescharacteristic
182C22×C3C22×Q811yescharacteristic
183C22×C3C2×#4111yescharacteristic
184C22×C3C2×#4112yes—
185C22×C3H(5,𝔽2)12yes—
186C2×D3—22no—
187C2×D3—22no—
188C2×D3—24no—
189C2×D3—24no—
190C2×D3—22no—
191C2×D3—24no—
192C2×D3—41no—
193C2×D3—41no—
194C2×D3—42no—
195C2×D3C22⋊Q814yes—
196C3×C4—22no—
197C3×C4—22no—
198C3×C4—42no—
199C3×C4—41no—
200C3×C4—42no—
201Dic3—22no—
202Dic3—24no—
203Dic3—41no—
204Dic3—42no—
205C23—32no—
206C23—32no—
207C23—32no—
208C23—32no—
209C23—62no—
210C23—62no—
211C23—61no—
212C23—62no—
213C23—61no—
214C23—62no—
215C23C23×D311yescharacteristic
216Q8—124no—
217D4—68no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—32no—
230C2×C4—34no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—61no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—124no—
240C2×C4—128no—
241C2×C4—124no—
242C2×C4D3×D412yes—
243C2×C4D3×Q812yes—
244C2×C3C2×C22⋊Q811yescharacteristic
245C2×C3#53211yescharacteristic
246C2×C3#53411yescharacteristic
247C2×C3#55112yes—
248C2×C3#55412yes—
249D3—24no—
250D3—42no—
251C22—34no—
252C22—62no—
253C22—62no—
254C22—64no—
255C22—64no—
256C22—64no—
257C22—62no—
258C22—121no—
259C22—121no—
260C22—122no—
261C22C2×D3×D411yescharacteristic
262C22C2×D3×Q811yescharacteristic
263C22D3×#4112yes—
264C22D3×#4111yescharacteristic
265C22#100912yes—
266C4—22no—
267C4—22no—
268C4—41no—
269C4—42no—
270C4—42no—
271C4—62no—
272C4—64no—
273C4—122no—
274C4—121no—
275C3#256411yescharacteristic, complemented, Sylow
276C2—64no—
277C2—122no—
278C2D3×C22⋊Q811yescharacteristic
279C2#565311yescharacteristic
280C2#566811yescharacteristic
281C2#569012yes—
282C2#570512yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#400C2
prime radicalC22×D12C22
hypercenter#400D3
hypocenterC3#2564
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#897C2semidirect
3. C3×#400C2semidirect
4. #4993C2semidirect
5. #4979C2semidirect
6. #4673C2semidirect
7. #4668C2semidirect
8. #4667C2semidirect
10. D3×#95C2semidirect
11. #4655C2semidirect
40. #400D3semidirect
62. C2×D12Q8semidirect
275. C3#2564semidirect

Maximal subgroups

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

subgroupindex
C2×#8972
C2×#8932
C3×#4002
#49932
#49792
#46732
#46682
#46672
#46652
D3×#952
#46552
#25643