Finite Group Databasebeta

← #77203

Subgroups — #77203

Order 384.

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

normal subgroups only

This group has 500 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×#947C211yesmaximal, characteristic
2C2×#887C211yesmaximal, characteristic, complemented
3C3×#395C211yesmaximal, characteristic, complemented
4#5230C211yesmaximal, characteristic, complemented
5#4993C212yesmaximal, complemented
6#4991C211yesmaximal, characteristic
7#4990C212yesmaximal, complemented
8#4673C212yesmaximal, complemented
9#4672C211yesmaximal, characteristic
10#4669C212yesmaximal, complemented
11#4651C211yesmaximal, characteristic, complemented
12#2632—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×D12C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2212yes—
18#947—24no—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2211yescharacteristic
27C2×Dic3⋊C4C2212yes—
28C2×C4×Dic3C2211yescharacteristic
29C2×C4×Dic3C2211yescharacteristic
30C2×C4×Dic3C2211yescharacteristic
31#887—24no—
32C3×#95C2211yescharacteristic
33C3×#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#124—31no—
39C2×C4×Q8—31no—
40#399—32no—
41#395—31no—
42#395D311yescharacteristic, complemented
43#394—31no—
44#394—31no—
45#393—32no—
46#388—32no—
47#388—32no—
48#383—31no—
49C23×D3C2312yes—
50C2×C3×Q8—22no—
51C2×C3×Q8—42no—
52C22×C3×C4C2312yes—
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2311yescharacteristic
62C2×D12—22no—
63C2×D12—42no—
64C2×Dic6—22no—
65C2×Dic6—22no—
66C2×Dic6—22no—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—42no—
69(C2×D3)⋊C4—24no—
70(C2×D3)⋊C4—24no—
71(C2×D3)⋊C4—24no—
72(C2×D3)⋊C4—24no—
73(C2×D3)⋊C4—42no—
74(C2×D3)⋊C4—42no—
75Dic3⋊C4—22no—
76Dic3⋊C4—22no—
77Dic3⋊C4—42no—
78C4×Dic3—22no—
79C4×Dic3—22no—
80C4×Dic3D414yes—
81C22×Q8—31no—
82C22×Q8C2×D311yescharacteristic
83C22×D4—31no—
84#124—64no—
85C4×Q8—64no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—31no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4C2×D311yescharacteristic
90C2×C4⋊C4C2×D312yes—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—32no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C42—31no—
98C2×C42—31no—
99C2×C42—31no—
100#95—32no—
101#95—32no—
102#95—31no—
103#95—32no—
104#95C2×D311yescharacteristic
105#95C2×D312yes—
106C23×C3C2411yescharacteristic
107C22×D3—22no—
108C22×D3—22no—
109C22×D3—22no—
110C22×D3—22no—
111C22×D3—22no—
112C22×D3—22no—
113C22×D3—22no—
114C3×Q8—44no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—22no—
118C2×C3×C4—21no—
119C2×C3×C4—22no—
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—
127C2×C3×C4#4112yes—
128C2×Dic3—21no—
129C2×Dic3—21no—
130C2×Dic3—21no—
131C2×Dic3—21no—
132C2×Dic3—22no—
133C2×Dic3—21no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3C2×D412yes—
139C2×Dic3C2×D412yes—
140C2×Dic3#4112yes—
141D12—44no—
142Dic6—28no—
143C24—32no—
144C2×Q8—22no—
145C2×Q8—42no—
146C2×Q8—62no—
147C2×Q8—62no—
148C2×Q8—62no—
149C2×D4—62no—
150C2×D4—122no—
151C22×C4—31no—
152C22×C4—31no—
153C22×C4—31no—
154C22×C4—32no—
155C22×C4—31no—
156C22×C4C22×D312yes—
157C22×C4C22×D312yes—
158C22×C4C22×D311yescharacteristic
159C22×C4C22×D311yescharacteristic
160C22×C4C22×D311yescharacteristic
161C4⋊C4—22no—
162C4⋊C4—42no—
163C4⋊C4—62no—
164C4⋊C4—62no—
165C4⋊C4—122no—
166C22⋊C4—64no—
167C22⋊C4—64no—
168C22⋊C4—64no—
169C22⋊C4—64no—
170C22⋊C4—122no—
171C22⋊C4—122no—
172C42—34no—
173C42—62no—
174C42—62no—
175C22×C3C22×D411yescharacteristic
176C22×C3C2×#4111yescharacteristic
177C22×C3C2×#4111yescharacteristic
178C22×C3H(5,𝔽2)12yes—
179C22×C32−1+412yes—
180C2×D3—42no—
181C2×D3—42no—
182C2×D3—42no—
183C2×D3—42no—
184C2×D3—42no—
185C2×D3—42no—
186C2×D3—42no—
187C3×C4—22no—
188C3×C4—22no—
189C3×C4—41no—
190C3×C4—42no—
191C3×C4—42no—
192Dic3—22no—
193Dic3—22no—
194Dic3—22no—
195Dic3—41no—
196Dic3—42no—
197C23—62no—
198C23—62no—
199C23—62no—
200C23—62no—
201C23—62no—
202C23—62no—
203C23—62no—
204C23C23×D311yescharacteristic
205Q8—44no—
206Q8—68no—
207D4—124no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—32no—
220C2×C4—32no—
221C2×C4—32no—
222C2×C4—61no—
223C2×C4—61no—
224C2×C4—61no—
225C2×C4—62no—
226C2×C4—61no—
227C2×C4—61no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4D3×D412yes—
233C2×C4#23912yes—
234C2×C3#52511yescharacteristic
235C2×C3#53511yescharacteristic
236C2×C3#53611yescharacteristic
237C2×C3#54212yes—
238C2×C3#54212yes—
239D3—82no—
240C22—122no—
241C22—122no—
242C22—122no—
243C22—122no—
244C22—122no—
245C22—122no—
246C22—122no—
247C22C2×D3×D411yescharacteristic
248C22#100412yes—
249C22C2×#23911yescharacteristic
250C22#100712yes—
251C22D3×#4111yescharacteristic
252C4—22no—
253C4—22no—
254C4—42no—
255C4—42no—
256C4—41no—
257C4—62no—
258C4—62no—
259C4—62no—
260C4—122no—
261C4—121no—
262C3#263211yescharacteristic, complemented, Sylow
263C2—242no—
264C2#559211yescharacteristic
265C2#562711yescharacteristic
266C2#564012yes—
267C2#572411yescharacteristic
268C2#573312yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#395C2
prime radicalC22×D12C22
hypercenter#395D3
hypocenterC3#2632
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#887C2semidirect
3. C3×#395C2semidirect
4. #5230C2semidirect
5. #4993C2semidirect
7. #4990C2semidirect
8. #4673C2semidirect
10. #4669C2semidirect
11. #4651C2semidirect
42. #395D3semidirect
262. C3#2632semidirect

Maximal subgroups

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

subgroupindex
C2×#9472
C2×#8872
C3×#3952
#52302
#49932
#49912
#49902
#46732
#46722
#46692
#46512
#26323