Finite Group Databasebeta

← #76805

Subgroups — #76805

Order 384.

492 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 139 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×Q8×Dic3C211yesmaximal, characteristic, complemented
2C2×#877C211yesmaximal, characteristic
3C3×#393C211yesmaximal, characteristic
4#5230C211yesmaximal, characteristic, complemented
5#4953C211yesmaximal, characteristic
6#4950C212yesmaximal
7#4945C212yesmaximal
8#4661C211yesmaximal, characteristic, complemented
9#4655C212yesmaximal, complemented
10#4654C212yesmaximal, complemented
11#4651C211yesmaximal, characteristic, complemented
12#2599—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C23×Dic3C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×#217C2211yescharacteristic
18C2×#217C2212yes—
19Q8×Dic3—24no—
20C2×C4⋊Dic3C2212yes—
21C2×C4⋊Dic3C2212yes—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×Dic3⋊C4C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2211yescharacteristic
26C2×C4×Dic3C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29#877—24no—
30C3×#95C2211yescharacteristic
31C3×#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4×Q8—31no—
39C2×#117—31no—
40#398—31no—
41#397—31no—
42#395—31no—
43#393D311yescharacteristic
44#388—32no—
45#387—32no—
46#387—32no—
47#385—32no—
48#383—31no—
49C24×C3C2311yescharacteristic
50C2×C3×Q8—22no—
51C2×C3×Q8—24no—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2312yes—
56C22×Dic3—22no—
57C22×Dic3—22no—
58C22×Dic3—22no—
59C22×Dic3C2312yes—
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2312yes—
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2311yescharacteristic
65C3×C22⋊C4—22no—
66C3×C22⋊C4—24no—
67#217—24no—
68#217—22no—
69C4⋊Dic3—22no—
70C4⋊Dic3—24no—
71C4⋊Dic3—24no—
72Dic3⋊C4—22no—
73Dic3⋊C4—24no—
74Dic3⋊C4—22no—
75C4×Dic3—22no—
76C4×Dic3—24no—
77C4×Dic3D414yes—
78C22×Q8C2×D311yescharacteristic
79C23×C4—31no—
80C4×Q8—64no—
81#117—64no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—31no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4—32no—
88C2×C22⋊C4—31no—
89C2×C22⋊C4—32no—
90C2×C22⋊C4C2×D311yescharacteristic
91C2×C22⋊C4C2×D312yes—
92C2×C42—31no—
93C2×C42—31no—
94C2×C42—32no—
95#95—32no—
96#95—32no—
97#95—31no—
98#95—32no—
99#95—32no—
100#95—32no—
101#95C2×D311yescharacteristic
102#95C2×D312yes—
103C23×C3—21no—
104C23×C3—22no—
105C23×C3—21no—
106C23×C3—22no—
107C23×C3#4112yes—
108C23×C3C2411yescharacteristic
109C3×Q8—28no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—21no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4C2×D412yes—
119C2×C3×C4#4114yes—
120C2×Dic3—28no—
121C2×Dic3—21no—
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—22no—
126C2×Dic3—21no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3C2×D412yes—
135C2×Dic3C2×D412yes—
136C2×Dic3#4112yes—
137C24C22×D311yescharacteristic
138C2×Q8—22no—
139C2×Q8—24no—
140C22×C4—31no—
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—32no—
144C22×C4—31no—
145C22×C4—32no—
146C22×C4—62no—
147C22×C4—62no—
148C22×C4—62no—
149C22×C4C22×D312yes—
150C22×C4C22×D311yescharacteristic
151C22×C4C22×D311yescharacteristic
152C22×C4C22×D312yes—
153C4⋊C4—62no—
154C4⋊C4—64no—
155C4⋊C4—64no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—64no—
159C22⋊C4—22no—
160C22⋊C4—24no—
161C22⋊C4—62no—
162C22⋊C4—64no—
163C42—34no—
164C42—62no—
165C42—64no—
166C22×C3—22no—
167C22×C3—24no—
168C22×C3—41no—
169C22×C3—41no—
170C22×C3—42no—
171C22×C3C22×D411yescharacteristic
172C22×C3C2×#4112yes—
173C22×C3C2×#4111yescharacteristic
174C22×C3C2×#4111yescharacteristic
175C22×C32−1+412yes—
176C3×C4—22no—
177C3×C4—24no—
178C3×C4—41no—
179C3×C4—42no—
180Dic3—22no—
181Dic3—22no—
182Dic3—22no—
183Dic3—41no—
184Dic3—42no—
185Dic3—42no—
186C23—21no—
187C23—22no—
188C23—21no—
189C23—22no—
190C23#23912yes—
191C23C23×D311yescharacteristic
192Q8—28no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—32no—
202C2×C4—32no—
203C2×C4—32no—
204C2×C4—68no—
205C2×C4—61no—
206C2×C4—61no—
207C2×C4—61no—
208C2×C4—61no—
209C2×C4—62no—
210C2×C4—61no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4D3×D412yes—
219C2×C4#23714yes—
220C2×C3—42no—
221C2×C3#52511yescharacteristic
222C2×C3#52511yescharacteristic
223C2×C3#53611yescharacteristic
224C2×C3#55212yes—
225C2×C3#55612yes—
226C22—24no—
227C22—22no—
228C22—42no—
229C22—41no—
230C22—41no—
231C22C2×D3×D411yescharacteristic
232C22C2×#23712yes—
233C22C2×#23911yescharacteristic
234C22D3×#4111yescharacteristic
235C22#101012yes—
236C4—22no—
237C4—24no—
238C4—42no—
239C4—41no—
240C4—62no—
241C4—62no—
242C4—62no—
243C4—122no—
244C4—121no—
245C4—122no—
246C3#259911yescharacteristic, complemented, Sylow
247C2—42no—
248C2#558611yescharacteristic
249C2#562711yescharacteristic
250C2#564212yes—
251C2#567112yes—
252C2#567511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
prime radicalC24×C3C23
hypercenter#393D3
hypocenterC3#2599
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic3C2semidirect
4. #5230C2semidirect
8. #4661C2semidirect
9. #4655C2semidirect
10. #4654C2semidirect
11. #4651C2semidirect
246. C3#2599semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic32
C2×#8772
C3×#3932
#52302
#49532
#49502
#49452
#46612
#46552
#46542
#46512
#25993