Finite Group Databasebeta

← #76810

Subgroups — #76810

Order 384.

488 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 143 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×#944C211yesmaximal, characteristic, complemented
2C2×#877C211yesmaximal, characteristic
3C3×#393C211yesmaximal, characteristic
4#5230C211yesmaximal, characteristic, complemented
5#4949C212yesmaximal
6#4947C211yesmaximal, characteristic
7#4942C212yesmaximal
8#4658C212yesmaximal, complemented
9#4656C211yesmaximal, characteristic, complemented
10#4650C211yesmaximal, characteristic, complemented
11#4647C212yesmaximal, complemented
12#2628—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C23×Dic3C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2212yes—
18C2×#217C2211yescharacteristic
19C2×#217C2212yes—
20#944—24no—
21C2×C4⋊Dic3C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2211yescharacteristic
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×C4×Dic3C2211yescharacteristic
29C2×C4×Dic3C2211yescharacteristic
30#877—24no—
31C3×#95C2212yes—
32C3×#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4⋊Q8—31no—
39C2×#117—31no—
40#400—32no—
41#399—32no—
42#395—31no—
43#395—31no—
44#393—31no—
45#393D311yescharacteristic
46#389—32no—
47#382—31no—
48#380—32no—
49C24×C3C2311yescharacteristic
50C2×C3×Q8—22no—
51C2×C3×Q8—42no—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C22×Dic3—22no—
57C22×Dic3—22no—
58C22×Dic3Q814yescomplemented
59C22×Dic3C2312yes—
60C22×Dic3C2312yes—
61C22×Dic3C2311yescharacteristic
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2311yescharacteristic
65C2×Dic6—22no—
66C2×Dic6—42no—
67C3×C22⋊C4—22no—
68C3×C22⋊C4—42no—
69#217—22no—
70#217—42no—
71C4⋊Dic3—42no—
72Dic3⋊C4—22no—
73Dic3⋊C4—22no—
74Dic3⋊C4—24no—
75Dic3⋊C4—24no—
76Dic3⋊C4—24no—
77Dic3⋊C4—42no—
78C4×Dic3—22no—
79C4×Dic3D414yes—
80C22×Q8—31no—
81C22×Q8C2×D311yescharacteristic
82C23×C4—31no—
83C4⋊Q8—64no—
84#117—64no—
85C2×C4⋊C4—31no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—32no—
92C2×C22⋊C4—31no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4C2×D311yescharacteristic
95C2×C22⋊C4C2×D312yes—
96C2×C42—31no—
97C2×C42—31no—
98#95—32no—
99#95—32no—
100#95—31no—
101#95—32no—
102#95—32no—
103#95C2×D311yescharacteristic
104#95C2×D312yes—
105C23×C3—21no—
106C23×C3—21no—
107C23×C3—22no—
108C23×C3—22no—
109C23×C3C2×Q812yes—
110C23×C3C2411yescharacteristic
111C3×Q8—44no—
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×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4C2×D412yes—
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—22no—
125C2×Dic3—28no—
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×Dic3C2×Q812yes—
137C2×Dic3C2×Q812yes—
138C2×Dic3#4112yes—
139Dic6—44no—
140C24C22×D311yescharacteristic
141C2×Q8—22no—
142C2×Q8—42no—
143C2×Q8—62no—
144C2×Q8—122no—
145C22×C4—31no—
146C22×C4—34no—
147C22×C4—31no—
148C22×C4—31no—
149C22×C4—32no—
150C22×C4—32no—
151C22×C4—31no—
152C22×C4—62no—
153C22×C4—62no—
154C22×C4C22×D311yescharacteristic
155C22×C4C22×D312yes—
156C22×C4C22×D312yes—
157C22×C4C22×D311yescharacteristic
158C4⋊C4—62no—
159C4⋊C4—64no—
160C4⋊C4—64no—
161C4⋊C4—64no—
162C4⋊C4—62no—
163C4⋊C4—122no—
164C4⋊C4—122no—
165C22⋊C4—22no—
166C22⋊C4—42no—
167C22⋊C4—62no—
168C22⋊C4—122no—
169C42—34no—
170C42—62no—
171C22×C3—22no—
172C22×C3—24no—
173C22×C3—41no—
174C22×C3—41no—
175C22×C3—42no—
176C22×C3C22×D411yescharacteristic
177C22×C3C22×Q811yescharacteristic
178C22×C3C2×#4111yescharacteristic
179C22×C3H(5,𝔽2)12yes—
180C22×C32−1+412yes—
181C3×C4—22no—
182C3×C4—42no—
183C3×C4—41no—
184C3×C4—42no—
185Dic3—22no—
186Dic3—22no—
187Dic3—42no—
188Dic3—41no—
189Dic3—42no—
190Dic3C22⋊Q814yes—
191C23—21no—
192C23—22no—
193C23—21no—
194C23—22no—
195C23D3×Q812yes—
196C23C23×D311yescharacteristic
197Q8—44no—
198Q8—124no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—32no—
209C2×C4—32no—
210C2×C4—32no—
211C2×C4—32no—
212C2×C4—32no—
213C2×C4—68no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—62no—
217C2×C4—61no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4D3×D412yes—
226C2×C3—42no—
227C2×C3C2×C22⋊Q811yescharacteristic
228C2×C3#53511yescharacteristic
229C2×C3#53611yescharacteristic
230C2×C3#54212yes—
231C2×C3#54412yes—
232C22—24no—
233C22—22no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22C2×D3×D411yescharacteristic
238C22#100412yes—
239C22C2×D3×Q811yescharacteristic
240C22D3×#4111yescharacteristic
241C22#101012yes—
242C4—22no—
243C4—34no—
244C4—42no—
245C4—41no—
246C4—42no—
247C4—62no—
248C4—62no—
249C4—122no—
250C4—122no—
251C4—121no—
252C3#262811yescharacteristic, complemented, Sylow
253C2—42no—
254C2#559211yescharacteristic
255C2#562312yes—
256C2D3×C22⋊Q811yescharacteristic
257C2#566912yes—
258C2#567511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
prime radicalC24×C3C23
hypercenter#393D3
hypocenterC3#2628
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#944C2semidirect
4. #5230C2semidirect
8. #4658C2semidirect
9. #4656C2semidirect
10. #4650C2semidirect
11. #4647C2semidirect
58. C22×Dic3Q8semidirect
252. C3#2628semidirect

Maximal subgroups

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

subgroupindex
C2×#9442
C2×#8772
C3×#3932
#52302
#49492
#49472
#49422
#46582
#46562
#46502
#46472
#26283