Finite Group Databasebeta

← #77202

Subgroups — #77202

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×#892C211yesmaximal, characteristic
3C3×#395C211yesmaximal, characteristic, complemented
4#5231C211yesmaximal, characteristic
5#4993C212yesmaximal
6#4979C211yesmaximal, characteristic, complemented
7#4976C212yesmaximal, complemented
8#4673C211yesmaximal, characteristic
9#4667C212yesmaximal
10#4660C212yesmaximal, complemented
11#4650C211yesmaximal, characteristic, complemented
12#2628—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2212yes—
18#944—24no—
19C2×(C2×D3)⋊C4C2211yescharacteristic
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×C4×Dic3C2211yescharacteristic
29C2×C4×Dic3C2211yescharacteristic
30#892—24no—
31C3×#95C2211yescharacteristic
32C3×#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4⋊Q8—31no—
39C2×#117—31no—
40#400—32no—
41#399—32no—
42#395—31no—
43#395D311yescharacteristic, complemented
44#393—31no—
45#393—31no—
46#389—32no—
47#382—31no—
48#380—32no—
49C23×D3C2311yescharacteristic
50C2×C3×Q8—22no—
51C2×C3×Q8—42no—
52C22×C3×C4C2312yes—
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2312yes—
59C22×Dic3C2312yes—
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2311yescharacteristic
62C2×C4×D3—22no—
63C2×C4×D3—22no—
64C2×C4×D3Q814yescomplemented
65C2×Dic6—22no—
66C2×Dic6—42no—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—42no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4—42no—
72(C2×D3)⋊C4—42no—
73C4⋊Dic3—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—32no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4C2×D311yescharacteristic
91C2×C4⋊C4C2×D312yes—
92C2×C22⋊C4—31no—
93C2×C22⋊C4—31no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C42—31no—
97C2×C42—31no—
98#95—32no—
99#95—31no—
100#95—32no—
101#95—32no—
102#95—32no—
103#95C2×D312yes—
104#95C2×D311yescharacteristic
105C23×C3C2411yescharacteristic
106C22×D3—21no—
107C22×D3—21no—
108C22×D3—22no—
109C22×D3—22no—
110C22×D3C2×Q812yes—
111C3×Q8—44no—
112C2×C3×C4—21no—
113C2×C3×C4—21no—
114C2×C3×C4—22no—
115C2×C3×C4—21no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4C2×D412yes—
124C2×C3×C4C2×Q812yes—
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×Dic3C2×Q812yes—
137C2×Dic3#4112yes—
138C4×D3—28no—
139Dic6—44no—
140C24—31no—
141C2×Q8—22no—
142C2×Q8—42no—
143C2×Q8—62no—
144C2×Q8—122no—
145C22×C4—31no—
146C22×C4—31no—
147C22×C4—34no—
148C22×C4—31no—
149C22×C4—32no—
150C22×C4—32no—
151C22×C4—62no—
152C22×C4—62no—
153C22×C4C22×D312yes—
154C22×C4C22×D312yes—
155C22×C4C22×D311yescharacteristic
156C22×C4C22×D311yescharacteristic
157C22×C4C22×D311yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—42no—
160C4⋊C4—62no—
161C4⋊C4—64no—
162C4⋊C4—64no—
163C4⋊C4—64no—
164C4⋊C4—122no—
165C22⋊C4—62no—
166C22⋊C4—62no—
167C22⋊C4—122no—
168C22⋊C4—122no—
169C42—34no—
170C42—62no—
171C22×C3C22×D411yescharacteristic
172C22×C3C22×Q811yescharacteristic
173C22×C3C2×#4111yescharacteristic
174C22×C3H(5,𝔽2)12yes—
175C22×C32−1+412yes—
176C2×D3—22no—
177C2×D3—24no—
178C2×D3—41no—
179C2×D3—41no—
180C2×D3—42no—
181C3×C4—22no—
182C3×C4—22no—
183C3×C4—41no—
184C3×C4—42no—
185C3×C4—42no—
186Dic3—22no—
187Dic3—42no—
188Dic3—41no—
189Dic3—42no—
190Dic3C22⋊Q814yes—
191C23—32no—
192C23—61no—
193C23—62no—
194C23—62no—
195C23—61no—
196C23C23×D311yescharacteristic
197Q8—44no—
198Q8—124no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—32no—
211C2×C4—32no—
212C2×C4—32no—
213C2×C4—32no—
214C2×C4—68no—
215C2×C4—62no—
216C2×C4—61no—
217C2×C4—62no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4D3×D412yes—
225C2×C4D3×Q812yes—
226C2×C3C2×C22⋊Q811yescharacteristic
227C2×C3#53511yescharacteristic
228C2×C3#53611yescharacteristic
229C2×C3#54212yes—
230C2×C3#54412yes—
231D3—42no—
232C22—64no—
233C22—62no—
234C22—122no—
235C22—121no—
236C22—121no—
237C22C2×D3×D411yescharacteristic
238C22#100412yes—
239C22C2×D3×Q811yescharacteristic
240C22#100712yes—
241C22D3×#4111yescharacteristic
242C4—22no—
243C4—22no—
244C4—34no—
245C4—42no—
246C4—42no—
247C4—41no—
248C4—62no—
249C4—122no—
250C4—121no—
251C4—122no—
252C3#262811yescharacteristic, complemented, Sylow
253C2—122no—
254C2#559211yescharacteristic
255C2D3×C22⋊Q811yescharacteristic
256C2#564012yes—
257C2#572411yescharacteristic
258C2#572912yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#395C2
prime radicalC23×D3C23
hypercenter#395D3
hypocenterC3#2628
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#944C2semidirect
3. C3×#395C2semidirect
6. #4979C2semidirect
7. #4976C2semidirect
10. #4660C2semidirect
11. #4650C2semidirect
43. #395D3semidirect
64. C2×C4×D3Q8semidirect
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×#8922
C3×#3952
#52312
#49932
#49792
#49762
#46732
#46672
#46602
#46502
#26283