Finite Group Databasebeta

← #78130

Subgroups — #78130

Order 384.

528 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 151 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×#892C211yesmaximal, characteristic
2C2×#887C211yesmaximal, characteristic, complemented
3C3×#400C211yesmaximal, characteristic, complemented
4#4993C211yesmaximal, characteristic
5#4979C211yesmaximal, characteristic, complemented
6#4676C212yesmaximal
7#4673C211yesmaximal, characteristic
8#4666C212yesmaximal
9#4657C212yesmaximal, complemented
10#4649C212yesmaximal, complemented
11#4649C211yesmaximal, characteristic, complemented
12#2396—31nomaximal, Sylow
13C22×C4×D3C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×(C2×D3)⋊C4C2212yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2211yescharacteristic
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×C4⋊Dic3C2211yescharacteristic
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×C4×Dic3C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28C2×C4×Dic3C2212yes—
29C2×C4×Dic3C2211yescharacteristic
30#892—24no—
31#887—24no—
32C3×#95C2211yescharacteristic
33C3×#95C2212yes—
34C3×#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×C4×Q8—31no—
39C2×#117—31no—
40#400D311yescharacteristic, complemented
41#399—31no—
42#398—32no—
43#396—32no—
44#395—31no—
45#393—31no—
46C4×C4⋊C4—32no—
47C4×C4⋊C4—31no—
48C4×C22⋊C4—32no—
49C23×D3C2311yescharacteristic
50C22×C3×C4C2312yes—
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×Dic3C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C22×Dic3C2311yescharacteristic
60C2×C4×D3—22no—
61C2×C4×D3—22no—
62C2×C4×D3Q814yescomplemented
63C2×Dic6—22no—
64C2×Dic6—24no—
65C3×C4⋊C4—22no—
66C3×C4⋊C4—22no—
67(C2×D3)⋊C4—22no—
68(C2×D3)⋊C4—22no—
69(C2×D3)⋊C4—24no—
70(C2×D3)⋊C4—24no—
71C4⋊Dic3—22no—
72Dic3⋊C4—24no—
73Dic3⋊C4—24no—
74Dic3⋊C4—24no—
75Dic3⋊C4—24no—
76C4×Dic3—24no—
77C4×Dic3—22no—
78C4×Dic3—24no—
79C4×Dic3D414yes—
80C22×Q8—31no—
81C23×C4—31no—
82C4×Q8—64no—
83#117—64no—
84C2×C4⋊C4—31no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4C2×D311yescharacteristic
90C2×C4⋊C4C2×D311yescharacteristic
91C2×C22⋊C4—31no—
92C2×C22⋊C4—31no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C42—31no—
96C2×C42—31no—
97C2×C42—32no—
98C2×C42—32no—
99#95—32no—
100#95—32no—
101#95—31no—
102#95C2×D312yes—
103#95C2×D311yescharacteristic
104#95C2×D312yes—
105C23×C3C2411yescharacteristic
106C22×D3—21no—
107C22×D3—22no—
108C22×D3—22no—
109C22×D3—21no—
110C22×D3C2×Q812yes—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
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×C4C2×D412yes—
123C2×C3×C4C2×Q812yes—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—21no—
130C2×Dic3—22no—
131C2×Dic3C2×D412yes—
132C2×Dic3C2×D412yes—
133C2×Dic3C2×Q812yes—
134C2×Dic3#4114yes—
135C2×Dic3#4114yes—
136C2×Dic3#4112yes—
137C4×D3—28no—
138Dic6—28no—
139C24—31no—
140C2×Q8—64no—
141C2×Q8—62no—
142C22×C4—31no—
143C22×C4—32no—
144C22×C4—32no—
145C22×C4—31no—
146C22×C4—34no—
147C22×C4—31no—
148C22×C4—62no—
149C22×C4—62no—
150C22×C4C22×D312yes—
151C22×C4C22×D311yescharacteristic
152C22×C4C22×D312yes—
153C22×C4C22×D311yescharacteristic
154C22×C4C22×D311yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—62no—
158C4⋊C4—64no—
159C4⋊C4—64no—
160C4⋊C4—64no—
161C4⋊C4—64no—
162C22⋊C4—62no—
163C22⋊C4—64no—
164C22⋊C4—64no—
165C22⋊C4—62no—
166C42—34no—
167C42—62no—
168C42—64no—
169C42—64no—
170C22×C3C22×D411yescharacteristic
171C22×C3C22×Q811yescharacteristic
172C22×C3C2×#4111yescharacteristic
173C22×C3C2×#4112yes—
174C22×C3C2×#4112yes—
175C2×D3—22no—
176C2×D3—24no—
177C2×D3—41no—
178C2×D3—42no—
179C2×D3—41no—
180C3×C4—22no—
181C3×C4—22no—
182C3×C4—41no—
183C3×C4—42no—
184C3×C4—42no—
185Dic3—24no—
186Dic3—24no—
187Dic3—22no—
188Dic3—41no—
189Dic3C22⋊Q814yes—
190C23—32no—
191C23—61no—
192C23—62no—
193C23—62no—
194C23—61no—
195C23C23×D311yescharacteristic
196Q8—68no—
197C2×C4—21no—
198C2×C4—21no—
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—34no—
213C2×C4—34no—
214C2×C4—68no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—61no—
221C2×C4—62no—
222C2×C4D3×D412yes—
223C2×C4D3×Q812yes—
224C2×C3C2×C22⋊Q811yescharacteristic
225C2×C3#52511yescharacteristic
226C2×C3#52912yes—
227C2×C3#53211yescharacteristic
228C2×C3#53312yes—
229D3—42no—
230C22—64no—
231C22—62no—
232C22—121no—
233C22—121no—
234C22—122no—
235C22C2×D3×D411yescharacteristic
236C22C2×D3×Q811yescharacteristic
237C22D3×#4112yes—
238C22D3×#4112yes—
239C22D3×#4111yescharacteristic
240C4—22no—
241C4—22no—
242C4—34no—
243C4—41no—
244C4—42no—
245C4—42no—
246C4—64no—
247C4—62no—
248C4—64no—
249C4—121no—
250C3#239611yescharacteristic, complemented, Sylow
251C2—122no—
252C2D3×C22⋊Q811yescharacteristic
253C2#565611yescharacteristic
254C2#566811yescharacteristic
255C2#568812yes—
256C2#570612yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#400C2
prime radicalC23×D3C23
hypercenter#400D3
hypocenterC3#2396
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#887C2semidirect
3. C3×#400C2semidirect
5. #4979C2semidirect
9. #4657C2semidirect
10. #4649C2semidirect
11. #4649C2semidirect
40. #400D3semidirect
62. C2×C4×D3Q8semidirect
250. C3#2396semidirect

Maximal subgroups

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

subgroupindex
C2×#8922
C2×#8872
C3×#4002
#49932
#49792
#46762
#46732
#46662
#46572
#46492
#23963