Finite Group Databasebeta

← #82264

Subgroups — #82264

Order 384.

488 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 165 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×C3×#117C211yesmaximal, characteristic
2C2×#869C211yesmaximal, characteristic, complemented
3#5212C212yesmaximal
4#4986C212yesmaximal, complemented
5#4979C212yesmaximal, complemented
6#4949C212yesmaximal
7#4947C212yesmaximal
8#4936C211yesmaximal, characteristic, complemented
9#4928C212yesmaximal, complemented
10#2628—31nomaximal, Sylow
11C23×C3×C4C2211yescharacteristic
12C22×Dic6C2212yes—
13C3×#117—24no—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×C3×C42C2211yescharacteristic
19C2×C3×C42C2211yescharacteristic
20C2×#217C2214yes—
21C2×C4⋊Dic3C2212yes—
22C2×C4⋊Dic3C2212yes—
23C2×C4⋊Dic3C2212yes—
24C2×Dic3⋊C4C2214yes—
25#869—24no—
26C3⋊#95C2214yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2214yes—
30C2×C4⋊Q8—31no—
31C2×#117D311yescharacteristic
32#400—32no—
33#399—32no—
34#395—32no—
35#393—32no—
36#389—32no—
37#382—31no—
38#380—32no—
39C24×C3C2311yescharacteristic
40C22×C3×C4—22no—
41C22×C3×C4—22no—
42C22×C3×C4Q814yescomplemented
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2311yescharacteristic
49C22×Dic3C2314yes—
50C22×Dic3C2314yes—
51C2×Dic6—24no—
52C2×Dic6—44no—
53C3×C4⋊C4—22no—
54C3×C4⋊C4—22no—
55C3×C22⋊C4—22no—
56C3×C22⋊C4—22no—
57C3×C42—22no—
58C3×C42D414yes—
59#217—44no—
60C4⋊Dic3—24no—
61C4⋊Dic3—24no—
62C4⋊Dic3—24no—
63Dic3⋊C4—44no—
64C22×Q8—32no—
65C23×C4C2×D311yescharacteristic
66C4⋊Q8—64no—
67#117—24no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—34no—
72C2×C4⋊C4C2×D311yescharacteristic
73C2×C4⋊C4C2×D311yescharacteristic
74C2×C22⋊C4—34no—
75C2×C22⋊C4C2×D311yescharacteristic
76C2×C22⋊C4C2×D311yescharacteristic
77C2×C42C2×D311yescharacteristic
78C2×C42C2×D311yescharacteristic
79#95—34no—
80#95—32no—
81#95—32no—
82#95—34no—
83C23×C3—21no—
84C23×C3—21no—
85C23×C3—22no—
86C23×C3—22no—
87C23×C3C2×Q812yes—
88C23×C3C2411yescharacteristic
89C2×C3×C4—21no—
90C2×C3×C4—21no—
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—28no—
94C2×C3×C4—21no—
95C2×C3×C4—21no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4C2×D412yes—
99C2×C3×C4C2×D412yes—
100C2×C3×C4C2×D412yes—
101C2×C3×C4C2×Q812yes—
102C2×C3×C4C2×Q812yes—
103C2×C3×C4#4112yes—
104C2×Dic3—24no—
105C2×Dic3—24no—
106C2×Dic3—24no—
107C2×Dic3—24no—
108C2×Dic3—24no—
109C2×Dic3—24no—
110Dic6—48no—
111C24C22×D311yescharacteristic
112C2×Q8—64no—
113C2×Q8—124no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—34no—
117C22×C4—34no—
118C22×C4Dic614yescomplemented
119C22×C4C22×D311yescharacteristic
120C22×C4C22×D311yescharacteristic
121C22×C4C22×D311yescharacteristic
122C22×C4C22×D311yescharacteristic
123C22×C4C22×D311yescharacteristic
124C22×C4C22×D311yescharacteristic
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—64no—
128C4⋊C4—64no—
129C4⋊C4—64no—
130C4⋊C4—124no—
131C22⋊C4—22no—
132C22⋊C4—22no—
133C22⋊C4—124no—
134C42—22no—
135C42#6714yes—
136C22×C3—22no—
137C22×C3—24no—
138C22×C3—41no—
139C22×C3—41no—
140C22×C3—42no—
141C22×C3C22×D411yescharacteristic
142C22×C3C22×Q811yescharacteristic
143C22×C3C2×#4111yescharacteristic
144C22×C3H(5,𝔽2)12yes—
145C22×C32−1+412yes—
146C3×C4—22no—
147C3×C4—22no—
148C3×C4—22no—
149C3×C4—41no—
150C3×C4—41no—
151C3×C4C22⋊Q814yes—
152Dic3—44no—
153Dic3—44no—
154C23—21no—
155C23—22no—
156C23—22no—
157C23—21no—
158C23C2×Dic612yes—
159C23C23×D311yescharacteristic
160Q8—128no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—28no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—64no—
171C2×C4—64no—
172C2×C4—64no—
173C2×C4—64no—
174C2×C4—64no—
175C2×C4—64no—
176C2×C4C2×Dic612yes—
177C2×C4C2×Dic612yes—
178C2×C4#23512yes—
179C2×C4C2×#6712yes—
180C2×C4C2×#6712yes—
181C2×C4C2×#6712yes—
182C2×C3—42no—
183C2×C3C2×C22⋊Q811yescharacteristic
184C2×C3#53511yescharacteristic
185C2×C3#53611yescharacteristic
186C2×C3#54212yes—
187C2×C3#54412yes—
188C22—24no—
189C22—22no—
190C22—42no—
191C22—41no—
192C22—41no—
193C22C22×Dic611yescharacteristic
194C22C2×#23511yescharacteristic
195C22#100912yes—
196C22#101012yes—
197C22C22×#6711yescharacteristic
198C4—22no—
199C4—22no—
200C4—22no—
201C4—41no—
202C4—41no—
203C4—124no—
204C4—124no—
205C4#92414yes—
206C3#262811yescharacteristic, complemented, Sylow
207C2—42no—
208C2#551912yes—
209C2#553412yes—
210C2C2×#92411yescharacteristic
211C2#582511yescharacteristic
212C2#583011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#117C2
prime radicalC24×C3C23
hypercenterC2×#117D3
hypocenterC3#2628
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#869C2semidirect
4. #4986C2semidirect
5. #4979C2semidirect
8. #4936C2semidirect
9. #4928C2semidirect
42. C22×C3×C4Q8semidirect
118. C22×C4Dic6semidirect
206. C3#2628semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1172
C2×#8692
#52122
#49862
#49792
#49492
#49472
#49362
#49282
#26283