Finite Group Databasebeta

← #82256

Subgroups — #82256

Order 384.

484 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 153 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×#924C211yesmaximal, characteristic
3C4×#217C211yesmaximal, characteristic
4#4977C211yesmaximal, characteristic, complemented
5#4975C211yesmaximal, characteristic, complemented
6#4973C211yesmaximal, characteristic, complemented
7#4973C211yesmaximal, characteristic, complemented
8#4952C214yesmaximal
9#4933C214yesmaximal, complemented
10#2499—31nomaximal, Sylow
11C23×C3×C4C2211yescharacteristic
12C22×Dic6C2211yescharacteristic
13C3×#117—24no—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C22⋊C4C2212yes—
17C2×C3×C42C2212yes—
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×C4⋊Dic3C2211yescharacteristic
21#924—24no—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2214yes—
24C2×C4×Dic3C2211yescharacteristic
25C2×C4×Dic3C2211yescharacteristic
26C3⋊#95C2212yes—
27C3⋊#95C2214yes—
28C3⋊#95C2214yes—
29C3⋊#95C2214yes—
30C2×C22⋊Q8—31no—
31C2×#117D311yescharacteristic
32#397—34no—
33#391—31no—
34#391—31no—
35#387—34no—
36#384—31no—
37#382—31no—
38C4×C22⋊C4—31no—
39C24×C3C2311yescharacteristic
40C22×C3×C4—22no—
41C22×C3×C4—22no—
42C22×C3×C4D414yes—
43C22×C3×C4C2312yes—
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C22×Dic3C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C2×Dic6—22no—
52C2×Dic6—22no—
53C2×Dic6—22no—
54C3×C4⋊C4—22no—
55C3×C4⋊C4—22no—
56C3×C22⋊C4—24no—
57C3×C42—24no—
58#217—24no—
59#217—24no—
60C4⋊Dic3—22no—
61Dic3⋊C4—24no—
62Dic3⋊C4—28no—
63C4×Dic3—22no—
64C4×Dic3—22no—
65C22×Q8—31no—
66C23×C4C2×D311yescharacteristic
67C22⋊Q8—64no—
68#117—24no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—34no—
72C2×C4⋊C4C2×D311yescharacteristic
73C2×C4⋊C4C2×D311yescharacteristic
74C2×C22⋊C4—32no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4C2×D312yes—
77C2×C42—31no—
78C2×C42—31no—
79C2×C42C2×D312yes—
80#95—32no—
81#95—34no—
82#95—34no—
83#95—34no—
84C23×C3—22no—
85C23×C3—21no—
86C23×C3—21no—
87C23×C3—22no—
88C23×C3C2×D412yes—
89C23×C3C2411yescharacteristic
90C2×C3×C4—21no—
91C2×C3×C4—21no—
92C2×C3×C4—28no—
93C2×C3×C4—22no—
94C2×C3×C4—22no—
95C2×C3×C4—24no—
96C2×C3×C4—24no—
97C2×C3×C4C2×D412yes—
98C2×C3×C4C2×D412yes—
99C2×C3×C4#4112yes—
100C2×C3×C4#4112yes—
101C2×Dic3—22no—
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—22no—
105C2×Dic3—22no—
106C2×Dic3—24no—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3#4114yes—
112Dic6—44no—
113C24C22×D311yescharacteristic
114C2×Q8—62no—
115C2×Q8—62no—
116C2×Q8—62no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—32no—
120C22×C4—32no—
121C22×C4—32no—
122C22×C4—32no—
123C22×C4#6714yes—
124C22×C4C22×D312yes—
125C22×C4C22×D312yes—
126C22×C4C22×D311yescharacteristic
127C22×C4C22×D311yescharacteristic
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—68no—
131C4⋊C4—64no—
132C4⋊C4—62no—
133C22⋊C4—24no—
134C22⋊C4—64no—
135C22⋊C4—64no—
136C42—24no—
137C42—62no—
138C42—62no—
139C22×C3—22no—
140C22×C3—22no—
141C22×C3—22no—
142C22×C3—42no—
143C22×C3—42no—
144C22×C3C22×D411yescharacteristic
145C22×C3C2×#4111yescharacteristic
146C22×C3C2×#4111yescharacteristic
147C22×C3C2×#4112yes—
148C22×C32−1+412yes—
149C3×C4—22no—
150C3×C4—42no—
151C3×C4—42no—
152C3×C4#12314yes—
153Dic3—24no—
154Dic3—42no—
155Dic3—42no—
156Dic3—42no—
157C23—21no—
158C23—21no—
159C23—22no—
160C23—22no—
161C23C2×#6712yes—
162C23C23×D311yescharacteristic
163Q8—124no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—28no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—34no—
172C2×C4—62no—
173C2×C4—62no—
174C2×C4—62no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4—62no—
178C2×C4—64no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4#23512yes—
183C2×C4#23512yes—
184C2×C4C2×#6712yes—
185C2×C4C2×#6712yes—
186C2×C3—42no—
187C2×C3C2×#12311yescharacteristic
188C2×C3#53211yescharacteristic
189C2×C3#53611yescharacteristic
190C2×C3#55212yes—
191C2×C3#55612yes—
192C22—22no—
193C22—22no—
194C22—22no—
195C22—42no—
196C22—42no—
197C22C2×#23511yescharacteristic
198C22C2×#23511yescharacteristic
199C22D3×#4112yes—
200C22#101012yes—
201C22C22×#6711yescharacteristic
202C4—22no—
203C4—42no—
204C4—42no—
205C4—64no—
206C4—122no—
207C4—122no—
208C4—122no—
209C4#92914yes—
210C3#249911yescharacteristic, complemented, Sylow
211C2—42no—
212C2#552912yes—
213C2#553312yes—
214C2C2×#92911yescharacteristic
215C2#582511yescharacteristic
216C2#583211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#117C2
prime radicalC24×C3C23
hypercenterC2×#117D3
hypocenterC3#2499
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
4. #4977C2semidirect
5. #4975C2semidirect
6. #4973C2semidirect
7. #4973C2semidirect
9. #4933C2semidirect
210. C3#2499semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1172
C2×#9242
C4×#2172
#49772
#49752
#49732
#49522
#49332
#24993