Finite Group Databasebeta

← #82220

Subgroups — #82220

Order 384.

410 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 137 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
3#5207C211yesmaximal, characteristic
4#5004C212yesmaximal, complemented
5#4996C214yesmaximal, complemented
6#4995C212yesmaximal, complemented
7#4971C212yesmaximal, complemented
8#4970C212yesmaximal, complemented
9#3095—31nomaximal, Sylow
10C23×C3×C4C2211yescharacteristic
11C22×Dic6C2211yescharacteristic
12C3×#117—21no—
13C3×#117—21no—
14C3×#117C2212yes—
15C3×#117C2212yes—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4—21no—
19C2×C3×C42—21no—
20C2×#217—21no—
21C2×C4⋊Dic3C2211yescharacteristic
22#924—21no—
23#924—21no—
24#924C2212yes—
25#924C2212yes—
26C2×Dic3⋊C4—21no—
27C2×#208C2212yes—
28#830C2212yes—
29#830C2212yes—
30#810C2214yescomplemented
31#810C2214yescomplemented
32#808C2214yescomplemented
33#807C2214yescomplemented
34C2×C22⋊Q8—31no—
35C2×#117D311yescharacteristic
36#485—34no—
37#484—32no—
38#483—32no—
39#428—32no—
40#419—32no—
41#407—31no—
42C24×C3D411yescharacteristic
43C22×C3×C4—24no—
44C22×C3×C4—21no—
45C22×C3×C4—21no—
46C22×C3×C4D411yescharacteristic
47C22×C3×C4D411yescharacteristic
48C22×C3×C4D411yescharacteristic
49C22×C3×C4C2311yescharacteristic
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C22×Dic3—21no—
53C22×Dic3—21no—
54C2×Dic6—24no—
55C2×Dic6—21no—
56C2×Dic6C2312yes—
57C3×C4⋊C4—21no—
58C3×C4⋊C4—21no—
59C3×C4⋊C4C2312yes—
60C3×C4⋊C4C2312yes—
61C3×C22⋊C4—22no—
62C3×C22⋊C4—22no—
63C3×C42—22no—
64C3×C42—22no—
65#217—22no—
66#217—22no—
67C4⋊Dic3—21no—
68C4⋊Dic3C2312yes—
69Dic3⋊C4—22no—
70Dic3⋊C4—22no—
71#208—24no—
72C2×C3⋊C8C2314yes—
73C22×Q8—31no—
74C23×C4C2×D311yescharacteristic
75C2×M16—32no—
76C22⋊Q8—32no—
77C22⋊Q8—32no—
78C22⋊Q8—61no—
79C22⋊Q8—61no—
80#117—21no—
81#117—21no—
82#117C2×D312yes—
83#117C2×D312yes—
84C2×C4⋊C4—31no—
85C2×C4⋊C4—61no—
86C2×C4⋊C4C2×D311yescharacteristic
87C2×C4⋊C4C2×D311yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—61no—
90C2×C42—21no—
91#107—34no—
92#106—34no—
93#103—34no—
94#103—34no—
95C22⋊C8—32no—
96C22⋊C8—32no—
97C23×C3—22no—
98C23×C3—22no—
99C23×C3—41no—
100C23×C3C2×D411yescharacteristic
101C23×C3C2×D411yescharacteristic
102C23×C3C2×D411yescharacteristic
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—42no—
111C2×C3×C4—42no—
112C2×C3×C4C2×D411yescharacteristic
113C2×C3×C4C2×D411yescharacteristic
114C2×C3×C4C2×D411yescharacteristic
115C2×C3×C4#4114yes—
116C2×C3×C4C2411yescharacteristic
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—42no—
120C2×Dic3—42no—
121Dic6—24no—
122Dic6—42no—
123C3⋊C8—24no—
124C24#6711yescharacteristic
125C2×Q8—32no—
126C2×Q8—61no—
127C2×Q8—64no—
128C22×C4—24no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—61no—
132C22×C4—61no—
133C22×C4#6711yescharacteristic
134C22×C4#6711yescharacteristic
135C22×C4#6711yescharacteristic
136C22×C4C22×D311yescharacteristic
137C22×C4C22×D311yescharacteristic
138C22×C4C22×D311yescharacteristic
139M16—64no—
140C2×C8—34no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—32no—
144C4⋊C4—61no—
145C4⋊C4—62no—
146C4⋊C4—62no—
147C4⋊C4C22×D312yes—
148C4⋊C4C22×D312yes—
149C22⋊C4—22no—
150C22⋊C4—22no—
151C22⋊C4—62no—
152C22⋊C4—62no—
153C42—22no—
154C42—22no—
155C22×C3—21no—
156C22×C3—21no—
157C22×C3—21no—
158C22×C3—21no—
159C22×C3—21no—
160C22×C3—21no—
161C22×C3—21no—
162C22×C3—44no—
163C22×C3#12312yes—
164C22×C3#13712yes—
165C22×C3C22×D411yescharacteristic
166C3×C4—21no—
167C3×C4—21no—
168C3×C4—42no—
169C3×C4—42no—
170C3×C4#12312yes—
171C3×C4C2×#4112yes—
172Dic3—42no—
173Dic3—42no—
174C23—22no—
175C23—22no—
176C23—41no—
177C23C2×#6711yescharacteristic
178C23C2×#6711yescharacteristic
179C23C2×#6711yescharacteristic
180Q8—64no—
181Q8—122no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—122no—
194C2×C4—122no—
195C2×C4#23514yes—
196C2×C4C2×#6711yescharacteristic
197C2×C4C2×#6711yescharacteristic
198C2×C4C2×#6711yescharacteristic
199C2×C4C23×D311yescharacteristic
200C8—64no—
201C2×C3—22no—
202C2×C3—22no—
203C2×C3—41no—
204C2×C3C2×#12311yescharacteristic
205C2×C3C2×#13711yescharacteristic
206C2×C3#57511yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—44no—
215C22#92912yes—
216C22#95112yes—
217C22C22×#6711yescharacteristic
218C4—21no—
219C4—21no—
220C4—42no—
221C4—42no—
222C4—122no—
223C4—122no—
224C4#92912yes—
225C4C2×#23512yes—
226C3#309511yescharacteristic, complemented, Sylow
227C2—22no—
228C2—22no—
229C2—41no—
230C2C2×#92911yescharacteristic
231C2#582211yescharacteristic
232C2C2×#95111yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#117C2
socleC22×C3C22×D4
prime radicalC24×C3D4
hypercenterC2×#117D3
hypocenterC3#3095
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
4. #5004C2semidirect
5. #4996C2semidirect
6. #4995C2semidirect
7. #4971C2semidirect
8. #4970C2semidirect
30. #810C22semidirect
31. #810C22semidirect
32. #808C22semidirect
33. #807C22semidirect
226. C3#3095semidirect

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
#52072
#50042
#49962
#49952
#49712
#49702
#30953