Finite Group Databasebeta

← #74057

Subgroups — #74057

Order 384.

484 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 177 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×#924C211yesmaximal, characteristic
2C3×#381C211yesmaximal, characteristic
3C4×#217C211yesmaximal, characteristic
4#4945C212yesmaximal
5#4944C212yesmaximal
6#4654C212yesmaximal, complemented
7#4652C212yesmaximal, complemented
8#4649C212yesmaximal, complemented
9#4646C212yesmaximal, complemented
10#2332—31nomaximal, Sylow
11C23×C3×C4C2211yescharacteristic
12C22×Dic6C2211yescharacteristic
13C2×C3×C22⋊C4C2211yescharacteristic
14C2×C3×C22⋊C4C2211yescharacteristic
15C2×#217C2212yes—
16C2×#217C2212yes—
17C2×C4⋊Dic3C2211yescharacteristic
18#924C418yescomplemented
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×C4×Dic3C2212yes—
23C2×C4×Dic3C2212yes—
24C2×C4×Dic3C2211yescharacteristic
25C2×C4×Dic3C2211yescharacteristic
26C3×#95C2212yes—
27C3×#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C2×C22⋊Q8—31no—
34#391—32no—
35#388—32no—
36#388—32no—
37#387—32no—
38#387—32no—
39#381D311yescharacteristic
40C4×C4⋊C4—32no—
41C4×C22⋊C4—31no—
42C24×C3C2311yescharacteristic
43C22×C3×C4—41no—
44C22×C3×C4—41no—
45C22×C3×C4C2×C414yes—
46C22×C3×C4C2312yes—
47C22×C3×C4C2312yes—
48C22×C3×C4C2311yescharacteristic
49C22×C3×C4C2311yescharacteristic
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C2×Dic6—41no—
55C2×Dic6—41no—
56C2×Dic6C2×C414yes—
57C3×C22⋊C4—41no—
58C3×C22⋊C4—41no—
59#217—24no—
60#217C2×C418yes—
61C4⋊Dic3C2×C414yes—
62Dic3⋊C4—24no—
63Dic3⋊C4—42no—
64Dic3⋊C4C2×C418yes—
65C4×Dic3—24no—
66C4×Dic3—41no—
67C4×Dic3—42no—
68C4×Dic3—41no—
69C22×Q8—31no—
70C23×C4C2×D311yescharacteristic
71C22⋊Q8—38no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—31no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4—32no—
76C2×C22⋊C4—32no—
77C2×C22⋊C4—32no—
78C2×C22⋊C4C2×D311yescharacteristic
79C2×C22⋊C4C2×D311yescharacteristic
80C2×C42—32no—
81C2×C42—31no—
82C2×C42—31no—
83C2×C42—32no—
84#95—32no—
85#95—32no—
86#95—32no—
87#95—32no—
88#95—32no—
89#95C2×D312yes—
90#95C2×D312yes—
91C23×C3—22no—
92C23×C3—21no—
93C23×C3—21no—
94C23×C3—22no—
95C23×C3C22×C412yes—
96C23×C3C2411yescharacteristic
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—44no—
108C2×C3×C4C22×C412yes—
109C2×C3×C4C22×C412yes—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3C22×C414yes—
120C2×Dic3C22×C414yes—
121C2×Dic3#4114yes—
122Dic6—44no—
123C24C22×D311yescharacteristic
124C2×Q8—34no—
125C2×Q8—121no—
126C2×Q8—121no—
127C22×C4—32no—
128C22×C4—32no—
129C22×C4—32no—
130C22×C4—32no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4C4×D314yes—
134C22×C4C22×D312yes—
135C22×C4C22×D312yes—
136C22×C4C22×D311yescharacteristic
137C22×C4C22×D311yescharacteristic
138C4⋊C4—38no—
139C4⋊C4—34no—
140C4⋊C4—64no—
141C4⋊C4—122no—
142C22⋊C4—38no—
143C22⋊C4—41no—
144C22⋊C4—41no—
145C22⋊C4—64no—
146C42—64no—
147C42—121no—
148C42—121no—
149C42—122no—
150C22×C3—22no—
151C22×C3—22no—
152C22×C3—22no—
153C22×C3—42no—
154C22×C3—42no—
155C22×C3C23×C411yescharacteristic
156C22×C3C2×#4112yes—
157C22×C3H(5,𝔽2)11yescharacteristic
158C22×C3H(5,𝔽2)11yescharacteristic
159C22×C32−1+412yes—
160C3×C4—22no—
161C3×C4—22no—
162C3×C4—42no—
163C3×C4—42no—
164Dic3—24no—
165Dic3—24no—
166Dic3—24no—
167Dic3—42no—
168C23—21no—
169C23—21no—
170C23—22no—
171C23—22no—
172C23C2×C4×D312yes—
173C23C23×D311yescharacteristic
174Q8—124no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—34no—
186C2×C4—34no—
187C2×C4—34no—
188C2×C4—44no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4C2×C4×D312yes—
199C2×C4C2×C4×D312yes—
200C2×C3—42no—
201C2×C3C4×#4111yescharacteristic
202C2×C3#51811yescharacteristic
203C2×C3#51911yescharacteristic
204C2×C3#53912yes—
205C2×C3#54212yes—
206C22—22no—
207C22—22no—
208C22—22no—
209C22—42no—
210C22—42no—
211C22C22×C4×D311yescharacteristic
212C22#100411yescharacteristic
213C22#100411yescharacteristic
214C22D3×#4112yes—
215C22#101012yes—
216C4—22no—
217C4—22no—
218C4—42no—
219C4—42no—
220C4—64no—
221C4—64no—
222C4—64no—
223C4—122no—
224C3#233211yescharacteristic, complemented, Sylow
225C2—42no—
226C2#548611yescharacteristic
227C2#551611yescharacteristic
228C2#552211yescharacteristic
229C2#566012yes—
230C2#566212yes—
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#381C2
socle, cosocleC23×C3C24
prime radicalC24×C3C23
hypercenter#381D3
hypocenterC3#2332
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
6. #4654C2semidirect
7. #4652C2semidirect
8. #4649C2semidirect
9. #4646C2semidirect
18. #924C4semidirect
224. C3#2332semidirect

Maximal subgroups

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

subgroupindex
C2×#9242
C3×#3812
C4×#2172
#49452
#49442
#46542
#46522
#46492
#46462
#23323