Finite Group Databasebeta

← #82373

Subgroups — #82373

Order 384.

624 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 181 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×C4×D4C211yesmaximal, characteristic
2C2×#935C212yesmaximal
3C22×C4⋊Dic3C211yesmaximal, characteristic, complemented
4#5212C212yesmaximal, complemented
5#5212C211yesmaximal, characteristic, complemented
6#4981C212yesmaximal, complemented
7#4953C212yesmaximal, complemented
8#4943C212yesmaximal, complemented
9#4928C212yesmaximal, complemented
10#2486—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C23×C3×C4C2211yescharacteristic
14C23×Dic3C2212yes—
15C3×C4×D4—24no—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C42C2211yescharacteristic
20C2×#217C2212yes—
21C2×#217C2214yes—
22#935—28no—
23C2×C4⋊Dic3—24no—
24C2×C4⋊Dic3C2212yes—
25C2×C4⋊Dic3C2212yes—
26C2×C4⋊Dic3C2212yes—
27C2×Dic3⋊C4C2214yes—
28C3⋊#95C2214yes—
29C3⋊#95C2214yes—
30C3⋊#95C2212yes—
31C2×#123—32no—
32C2×C4×D4D311yescharacteristic
33C22×C4⋊C4—31no—
34#398—32no—
35#397—32no—
36#389—32no—
37#381—32no—
38#380—32no—
39#380—31no—
40C24×C3C2311yescharacteristic
41C24×C3C2311yescharacteristic
42C2×C3×D4—22no—
43C2×C3×D4—22no—
44C2×C3×D4D414yes—
45C22×C3×C4—24no—
46C22×C3×C4—22no—
47C22×C3×C4—24no—
48C22×C3×C4D414yes—
49C22×C3×C4C2311yescharacteristic
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×Dic3—24no—
55C22×Dic3—24no—
56C22×Dic3—24no—
57C22×Dic3C2314yes—
58C22×Dic3C2314yes—
59C3×C4⋊C4—22no—
60C3×C22⋊C4—22no—
61C3×C22⋊C4—22no—
62C3×C42—22no—
63#217—24no—
64#217—28no—
65C4⋊Dic3—24no—
66C4⋊Dic3—24no—
67C4⋊Dic3—24no—
68Dic3⋊C4—28no—
69C22×D4C2×D311yescharacteristic
70C23×C4—32no—
71C23×C4C2×D311yescharacteristic
72C23×C4C2×D311yescharacteristic
73#123—68no—
74C4×D4—24no—
75C2×C4⋊C4—32no—
76C2×C4⋊C4—32no—
77C2×C4⋊C4—34no—
78C2×C4⋊C4—32no—
79C2×C4⋊C4—64no—
80C2×C4⋊C4C2×D311yescharacteristic
81C2×C22⋊C4—32no—
82C2×C22⋊C4—34no—
83C2×C22⋊C4C2×D311yescharacteristic
84C2×C22⋊C4C2×D311yescharacteristic
85C2×C42C2×D311yescharacteristic
86#95—34no—
87#95—32no—
88#95—34no—
89C23×C3—22no—
90C23×C3—22no—
91C23×C3—21no—
92C23×C3—22no—
93C23×C3—21no—
94C23×C3—22no—
95C23×C3C2×D412yes—
96C23×C3C2×D412yes—
97C23×C3C2×D412yes—
98C23×C3#4112yes—
99C23×C3C2411yescharacteristic
100C3×D4—28no—
101C2×C3×C4—21no—
102C2×C3×C4—22no—
103C2×C3×C4—21no—
104C2×C3×C4—28no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—44no—
110C2×C3×C4C2×D412yes—
111C2×C3×C4C2×D412yes—
112C2×C3×C4C2×D412yes—
113C2×C3×C4#4114yes—
114C2×Dic3—24no—
115C2×Dic3—24no—
116C2×Dic3—24no—
117C2×Dic3—24no—
118C2×Dic3—24no—
119C2×Dic3—24no—
120C2×Dic3—48no—
121C24C22×D311yescharacteristic
122C24C22×D311yescharacteristic
123C2×D4—22no—
124C2×D4—22no—
125C2×D4D1214yes—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—34no—
130C22×C4—34no—
131C22×C4—64no—
132C22×C4—64no—
133C22×C4—64no—
134C22×C4#6714yes—
135C22×C4C22×D311yescharacteristic
136C22×C4C22×D311yescharacteristic
137C22×C4C22×D311yescharacteristic
138C22×C4C22×D311yescharacteristic
139C22×C4C22×D311yescharacteristic
140C4⋊C4—22no—
141C4⋊C4—68no—
142C4⋊C4—64no—
143C4⋊C4—64no—
144C4⋊C4—64no—
145C22⋊C4—22no—
146C22⋊C4—22no—
147C22⋊C4—64no—
148C22⋊C4—68no—
149C42—22no—
150C22×C3—22no—
151C22×C3—22no—
152C22×C3—24no—
153C22×C3—24no—
154C22×C3—22no—
155C22×C3—24no—
156C22×C3—41no—
157C22×C3—41no—
158C22×C3—42no—
159C22×C3#12114yes—
160C22×C3C22×D411yescharacteristic
161C22×C3C22×D411yescharacteristic
162C22×C3C2×#4111yescharacteristic
163C22×C3C2×#4112yes—
164C22×C32−1+412yes—
165C3×C4—22no—
166C3×C4—22no—
167C3×C4—41no—
168C3×C4—41no—
169C3×C4#12314yescomplemented
170Dic3—44no—
171Dic3—44no—
172C23—22no—
173C23—22no—
174C23—21no—
175C23—22no—
176C23—22no—
177C23—21no—
178C23C2×D1212yes—
179C23C2×D1212yes—
180C23#23512yes—
181C23C2×#6712yes—
182C23C23×D311yescharacteristic
183D4—28no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—28no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—44no—
193C2×C4—64no—
194C2×C4—64no—
195C2×C4—64no—
196C2×C4—64no—
197C2×C4—64no—
198C2×C4—64no—
199C2×C4—128no—
200C2×C4C2×D1212yes—
201C2×C4#23714yes—
202C2×C4C2×#6712yes—
203C2×C4C2×#6712yes—
204C2×C3—24no—
205C2×C3—42no—
206C2×C3C2×#12111yescharacteristic
207C2×C3C2×#12311yescharacteristic
208C2×C3#53611yescharacteristic
209C2×C3#54712yes—
210C2×C3#55212yes—
211C22—22no—
212C22—24no—
213C22—24no—
214C22—22no—
215C22—22no—
216C22—24no—
217C22—41no—
218C22—42no—
219C22—41no—
220C22#93014yes—
221C22C22×D1211yescharacteristic
222C22C2×#23511yescharacteristic
223C22C2×#23712yes—
224C22#101012yes—
225C22C22×#6711yescharacteristic
226C4—22no—
227C4—22no—
228C4—41no—
229C4—41no—
230C4—124no—
231C4—124no—
232C4#93514yescomplemented
233C3#248611yescharacteristic, complemented, Sylow
234C2—24no—
235C2—42no—
236C2#554112yes—
237C2#555512yes—
238C2C2×#93011yescharacteristic
239C2C2×#93511yescharacteristic
240C2#582511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
prime radicalC22×C3×D4C22
hypercenterC2×C4×D4D3
hypocenterC3#2486
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
3. C22×C4⋊Dic3C2semidirect
4. #5212C2semidirect
5. #5212C2semidirect
6. #4981C2semidirect
7. #4953C2semidirect
8. #4943C2semidirect
9. #4928C2semidirect
169. C3×C4#123semidirect
232. C4#935semidirect
233. C3#2486semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C2×#9352
C22×C4⋊Dic32
#52122
#49812
#49532
#49432
#49282
#24863