Finite Group Databasebeta

← #82114

Subgroups — #82114

Order 384.

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

normal subgroups only

This group has 700 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
1C22×C3×C22⋊C4C211yesmaximal, characteristic
2C22×#217C211yesmaximal, characteristic
3#5247C211yesmaximal, characteristic
4#5211C212yesmaximal, complemented
5#5210C212yesmaximal, complemented
6#4952C214yesmaximal, complemented
7#4946C212yesmaximal, complemented
8#4943C212yesmaximal, complemented
9#2483—31nomaximal, Sylow
10C25×C3C2211yescharacteristic
11C23×C3×C4C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C23×Dic3C2211yescharacteristic
14C23×Dic3C2211yescharacteristic
15C2×C3×C22⋊C4—24no—
16C2×C3×C22⋊C4C2212yes—
17C2×C3×C22⋊C4C2212yes—
18C2×#217—24no—
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×Dic3⋊C4C2214yescomplemented
25C3⋊#95C2214yescomplemented
26C3⋊#95C2214yescomplemented
27C3⋊#95C2214yescomplemented
28C22×C22⋊C4—31no—
29C22×C22⋊C4D311yescharacteristic
30#397—34no—
31#385—32no—
32#385—32no—
33#381—32no—
34#381—32no—
35#379—31no—
36C24×C3—24no—
37C24×C3—24no—
38C24×C3—42no—
39C24×C3D414yes—
40C24×C3C2311yescharacteristic
41C24×C3C2311yescharacteristic
42C24×C3C2311yescharacteristic
43C22×C3×C4—22no—
44C22×C3×C4—22no—
45C22×C3×C4—24no—
46C22×C3×C4—42no—
47C22×C3×C4C2312yes—
48C22×C3×C4C2312yes—
49C22×Dic3—22no—
50C22×Dic3—22no—
51C22×Dic3—24no—
52C22×Dic3—42no—
53C22×Dic3C2312yes—
54C22×Dic3C2314yes—
55C22×Dic3C2312yes—
56C3×C22⋊C4—24no—
57C3×C22⋊C4—24no—
58#217—24no—
59#217—24no—
60#217—24no—
61#217—24no—
62#217—42no—
63Dic3⋊C4—44no—
64C25C2×D311yescharacteristic
65C23×C4—31no—
66C23×C4—31no—
67C23×C4C2×D311yescharacteristic
68C23×C4C2×D311yescharacteristic
69C2×C4⋊C4—34no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—32no—
72C2×C22⋊C4—32no—
73C2×C22⋊C4—32no—
74C2×C22⋊C4—32no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4—64no—
77C2×C22⋊C4C2×D312yes—
78C2×C22⋊C4C2×D312yes—
79#95—34no—
80#95—34no—
81#95—34no—
82C23×C3—22no—
83C23×C3—22no—
84C23×C3—22no—
85C23×C3—22no—
86C23×C3—28no—
87C23×C3—28no—
88C23×C3—22no—
89C23×C3—22no—
90C23×C3—22no—
91C23×C3—44no—
92C23×C3—416no—
93C23×C3C2×D412yes—
94C23×C3C2×D412yes—
95C23×C3C2×D412yes—
96C23×C3#4114yes—
97C23×C3#4114yes—
98C23×C3C2411yescharacteristic
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—24no—
102C2×C3×C4—24no—
103C2×C3×C4—44no—
104C2×C3×C4—44no—
105C2×Dic3—22no—
106C2×Dic3—22no—
107C2×Dic3—24no—
108C2×Dic3—24no—
109C2×Dic3—24no—
110C2×Dic3—24no—
111C2×Dic3—24no—
112C2×Dic3—44no—
113C2×Dic3—44no—
114C24—24no—
115C24—24no—
116C24—42no—
117C24#6714yes—
118C24C22×D311yescharacteristic
119C24C22×D311yescharacteristic
120C24C22×D311yescharacteristic
121C22×C4—24no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—32no—
125C22×C4—34no—
126C22×C4—32no—
127C22×C4—42no—
128C22×C4—64no—
129C22×C4—62no—
130C22×C4—62no—
131C22×C4—122no—
132C22×C4C22×D312yes—
133C22×C4C22×D312yes—
134C4⋊C4—124no—
135C22⋊C4—24no—
136C22⋊C4—24no—
137C22⋊C4—64no—
138C22⋊C4—64no—
139C22⋊C4—64no—
140C22⋊C4—64no—
141C22⋊C4—122no—
142C22×C3—22no—
143C22×C3—22no—
144C22×C3—22no—
145C22×C3—22no—
146C22×C3—28no—
147C22×C3—28no—
148C22×C3—22no—
149C22×C3—22no—
150C22×C3—22no—
151C22×C3—416no—
152C22×C3—44no—
153C22×C3#12314yes—
154C22×C3#12314yes—
155C22×C3C22×D411yescharacteristic
156C22×C3C2×#4112yes—
157C22×C3C2×#4112yes—
158C22×C3H(5,𝔽2)12yes—
159C3×C4—42no—
160C3×C4—42no—
161Dic3—42no—
162Dic3—42no—
163Dic3—44no—
164C23—22no—
165C23—22no—
166C23—22no—
167C23—22no—
168C23—28no—
169C23—28no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—44no—
174C23—416no—
175C23#23514yes—
176C23#23714yes—
177C23C2×#6712yes—
178C23C2×#6712yes—
179C23C2×#6712yes—
180C23C23×D311yescharacteristic
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—44no—
186C2×C4—44no—
187C2×C4—64no—
188C2×C4—64no—
189C2×C4—64no—
190C2×C4—64no—
191C2×C4—64no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—124no—
195C2×C4—124no—
196C2×C3—24no—
197C2×C3—24no—
198C2×C3—42no—
199C2×C3C2×#12311yescharacteristic
200C2×C3C2×#12311yescharacteristic
201C2×C3#53411yescharacteristic
202C2×C3#55114yes—
203C22—28no—
204C22—28no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—416no—
213C22—44no—
214C22#92914yes—
215C22#93514yes—
216C22C2×#23512yes—
217C22C2×#23712yes—
218C22#100412yes—
219C22C22×#6711yescharacteristic
220C4—42no—
221C4—42no—
222C4—124no—
223C4—122no—
224C4—122no—
225C3#248311yescharacteristic, complemented, Sylow
226C2—24no—
227C2—24no—
228C2—42no—
229C2#549514yes—
230C2C2×#92911yescharacteristic
231C2C2×#93511yescharacteristic
232C2#580411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C22⋊C4C2
prime radicalC25×C3C22
hypercenterC22×C22⋊C4D3
hypocenterC3#2483
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #5211C2semidirect
5. #5210C2semidirect
6. #4952C2semidirect
7. #4946C2semidirect
8. #4943C2semidirect
24. C2×Dic3⋊C4C22semidirect
25. C3⋊#95C22semidirect
26. C3⋊#95C22semidirect
27. C3⋊#95C22semidirect
225. C3#2483semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊C42
C22×#2172
#52472
#52112
#52102
#49522
#49462
#49432
#24833