Finite Group Databasebeta

← #82992

Subgroups — #82992

Order 384.

464 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 133 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×#124C211yesmaximal, characteristic
2C2×#870C211yesmaximal, characteristic, complemented
3#5230C212yesmaximal, complemented
4#5222C212yesmaximal
5#4953C212yesmaximal, complemented
6#4949C212yesmaximal, complemented
7#4948C214yesmaximal, complemented
8#4936C211yesmaximal, characteristic, complemented
9#2820—31nomaximal, Sylow
10C22×C3×Q8C2211yescharacteristic
11C22×C3×D4C2211yescharacteristic
12C3×#124—24no—
13C2×C3×C22⋊C4C2212yes—
14C2×C3×C22⋊C4C2212yes—
15C2×C3×C42C2211yescharacteristic
16C2×#217C2214yes—
17C2×#217C2214yes—
18C2×C4⋊Dic3C2211yescharacteristic
19C2×C4⋊Dic3C2211yescharacteristic
20C2×C4⋊Dic3C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2214yes—
24#870—24no—
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2214yes—
28C2×#125—31no—
29C2×#124D311yescharacteristic
30#399—32no—
31#397—32no—
32#396—34no—
33#395—32no—
34#394—32no—
35#382—31no—
36C24×C3C2312yes—
37C2×C3×Q8—22no—
38C2×C3×Q8—42no—
39C2×C3×D4—22no—
40C2×C3×D4—42no—
41C22×C3×C4C2312yes—
42C22×C3×C4C2311yescharacteristic
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C22×Dic3C2314yes—
46C22×Dic3C2312yes—
47C22×Dic3C2312yes—
48C3×C22⋊C4—24no—
49C3×C22⋊C4—24no—
50C3×C42D414yes—
51#217—44no—
52#217—44no—
53C4⋊Dic3—22no—
54C4⋊Dic3—22no—
55C4⋊Dic3—42no—
56Dic3⋊C4—24no—
57Dic3⋊C4—24no—
58Dic3⋊C4—44no—
59C22×Q8C2×D311yescharacteristic
60C22×D4C2×D311yescharacteristic
61#125—64no—
62#124—24no—
63C2×C4⋊C4—32no—
64C2×C4⋊C4—32no—
65C2×C4⋊C4—31no—
66C2×C4⋊C4—31no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—34no—
69C2×C22⋊C4—34no—
70C2×C22⋊C4—34no—
71C2×C22⋊C4C2×D312yes—
72C2×C22⋊C4C2×D312yes—
73C2×C42C2×D311yescharacteristic
74#95—32no—
75#95—32no—
76#95—34no—
77C23×C3—22no—
78C23×C3—22no—
79C23×C3—24no—
80C23×C3—24no—
81C23×C3—22no—
82C23×C3C2411yescharacteristic
83C3×Q8—44no—
84C3×D4—44no—
85C2×C3×C4—21no—
86C2×C3×C4—22no—
87C2×C3×C4—22no—
88C2×C3×C4—21no—
89C2×C3×C4—22no—
90C2×C3×C4—24no—
91C2×C3×C4C2×D412yes—
92C2×C3×C4C2×D412yes—
93C2×C3×C4C2×D412yes—
94C2×Dic3—22no—
95C2×Dic3—22no—
96C2×Dic3—24no—
97C2×Dic3—24no—
98C2×Dic3—24no—
99C2×Dic3—22no—
100C2×Dic3—22no—
101C2×Dic3—24no—
102C24C22×D312yes—
103C2×Q8—22no—
104C2×Q8—42no—
105C2×D4—22no—
106C2×D4—42no—
107C22×C4—34no—
108C22×C4—32no—
109C22×C4—32no—
110C22×C4C22×D312yes—
111C22×C4C22×D311yescharacteristic
112C22×C4C22×D311yescharacteristic
113C22×C4C22×D311yescharacteristic
114C4⋊C4—64no—
115C4⋊C4—64no—
116C4⋊C4—62no—
117C4⋊C4—62no—
118C4⋊C4—124no—
119C4⋊C4—122no—
120C22⋊C4—24no—
121C22⋊C4—24no—
122C22⋊C4—124no—
123C22⋊C4—124no—
124C42#6714yes—
125C22×C3—42no—
126C22×C3—42no—
127C22×C3—42no—
128C22×C3—44no—
129C22×C3—44no—
130C22×C3C22×D411yescharacteristic
131C22×C3H(5,𝔽2)12yes—
132C22×C3H(5,𝔽2)11yescharacteristic
133C22×C32−1+412yes—
134C22×C32−1+411yescharacteristic
135C3×C4—22no—
136C3×C4—22no—
137C3×C4—22no—
138C3×C4—42no—
139Dic3—42no—
140Dic3—42no—
141Dic3—44no—
142C23—24no—
143C23—22no—
144C23—22no—
145C23—24no—
146C23—22no—
147C23C23×D311yescharacteristic
148Q8—44no—
149D4—44no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—24no—
155C2×C4—22no—
156C2×C4—64no—
157C2×C4—64no—
158C2×C4—62no—
159C2×C4—64no—
160C2×C4—64no—
161C2×C4—62no—
162C2×C4—62no—
163C2×C4—62no—
164C2×C4C2×#6712yes—
165C2×C4C2×#6712yes—
166C2×C4C2×#6712yes—
167C2×C3—82no—
168C2×C3#53511yescharacteristic
169C2×C3#53611yescharacteristic
170C2×C3#53711yescharacteristic
171C2×C3#56212yes—
172C2×C3#56312yes—
173C22—44no—
174C22—44no—
175C22—42no—
176C22—42no—
177C22—42no—
178C22#100412yes—
179C22#100911yescharacteristic
180C22#101011yescharacteristic
181C22#101012yes—
182C22C22×#6711yescharacteristic
183C4—22no—
184C4—22no—
185C4—22no—
186C4—42no—
187C4—124no—
188C4—122no—
189C4—122no—
190C3#282011yescharacteristic, complemented, Sylow
191C2—82no—
192C2#567212yes—
193C2#568412yes—
194C2#580511yescharacteristic
195C2#582511yescharacteristic
196C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#124C2
prime radicalC22×C3×D4C22
hypercenterC2×#124D3
hypocenterC3#2820
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#870C2semidirect
3. #5230C2semidirect
5. #4953C2semidirect
6. #4949C2semidirect
7. #4948C2semidirect
8. #4936C2semidirect
190. C3#2820semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1242
C2×#8702
#52302
#52222
#49532
#49492
#49482
#49362
#28203