Finite Group Databasebeta

← #82200

Subgroups — #82200

Order 384.

526 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 149 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×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#929C211yesmaximal, characteristic
3#5213C212yesmaximal
4#5211C211yesmaximal, characteristic, complemented
5#5210C212yesmaximal, complemented
6#4993C212yesmaximal, complemented
7#4992C212yesmaximal, complemented
8#4985C212yesmaximal, complemented
9#4983C212yesmaximal, complemented
10#2623—31nomaximal, Sylow
11C23×C3×C4C2211yescharacteristic
12C23×C3×C4C2212yes—
13C22×#67C2211yescharacteristic
14C2×C3×C4⋊C4—24no—
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C4⋊C4C2212yes—
17C2×#217C2211yescharacteristic
18C2×#217C2212yes—
19#929—24no—
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C4C2211yescharacteristic
22C2×(C2×D3)⋊C4C2214yes—
23C2×Dic3⋊C4C2211yescharacteristic
24C2×Dic3⋊C4C2211yescharacteristic
25C2×Dic3⋊C4C2214yes—
26C3⋊#95C2214yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2214yes—
30C2×#123—31no—
31C22×C4⋊C4D311yescharacteristic, complemented
32#400—32no—
33#399—32no—
34#398—32no—
35#397—32no—
36#386—32no—
37#385—32no—
38#381—31no—
39C24×C3C2311yescharacteristic
40C23×D3C2311yescharacteristic
41C22×C3×C4—24no—
42C22×C3×C4—24no—
43C22×C3×C4—44no—
44C22×C3×C4D414yes—
45C22×C3×C4C2314yes—
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C2×#67—22no—
49C2×#67—24no—
50C22×Dic3C2311yescharacteristic
51C22×Dic3C2311yescharacteristic
52C22×Dic3C2314yes—
53C22×Dic3C2311yescharacteristic
54C3×C4⋊C4—24no—
55C3×C4⋊C4—24no—
56#217—22no—
57#217—24no—
58(C2×D3)⋊C4—22no—
59(C2×D3)⋊C4—22no—
60(C2×D3)⋊C4—44no—
61Dic3⋊C4—22no—
62Dic3⋊C4—22no—
63Dic3⋊C4—44no—
64C22×D4—31no—
65C23×C4C2×D311yescharacteristic
66C23×C4C2×D312yes—
67#123—64no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4—31no—
71C2×C4⋊C4—34no—
72C2×C4⋊C4C2×D312yes—
73C2×C4⋊C4C2×D312yes—
74C2×C22⋊C4—31no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4—31no—
77C2×C22⋊C4—31no—
78C2×C22⋊C4—34no—
79#95—32no—
80#95—32no—
81#95—34no—
82#95—34no—
83C23×C3—22no—
84C23×C3—22no—
85C23×C3C2×D412yes—
86C23×C3#4114yes—
87C23×C3C2411yescharacteristic
88C22×D3—21no—
89C22×D3—22no—
90C22×D3—22no—
91C22×D3—22no—
92C2×C3×C4—28no—
93C2×C3×C4—22no—
94C2×C3×C4—22no—
95C2×C3×C4—24no—
96C2×C3×C4—24no—
97C2×C3×C4—24no—
98C2×C3×C4—48no—
99C2×C3×C4C2×D412yes—
100C2×C3×C4C2×D412yes—
101#67—44no—
102C2×Dic3—21no—
103C2×Dic3—21no—
104C2×Dic3—21no—
105C2×Dic3—24no—
106C2×Dic3—22no—
107C2×Dic3—22no—
108C2×Dic3—24no—
109C2×Dic3—22no—
110C2×Dic3—24no—
111C24—31no—
112C24C22×D311yescharacteristic
113C2×D4—64no—
114C2×D4—62no—
115C22×C4—24no—
116C22×C4—24no—
117C22×C4—34no—
118C22×C4—31no—
119C22×C4—31no—
120C22×C4—31no—
121C22×C4—44no—
122C22×C4#6714yes—
123C22×C4C22×D314yes—
124C22×C4C22×D311yescharacteristic
125C22×C4C22×D311yescharacteristic
126C4⋊C4—24no—
127C4⋊C4—24no—
128C4⋊C4—62no—
129C4⋊C4—62no—
130C4⋊C4—124no—
131C22⋊C4—64no—
132C22⋊C4—62no—
133C22⋊C4—62no—
134C22⋊C4—62no—
135C22⋊C4—124no—
136C22×C3—22no—
137C22×C3—22no—
138C22×C3—28no—
139C22×C3#12314yes—
140C22×C3C22×D411yescharacteristic
141C22×C3C2×#4112yes—
142C22×C3H(5,𝔽2)12yes—
143C22×C32−1+412yes—
144C2×D3—41no—
145C2×D3—41no—
146C2×D3—41no—
147C2×D3—44no—
148C3×C4—22no—
149C3×C4—22no—
150C3×C4—44no—
151Dic3—41no—
152Dic3—41no—
153Dic3—41no—
154Dic3—44no—
155C23—22no—
156C23—22no—
157C23—62no—
158C23—62no—
159C23—62no—
160C23—61no—
161C23#23514yes—
162C23C2×#6712yes—
163C23C23×D311yescharacteristic
164D4—124no—
165C2×C4—28no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—48no—
172C2×C4—64no—
173C2×C4—62no—
174C2×C4—64no—
175C2×C4—62no—
176C2×C4—64no—
177C2×C4—62no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4—61no—
181C2×C4C2×#6712yes—
182C2×C4C2×#6712yes—
183C2×C3—24no—
184C2×C3C2×#12311yescharacteristic
185C2×C3#53511yescharacteristic
186C2×C3#53611yescharacteristic
187C2×C3#53914yes—
188D3—81no—
189C22—28no—
190C22—22no—
191C22—22no—
192C22—124no—
193C22—121no—
194C22—121no—
195C22—121no—
196C22#92914yes—
197C22C2×#23512yes—
198C22#100412yes—
199C22#100712yes—
200C22C22×#6711yescharacteristic
201C4—22no—
202C4—22no—
203C4—44no—
204C4—124no—
205C4—121no—
206C4—121no—
207C4—121no—
208C3#262311yescharacteristic, complemented, Sylow
209C2—24no—
210C2—241no—
211C2#551514yes—
212C2C2×#92911yescharacteristic
213C2#580511yescharacteristic
214C2#581611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C4⋊C4C2
prime radicalC22×#67C22
hypercenterC22×C4⋊C4D3
hypocenterC3#2623
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊C4C2semidirect
4. #5211C2semidirect
5. #5210C2semidirect
6. #4993C2semidirect
7. #4992C2semidirect
8. #4985C2semidirect
9. #4983C2semidirect
31. C22×C4⋊C4D3semidirect
208. C3#2623semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊C42
C2×#9292
#52132
#52112
#52102
#49932
#49922
#49852
#49832
#26233