Finite Group Databasebeta

← #10596052

Subgroups — #10596052

Order 576.

642 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 185 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
1C32×C4×Q8C211yesmaximal, characteristic, complemented
2#64155C211yesmaximal, characteristic, complemented
3#64155C211yesmaximal, characteristic, complemented
4#64103C212yesmaximal
5#64101C211yesmaximal, characteristic, complemented
6C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
7#64098C212yesmaximal, complemented
8#64085C212yesmaximal, complemented
9C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
10#64081C212yesmaximal, complemented
11#64079C211yesmaximal, characteristic, complemented
12#5581—34nomaximal
13C2×C32×Q8C2211yescharacteristic
14C2×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C4×C3⋊D3C2212yes—
16C2×C4×C3⋊D3C2211yescharacteristic
17C2×C4×C3⋊D3C2211yescharacteristic
18C32×C4⋊C4C2212yes—
19C32×C4⋊C4C2211yescharacteristic
20C32×C42C2211yescharacteristic
21C32×C42C2212yes—
22#3752C2211yescharacteristic
23#3752C2212yes—
24#3752C2211yescharacteristic
25#3752C2212yes—
26#3752C2212yes—
27C4⋊(C3⋊Dic3)C2211yescharacteristic
28C4⋊(C3⋊Dic3)C2211yescharacteristic
29C4⋊(C3⋊Dic3)C2211yescharacteristic
30#3750C2212yes—
31#3750C2212yes—
32#3750C2212yes—
33#3750C2212yes—
34#3750C2212yes—
35C4×C3⋊Dic3C2212yes—
36C3×C4×Q8D314yescomplemented
37#946—34no—
38#946—34no—
39#894—38no—
40#892—34no—
41D3×C4⋊C4—34no—
42#889—38no—
43#876—38no—
44C4×D12—34no—
45#872—38no—
46#870—34no—
47C22×C3⋊D3C2311yescharacteristic
48C22×C3⋊D3C2311yescharacteristic
49C32×Q8—22no—
50C2×C32×C4C2312yes—
51C2×C32×C4C2311yescharacteristic
52C2×C32×C4C2311yescharacteristic
53C2×C32×C4C2312yes—
54C2×C32×C4C2311yescharacteristic
55C2×C3⋊Dic3C2311yescharacteristic
56C2×C3⋊Dic3C2312yes—
57C2×C3⋊Dic3C2312yes—
58C2×C3⋊Dic3C2311yescharacteristic
59C4⋊(C3⋊D3)—22no—
60C4×C3⋊D3—24no—
61C4×C3⋊D3—22no—
62C4×C3⋊D3—22no—
63#552—91noSylow
64C2×C3×Q8C2×D314yes—
65C2×D12—34no—
66C2×C4×D3—38no—
67C2×C4×D3—34no—
68C2×C4×D3—34no—
69C3×C4⋊C4C2×D318yes—
70C3×C4⋊C4C2×D314yes—
71C3×C42C2×D314yes—
72C3×C42C2×D318yes—
73(C2×D3)⋊C4—34no—
74(C2×D3)⋊C4—38no—
75(C2×D3)⋊C4—34no—
76(C2×D3)⋊C4—38no—
77(C2×D3)⋊C4—38no—
78C4⋊Dic3—34no—
79C4⋊Dic3—34no—
80C4⋊Dic3—34no—
81Dic3⋊C4—38no—
82Dic3⋊C4—38no—
83Dic3⋊C4—38no—
84Dic3⋊C4—38no—
85Dic3⋊C4—38no—
86C4×Dic3—38no—
87C22×C32C2411yescharacteristic
88C2×C3⋊D3—21no—
89C2×C3⋊D3—21no—
90C2×C3⋊D3—21no—
91C2×C3⋊D3—21no—
92C2×C3⋊D3—21no—
93C2×C3⋊D3#4112yes—
94C32×C4—22no—
95C32×C4—21no—
96C32×C4—22no—
97C32×C4—21no—
98C32×C4#4112yes—
99C3⋊Dic3—21no—
100C3⋊Dic3—21no—
101C3⋊Dic3—22no—
102C3⋊Dic3—22no—
103#126—92no—
104#125—92no—
105#125—91no—
106#123—92no—
107C22⋊Q8—91no—
108C22⋊Q8—91no—
109C4×Q8C3⋊D311yescharacteristic, complemented
110C4×D4—91no—
111#117—91no—
112#117—92no—
113C2×C4⋊C4—91no—
114C22×D3—34no—
115C22×D3—34no—
116C3×Q8—28no—
117C2×C3×C4C22×D318yes—
118C2×C3×C4C22×D314yes—
119C2×C3×C4C22×D314yes—
120C2×C3×C4C22×D318yes—
121C2×C3×C4C22×D314yes—
122C2×Dic3—34no—
123C2×Dic3—38no—
124C2×Dic3—34no—
125C2×Dic3—38no—
126D12—68no—
127C4×D3—616no—
128C4×D3—68no—
129C4×D3—68no—
130C2×C32C2×#4111yescharacteristic
131C2×C32C2×#4111yescharacteristic
132C2×C322−1+411yescharacteristic
133C3⋊D3—22no—
134C3⋊D3—41no—
135C2×Q8C2×C3⋊D311yescharacteristic
136C2×D4—91no—
137C22×C4—92no—
138C22×C4—91no—
139C22×C4—91no—
140C4⋊C4—91no—
141C4⋊C4—92no—
142C4⋊C4—92no—
143C4⋊C4—91no—
144C4⋊C4—92no—
145C4⋊C4—91no—
146C4⋊C4—92no—
147C4⋊C4—92no—
148C4⋊C4C2×C3⋊D311yescharacteristic
149C4⋊C4C2×C3⋊D312yes—
150C22⋊C4—91no—
151C22⋊C4—92no—
152C22⋊C4—92no—
153C22⋊C4—91no—
154C22⋊C4—92no—
155C42—92no—
156C42C2×C3⋊D312yes—
157C42C2×C3⋊D311yescharacteristic
158C22×C3C23×D314yes—
159C2×D3—38no—
160C2×D3—64no—
161C2×D3—64no—
162C2×D3—64no—
163C2×D3—64no—
164C2×D3—64no—
165C3×C4—28no—
166C3×C4—24no—
167C3×C4—28no—
168C3×C4—24no—
169C3×C4#23518yes—
170Dic3—64no—
171Dic3—64no—
172Dic3—68no—
173Dic3—68no—
174C32#55211yescharacteristic, complemented, Sylow
175C23—91no—
176C23—91no—
177Q8—22no—
178D4—182no—
179C2×C4—91no—
180C2×C4—92no—
181C2×C4—91no—
182C2×C4—92no—
183C2×C4—184no—
184C2×C4—182no—
185C2×C4—182no—
186C2×C4C22×C3⋊D312yes—
187C2×C4C22×C3⋊D311yescharacteristic
188C2×C4C22×C3⋊D312yes—
189C2×C4C22×C3⋊D311yescharacteristic
190C2×C4C22×C3⋊D311yescharacteristic
191C2×C3C2×#23514yes—
192C2×C3#100714yes—
193C2×C3D3×#4114yes—
194D3—68no—
195D3—124no—
196C22—92no—
197C22—181no—
198C22—181no—
199C22—181no—
200C22—181no—
201C22—181no—
202C22C23×C3⋊D311yescharacteristic
203C4—22no—
204C4—21no—
205C4—22no—
206C4—21no—
207C4—181no—
208C4—181no—
209C4—182no—
210C4—182no—
211C4#382812yes—
212C3#558114yescomplemented
213C2—182no—
214C2—361no—
215C2C2×#382811yescharacteristic
216C2#6436411yescharacteristic
217C2#41×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C4×Q8C2
prime radicalC2×C4⋊(C3⋊D3)C22
hypercenterC4×Q8C3⋊D3
hypocenterC32#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C32×C4×Q8C2semidirect
2. #64155C2semidirect
3. #64155C2semidirect
5. #64101C2semidirect
6. C4⋊C4×C3⋊D3C2semidirect
7. #64098C2semidirect
8. #64085C2semidirect
10. #64081C2semidirect
11. #64079C2semidirect
36. C3×C4×Q8D3semidirect
109. C4×Q8C3⋊D3semidirect
174. C32#552semidirect
212. C3#5581semidirect

Maximal subgroups

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

subgroupindex
C32×C4×Q82
#641552
#641032
#641012
C4⋊C4×C3⋊D32
#640982
#640852
C4×C4⋊(C3⋊D3)2
#640812
#640792
#55813