Finite Group Databasebeta

← #10596039

Subgroups — #10596039

Order 576.

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

normal subgroups only

This group has 636 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#64156C211yesmaximal, characteristic
3#64153C211yesmaximal, characteristic, complemented
4#64107C212yesmaximal, complemented
5#64105C212yesmaximal, complemented
6#64102C211yesmaximal, characteristic
7#64096C211yesmaximal, characteristic, complemented
8#64085C212yesmaximal, complemented
9#64084C211yesmaximal, characteristic
10#64081C212yesmaximal, complemented
11C4×#634C211yesmaximal, characteristic, complemented
12#5568—34nomaximal
13C2×C32×Q8C2211yescharacteristic
14C2×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C4×C3⋊D3C2212yes—
16C2×#634C2211yescharacteristic
17C2×#634C2211yescharacteristic
18C32×C4⋊C4C2211yescharacteristic
19C32×C4⋊C4C2212yes—
20C32×C42C2211yescharacteristic
21C32×C42C2212yes—
22#3752C2212yes—
23#3752C2212yes—
24#3752C2212yes—
25#3752C2212yes—
26#3752C2212yes—
27C4⋊(C3⋊Dic3)C2211yescharacteristic
28#3750C2211yescharacteristic
29#3750C2211yescharacteristic
30#3750C2212yes—
31#3750C2212yes—
32#3750C2212yes—
33C4×C3⋊Dic3C2211yescharacteristic
34C4×C3⋊Dic3C2211yescharacteristic
35C4×C3⋊Dic3C2212yes—
36C3×C4×Q8D314yescomplemented
37#947—34no—
38#944—34no—
39#898—38no—
40#896—38no—
41#893—34no—
42#887—34no—
43#876—38no—
44#875—34no—
45#872—38no—
46C4×Dic6—34no—
47C22×C3⋊D3C2312yes—
48C32×Q8—22no—
49C2×C32×C4C2312yes—
50C2×C32×C4C2311yescharacteristic
51C2×C32×C4C2312yes—
52C2×C32×C4C2311yescharacteristic
53C2×C32×C4C2311yescharacteristic
54C2×C3⋊Dic3C2311yescharacteristic
55C2×C3⋊Dic3C2311yescharacteristic
56C2×C3⋊Dic3C2311yescharacteristic
57C2×C3⋊Dic3C2312yes—
58C2×C3⋊Dic3C2311yescharacteristic
59C4⋊(C3⋊D3)—22no—
60C4×C3⋊D3—24no—
61#634—22no—
62#634—22no—
63#556—91noSylow
64C2×C3×Q8C2×D314yes—
65C2×D12—34no—
66C2×C4×D3—38no—
67C2×Dic6—34no—
68C2×Dic6—34no—
69C3×C4⋊C4C2×D314yes—
70C3×C4⋊C4C2×D318yes—
71C3×C42C2×D314yes—
72C3×C42C2×D318yes—
73(C2×D3)⋊C4—38no—
74(C2×D3)⋊C4—38no—
75(C2×D3)⋊C4—38no—
76(C2×D3)⋊C4—38no—
77(C2×D3)⋊C4—38no—
78C4⋊Dic3—34no—
79Dic3⋊C4—34no—
80Dic3⋊C4—34no—
81Dic3⋊C4—38no—
82Dic3⋊C4—38no—
83Dic3⋊C4—38no—
84C4×Dic3—34no—
85C4×Dic3—34no—
86C4×Dic3—38no—
87C22×C32C2411yescharacteristic
88C2×C3⋊D3—22no—
89C2×C3⋊D3—22no—
90C2×C3⋊D3—22no—
91C32×C4—21no—
92C32×C4—22no—
93C32×C4—22no—
94C32×C4—21no—
95C32×C4#4112yes—
96C3⋊Dic3—22no—
97C3⋊Dic3—21no—
98C3⋊Dic3—21no—
99C3⋊Dic3—21no—
100C3⋊Dic3#4112yes—
101C4⋊Q8—91no—
102#126—92no—
103#126—92no—
104#124—91no—
105#124—91no—
106C22⋊Q8—92no—
107C4×Q8—91no—
108C4×Q8—91no—
109C4×Q8C3⋊D311yescharacteristic, complemented
110C4×D4—91no—
111#117—92no—
112C22×D3—38no—
113C3×Q8—28no—
114C2×C3×C4C22×D318yes—
115C2×C3×C4C22×D314yes—
116C2×C3×C4C22×D318yes—
117C2×C3×C4C22×D314yes—
118C2×C3×C4C22×D314yes—
119C2×Dic3—34no—
120C2×Dic3—34no—
121C2×Dic3—34no—
122C2×Dic3—38no—
123C2×Dic3—34no—
124D12—68no—
125C4×D3—616no—
126Dic6—68no—
127Dic6—68no—
128C2×C32C2×#4111yescharacteristic
129C2×C32C2×#4111yescharacteristic
130C2×C322−1+411yescharacteristic
131C3⋊D3—42no—
132C2×Q8—91no—
133C2×Q8—91no—
134C2×Q8C2×C3⋊D311yescharacteristic
135C2×D4—91no—
136C22×C4—92no—
137C4⋊C4—91no—
138C4⋊C4—92no—
139C4⋊C4—92no—
140C4⋊C4—91no—
141C4⋊C4—91no—
142C4⋊C4—92no—
143C4⋊C4C2×C3⋊D311yescharacteristic
144C4⋊C4C2×C3⋊D312yes—
145C22⋊C4—92no—
146C22⋊C4—92no—
147C22⋊C4—92no—
148C22⋊C4—92no—
149C22⋊C4—92no—
150C42—92no—
151C42—91no—
152C42—91no—
153C42C2×C3⋊D312yes—
154C42C2×C3⋊D311yescharacteristic
155C22×C3C23×D314yes—
156C2×D3—68no—
157C2×D3—68no—
158C2×D3—68no—
159C3×C4—28no—
160C3×C4—24no—
161C3×C4—24no—
162C3×C4—28no—
163C3×C4#23518yes—
164Dic3—38no—
165Dic3—68no—
166Dic3—64no—
167Dic3—64no—
168Dic3—64no—
169C32#55611yescharacteristic, complemented, Sylow
170C23—92no—
171Q8—22no—
172Q8—182no—
173Q8—182no—
174D4—182no—
175C2×C4—91no—
176C2×C4—91no—
177C2×C4—91no—
178C2×C4—92no—
179C2×C4—91no—
180C2×C4—184no—
181C2×C4C22×C3⋊D312yes—
182C2×C4C22×C3⋊D311yescharacteristic
183C2×C4C22×C3⋊D312yes—
184C2×C4C22×C3⋊D311yescharacteristic
185C2×C4C22×C3⋊D311yescharacteristic
186C2×C3C2×#23514yes—
187C2×C3#100714yes—
188C2×C3D3×#4114yes—
189D3—128no—
190C22—182no—
191C22—182no—
192C22—182no—
193C22C23×C3⋊D311yescharacteristic
194C4—22no—
195C4—21no—
196C4—22no—
197C4—21no—
198C4—92no—
199C4—182no—
200C4—181no—
201C4—181no—
202C4—181no—
203C4#382812yes—
204C3#556814yescomplemented
205C2—362no—
206C2C2×#382811yescharacteristic
207C2#6436411yescharacteristic
208C2#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#556
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
3. #64153C2semidirect
4. #64107C2semidirect
5. #64105C2semidirect
7. #64096C2semidirect
8. #64085C2semidirect
10. #64081C2semidirect
11. C4×#634C2semidirect
36. C3×C4×Q8D3semidirect
109. C4×Q8C3⋊D3semidirect
169. C32#556semidirect
204. C3#5568semidirect

Maximal subgroups

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

subgroupindex
C32×C4×Q82
#641562
#641532
#641072
#641052
#641022
#640962
#640852
#640842
#640812
C4×#6342
#55683