Finite Group Databasebeta

← #10593984

Subgroups — #10593984

Order 576.

518 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 128 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
1#64081C211yesmaximal, characteristic, complemented
2C3×C4×D12C212yesmaximal, complemented
3#63916C212yesmaximal, complemented
4#63914C211yesmaximal, characteristic, complemented
5#63906C214yesmaximal, complemented
6#63905C214yesmaximal, complemented
7#63872C211yesmaximal, characteristic, complemented
8#5562—32nomaximal
9C2×C4×C3⋊D3C2211yescharacteristic
10C2×C3×D12C2212yes—
11C2×C3×C4×D3C2214yes—
12C2×#626C2214yes—
13C2×#625C2214yes—
14C32×C42C2211yescharacteristic
15#3752C2212yes—
16#3750C2212yes—
17C4×C3⋊Dic3C2211yescharacteristic
18C3×(C2×D3)⋊C4C2214yes—
19C3×C4⋊Dic3C2212yes—
20Dic3⋊Dic3C2214yes—
21(C2×D3)⋊Dic3C2214yes—
22C3×C4×D4—32no—
23#940—32no—
24#938—32no—
25#930—34no—
26#929—34no—
27#895—32no—
28#890—32no—
29#884—34no—
30#883—34no—
31C4×D12D312yescomplemented
32#872—32no—
33#872—61no—
34C22×C3⋊D3C2311yescharacteristic
35C22×C3×D3C2314yes—
36C2×C32×C4C2312yes—
37C2×C32×C4C2311yescharacteristic
38C2×C3⋊Dic3C2311yescharacteristic
39C2×C3⋊Dic3C2312yes—
40C4×C3⋊D3—22no—
41C2×C3×Dic3C2314yes—
42C3×D12—24no—
43C3×C4×D3—44no—
44#626—44no—
45#625—44no—
46#540—91noSylow
47C2×C3×D4—32no—
48C22×C3×C4—34no—
49C2×#67—34no—
50C2×#67—34no—
51C2×#67—34no—
52C2×D12—34no—
53C2×D12C2×D312yes—
54C2×C4×D3—32no—
55C2×C4×D3—61no—
56C2×C4×D3C2×D314yes—
57C3×C4⋊C4—32no—
58C3×C22⋊C4—34no—
59C3×C42—21no—
60C3×C42C2×D312yes—
61#217—34no—
62(C2×D3)⋊C4—34no—
63(C2×D3)⋊C4—32no—
64(C2×D3)⋊C4—32no—
65(C2×D3)⋊C4—62no—
66(C2×D3)⋊C4C2×D314yes—
67C4⋊Dic3—34no—
68C4⋊Dic3C2×D312yes—
69Dic3⋊C4—32no—
70Dic3⋊C4—32no—
71Dic3⋊C4—34no—
72Dic3⋊C4—62no—
73C4×Dic3—32no—
74C4×Dic3—61no—
75C22×C32C2411yescharacteristic
76C2×C3⋊D3—21no—
77C2×C3⋊D3—22no—
78C2×C3×D3—24no—
79C2×C3×D3—24no—
80C2×C3×D3—24no—
81C32×C4—22no—
82C32×C4#4112yes—
83C3⋊Dic3—21no—
84C3⋊Dic3—22no—
85C3×Dic3—24no—
86#127—91no—
87#125—91no—
88#123—94no—
89#121—94no—
90#121—92no—
91C4×D4—32no—
92#117—91no—
93C23×C3—34no—
94C22×D3—32no—
95C22×D3—61no—
96C22×D3C22×D314yes—
97C3×D4—64no—
98C2×C3×C4—21no—
99C2×C3×C4—22no—
100C2×C3×C4—34no—
101C2×C3×C4—124no—
102C2×C3×C4C22×D312yes—
103C2×C3×C4C22×D312yes—
104C2×C3×C4C22×D312yes—
105#67—124no—
106#67—124no—
107#67—124no—
108C2×Dic3—32no—
109C2×Dic3—32no—
110C2×Dic3—32no—
111C2×Dic3—61no—
112C2×Dic3—62no—
113C2×Dic3C22×D314yes—
114D12—24no—
115D12—124no—
116C4×D3—44no—
117C4×D3—64no—
118C4×D3—64no—
119C2×C32C2×#4111yescharacteristic
120C2×C32H(5,𝔽2)12yes—
121C3⋊D3—41no—
122C3×D3—44no—
123C2×D4—32no—
124C2×D4—94no—
125C2×D4—94no—
126C22×C4—34no—
127C22×C4—91no—
128C4⋊C4—32no—
129C4⋊C4—94no—
130C4⋊C4—92no—
131C22⋊C4—34no—
132C22⋊C4—94no—
133C22⋊C4—92no—
134C42—91no—
135C42D3211yescharacteristic
136C22×C3—21no—
137C22×C3—64no—
138C22×C3—64no—
139C22×C3—64no—
140C22×C3C23×D312yes—
141C2×D3—24no—
142C2×D3—24no—
143C2×D3—24no—
144C2×D3—62no—
145C2×D3—62no—
146C2×D3—62no—
147C2×D3—62no—
148C2×D3—122no—
149C3×C4—22no—
150C3×C4—22no—
151C3×C4—22no—
152C3×C4—42no—
153C3×C4—64no—
154C3×C4#23514yes—
155Dic3—24no—
156Dic3—62no—
157Dic3—62no—
158Dic3—62no—
159Dic3—62no—
160Dic3—122no—
161C32#54011yescharacteristic, complemented, Sylow
162C23—34no—
163C23—91no—
164D4—64no—
165D4—364no—
166D4—364no—
167C2×C4—34no—
168C2×C4—92no—
169C2×C4—91no—
170C2×C4—124no—
171C2×C4—182no—
172C2×C4C2×D3212yes—
173C2×C4C2×D3211yescharacteristic
174C2×C3—21no—
175C2×C3—22no—
176C2×C3—124no—
177C2×C3C2×#23512yes—
178C2×C3#100412yes—
179C2×C3#100912yes—
180D3—44no—
181D3—122no—
182D3—122no—
183C22—64no—
184C22—64no—
185C22—64no—
186C22—181no—
187C22—182no—
188C22C22×D3211yescharacteristic
189C4—22no—
190C4—64no—
191C4—181no—
192C4—182no—
193C4#379812yes—
194C3—21no—
195C3#556212yescomplemented
196C2—124no—
197C2—361no—
198C2C2×#379811yescharacteristic
199C2#6430712yes—
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
hypercenterC42D32
hypocenterC32#540
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #64081C2semidirect
2. C3×C4×D12C2semidirect
3. #63916C2semidirect
4. #63914C2semidirect
5. #63906C2semidirect
6. #63905C2semidirect
7. #63872C2semidirect
31. C4×D12D3semidirect
161. C32#540semidirect
195. C3#5562semidirect

Maximal subgroups

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

subgroupindex
#640812
C3×C4×D122
#639162
#639142
#639062
#639052
#638722
#55623