Finite Group Databasebeta

← #10594484

Subgroups — #10594484

Order 576.

503 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 138 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#64102C211yesmaximal, characteristic
2C3×#890C212yesmaximal, complemented
3#63919C211yesmaximal, characteristic, complemented
4C4×#627C211yesmaximal, characteristic, complemented
5#63900C212yesmaximal, complemented
6#63895C214yesmaximal, complemented
7#63894C212yesmaximal
8#63865C212yesmaximal, complemented
9#5690—32nomaximal
10C2×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C4×C3⋊D3C2212yes—
12C2×#627C2211yescharacteristic
13C2×#627C2212yes—
14C2×#624C2214yes—
15C32×C4⋊C4C2211yescharacteristic
16#3752C2212yes—
17C4×C3⋊Dic3C2211yescharacteristic
18C3×C4⋊Dic3C2212yes—
19C3×C4⋊Dic3C2214yes—
20C3×Dic3⋊C4C2214yes—
21C3×C4×Dic3C2212yes—
22#3724C2212yes—
23#3724C2211yescharacteristic
24Dic3⋊Dic3C2212yes—
25#3722C2214yes—
26C3×#125—32no—
27#897—34no—
28#896—34no—
29#896—34no—
30#893—32no—
31#893—32no—
32#893—61no—
33D3×C4⋊C4—34no—
34#890D312yescomplemented
35#888—32no—
36C4×D12—32no—
37C4×Dic6—32no—
38C22×C3⋊D3C2312yes—
39C2×C32×C4C2312yes—
40C2×C32×C4C2311yescharacteristic
41C2×C3⋊Dic3C2311yescharacteristic
42C2×C3⋊Dic3C2311yescharacteristic
43C4⋊(C3⋊D3)Q814yescomplemented
44C4×C3⋊D3—24no—
45C2×C3×Dic3C2312yes—
46C2×C3×Dic3C2314yes—
47C2×C3×Dic3C2312yes—
48#627—22no—
49#627—42no—
50#624—44no—
51#554—91noSylow
52C2×D12—32no—
53C2×D12—61no—
54C2×C4×D3—34no—
55C2×C4×D3—34no—
56C2×C4×D3—34no—
57C2×C4×D3—62no—
58C2×Dic6—32no—
59C2×Dic6—34no—
60C3×C4⋊C4—21no—
61C3×C4⋊C4—32no—
62C3×C4⋊C4—34no—
63C3×C4⋊C4—34no—
64C3×C4⋊C4C2×D312yes—
65C3×C42—32no—
66(C2×D3)⋊C4—34no—
67(C2×D3)⋊C4—34no—
68(C2×D3)⋊C4—34no—
69(C2×D3)⋊C4—62no—
70C4⋊Dic3—32no—
71C4⋊Dic3C2×D312yes—
72C4⋊Dic3C2×D314yes—
73Dic3⋊C4—32no—
74Dic3⋊C4—32no—
75Dic3⋊C4—34no—
76Dic3⋊C4C2×D314yes—
77C4×Dic3—32no—
78C4×Dic3—61no—
79C4×Dic3C2×D312yes—
80C22×C32C2411yescharacteristic
81C2×C3⋊D3—22no—
82C2×C3⋊D3—22no—
83C2×C3⋊D3C2×Q814yes—
84C32×C4—22no—
85C32×C4C2×Q812yes—
86C3⋊Dic3—21no—
87C3⋊Dic3#4112yes—
88C3×Dic3—22no—
89C3×Dic3—22no—
90C3×Dic3—24no—
91C4⋊Q8—91no—
92#125—32no—
93C22⋊Q8—92no—
94C22⋊Q8—94no—
95C4×Q8—91no—
96C4×D4—92no—
97C4×D4—91no—
98C2×C4⋊C4—92no—
99C22×D3—34no—
100C22×D3—62no—
101C2×C3×C4—21no—
102C2×C3×C4—22no—
103C2×C3×C4—32no—
104C2×C3×C4—34no—
105C2×C3×C4—32no—
106C2×C3×C4C22×D314yes—
107C2×C3×C4C22×D312yes—
108C2×Dic3—32no—
109C2×Dic3—32no—
110C2×Dic3—61no—
111C2×Dic3—61no—
112C2×Dic3C22×D312yes—
113C2×Dic3C22×D312yes—
114C2×Dic3C22×D314yes—
115D12—38no—
116D12—64no—
117C4×D3—68no—
118C4×D3—124no—
119C4×D3—124no—
120C4×D3—124no—
121Dic6—64no—
122Dic6—124no—
123C2×C32C22×Q811yescharacteristic
124C2×C32C2×#4111yescharacteristic
125C2×C32H(5,𝔽2)11yescharacteristic
126C3⋊D3—24no—
127C2×Q8—91no—
128C2×Q8—92no—
129C2×D4—91no—
130C22×C4—94no—
131C22×C4—92no—
132C4⋊C4—34no—
133C4⋊C4—32no—
134C4⋊C4—34no—
135C4⋊C4—91no—
136C4⋊C4—92no—
137C4⋊C4—92no—
138C4⋊C4D3211yescharacteristic
139C22⋊C4—94no—
140C22⋊C4—92no—
141C42—32no—
142C42—91no—
143C22×C3—21no—
144C22×C3C23×D312yes—
145C2×D3—38no—
146C2×D3—64no—
147C2×D3—64no—
148C2×D3—64no—
149C2×D3—122no—
150C2×D3—122no—
151C3×C4—24no—
152C3×C4—22no—
153C3×C4—42no—
154C3×C4—62no—
155C3×C4—64no—
156C3×C4—62no—
157C3×C4D3×Q814yes—
158Dic3—22no—
159Dic3—24no—
160Dic3—22no—
161Dic3—34no—
162Dic3—62no—
163Dic3—62no—
164Dic3—121no—
165C32#55411yescharacteristic, complemented, Sylow
166C23—92no—
167Q8—182no—
168Q8—362no—
169D4—94no—
170C2×C4—32no—
171C2×C4—32no—
172C2×C4—34no—
173C2×C4—91no—
174C2×C4—91no—
175C2×C4—184no—
176C2×C4—364no—
177C2×C4C2×D3212yes—
178C2×C4C2×D3211yescharacteristic
179C2×C3—21no—
180C2×C3—21no—
181C2×C3—21no—
182C2×C3C2×D3×Q812yes—
183C2×C3D3×#4112yes—
184C2×C3#100912yes—
185D3—68no—
186D3—124no—
187C22—94no—
188C22—182no—
189C22—182no—
190C22C22×D3211yescharacteristic
191C4—22no—
192C4—62no—
193C4—62no—
194C4—64no—
195C4—92no—
196C4—181no—
197C4#379712yes—
198C3—21no—
199C3#569012yescomplemented
200C2—184no—
201C2C2×#379711yescharacteristic
202C2#6431211yescharacteristic
203C2#6432011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×C4⋊(C3⋊D3)C22
hypercenterC4⋊C4D32
hypocenterC32#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#890C2semidirect
3. #63919C2semidirect
4. C4×#627C2semidirect
5. #63900C2semidirect
6. #63895C2semidirect
8. #63865C2semidirect
34. #890D3semidirect
43. C4⋊(C3⋊D3)Q8semidirect
165. C32#554semidirect
199. C3#5690semidirect

Maximal subgroups

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

subgroupindex
#641022
C3×#8902
#639192
C4×#6272
#639002
#638952
#638942
#638652
#56903