Finite Group Databasebeta

← #10596141

Subgroups — #10596141

Order 576.

708 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 175 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×#124C211yesmaximal, characteristic, complemented
2#64156C211yesmaximal, characteristic, complemented
3#64153C211yesmaximal, characteristic, complemented
4#64148C212yesmaximal, complemented
5#64094C212yesmaximal, complemented
6#64093C212yesmaximal, complemented
7#64090C212yesmaximal, complemented
8#64086C212yesmaximal, complemented
9#64081C212yesmaximal, complemented
10#5670—34nomaximal
11C2×C32×Q8C2211yescharacteristic
12C2×C32×D4C2211yescharacteristic
13C2×#638C2212yes—
14C2×#638C2212yes—
15C22×C3⋊Dic3C2212yes—
16C2×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C4×C3⋊D3C2212yes—
18C2×#634C2211yescharacteristic
19C32×C22⋊C4C2212yes—
20C32×C22⋊C4C2212yes—
21C32×C42C2211yescharacteristic
22#3757C2212yes—
23#3752C2212yes—
24#3752C2212yes—
25#3752C2212yes—
26#3750C2212yes—
27#3750C2212yes—
28#3750C2212yes—
29C4×C3⋊Dic3C2212yes—
30C4×C3⋊Dic3C2211yescharacteristic
31C4×C3⋊Dic3C2211yescharacteristic
32C3×#124D314yescomplemented
33#947—34no—
34#944—34no—
35#939—38no—
36#885—38no—
37#884—38no—
38#67⋊C4—38no—
39#877—38no—
40#872—38no—
41C23×C32C2312yes—
42C22×C3⋊D3C2312yes—
43C32×Q8—41no—
44C32×D4—41no—
45C2×C32×C4C2311yescharacteristic
46C2×C32×C4C2312yes—
47C2×C32×C4C2311yescharacteristic
48C2×C32×C4C2311yescharacteristic
49#638—24no—
50#638—42no—
51C2×C3⋊Dic3—24no—
52C2×C3⋊Dic3C2312yes—
53C2×C3⋊Dic3C2312yes—
54C2×C3⋊Dic3C2312yes—
55C4⋊(C3⋊D3)—41no—
56C4×C3⋊D3—24no—
57#634—41no—
58#555—91noSylow
59C2×C3×Q8C2×D314yes—
60C2×C3×D4C2×D314yes—
61C2×#67—38no—
62C2×#67—38no—
63C22×Dic3—38no—
64C2×D12—34no—
65C2×C4×D3—38no—
66C2×Dic6—34no—
67C3×C22⋊C4C2×D318yes—
68C3×C22⋊C4C2×D318yes—
69C3×C42C2×D314yes—
70#217—38no—
71(C2×D3)⋊C4—38no—
72(C2×D3)⋊C4—38no—
73(C2×D3)⋊C4—38no—
74Dic3⋊C4—38no—
75Dic3⋊C4—38no—
76Dic3⋊C4—38no—
77C4×Dic3—38no—
78C4×Dic3—34no—
79C4×Dic3—34no—
80C22×C32—22no—
81C22×C32—22no—
82C22×C32—22no—
83C22×C32C2411yescharacteristic
84C2×C3⋊D3—22no—
85C2×C3⋊D3—22no—
86C2×C3⋊D3—22no—
87C32×C4—22no—
88C32×C4—21no—
89C32×C4—21no—
90C32×C4—21no—
91C3⋊Dic3—22no—
92C3⋊Dic3—22no—
93C3⋊Dic3#4114yes—
94C4⋊Q8—91no—
95#124—91no—
96#124—92no—
97#124C3⋊D311yescharacteristic, complemented
98#121—92no—
99#121—92no—
100C4×D4—92no—
101#117—92no—
102#117—92no—
103C23×C3C22×D318yes—
104C22×D3—38no—
105C3×Q8—44no—
106C3×D4—44no—
107C2×C3×C4C22×D314yes—
108C2×C3×C4C22×D318yes—
109C2×C3×C4C22×D314yes—
110C2×C3×C4C22×D314yes—
111#67—616no—
112#67—128no—
113C2×Dic3—38no—
114C2×Dic3—38no—
115C2×Dic3—38no—
116C2×Dic3—616no—
117D12—124no—
118C4×D3—616no—
119Dic6—124no—
120C2×C32—42no—
121C2×C32C2×#4112yes—
122C2×C32H(5,𝔽2)11yescharacteristic
123C3⋊D3—42no—
124C2×Q8—91no—
125C2×Q8C2×C3⋊D311yescharacteristic
126C2×D4—92no—
127C2×D4—92no—
128C2×D4—91no—
129C2×D4C2×C3⋊D311yescharacteristic
130C22×C4—92no—
131C22×C4—92no—
132C4⋊C4—92no—
133C4⋊C4—92no—
134C4⋊C4—92no—
135C22⋊C4—92no—
136C22⋊C4—92no—
137C22⋊C4—92no—
138C22⋊C4—92no—
139C22⋊C4C2×C3⋊D312yes—
140C22⋊C4C2×C3⋊D312yes—
141C42—91no—
142C42—92no—
143C42—91no—
144C42C2×C3⋊D311yescharacteristic
145C22×C3—28no—
146C22×C3—28no—
147C22×C3—28no—
148C22×C3C23×D314yes—
149C2×D3—68no—
150C2×D3—68no—
151C2×D3—68no—
152C3×C4—24no—
153C3×C4—24no—
154C3×C4—24no—
155C3×C4—28no—
156Dic3—316no—
157Dic3—68no—
158Dic3—68no—
159C32#55511yescharacteristic, complemented, Sylow
160C23—92no—
161C23C22×C3⋊D312yes—
162Q8—41no—
163Q8—361no—
164D4—41no—
165D4—184no—
166D4—361no—
167D4—362no—
168C2×C4—92no—
169C2×C4—92no—
170C2×C4—92no—
171C2×C4—184no—
172C2×C4—184no—
173C2×C4C22×C3⋊D312yes—
174C2×C4C22×C3⋊D311yescharacteristic
175C2×C4C22×C3⋊D311yescharacteristic
176C2×C4C22×C3⋊D311yescharacteristic
177C2×C3—48no—
178C2×C3#100414yes—
179C2×C3D3×#4118yes—
180D3—128no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—182no—
185C22—182no—
186C22—182no—
187C22C23×C3⋊D311yescharacteristic
188C4—21no—
189C4—21no—
190C4—21no—
191C4—22no—
192C4—94no—
193C4—182no—
194C4—182no—
195C3#567014yescomplemented
196C2—42no—
197C2—362no—
198C2#6436111yescharacteristic
199C2#41×C3⋊D312yes—
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#124C2
hypercenter#124C3⋊D3
hypocenterC32#555
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C32×#124C2semidirect
2. #64156C2semidirect
3. #64153C2semidirect
4. #64148C2semidirect
5. #64094C2semidirect
6. #64093C2semidirect
7. #64090C2semidirect
8. #64086C2semidirect
9. #64081C2semidirect
32. C3×#124D3semidirect
97. #124C3⋊D3semidirect
159. C32#555semidirect
195. C3#5670semidirect

Maximal subgroups

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

subgroupindex
C32×#1242
#641562
#641532
#641482
#640942
#640932
#640902
#640862
#640812
#56703