Finite Group Databasebeta

← #10596116

Subgroups — #10596116

Order 576.

654 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 183 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×C22⋊Q8C211yesmaximal, characteristic, complemented
2#64155C212yesmaximal, complemented
3#64139C211yesmaximal, characteristic
4C2×#3750C211yesmaximal, characteristic, complemented
5#64107C212yesmaximal, complemented
6#64103C212yesmaximal, complemented
7#64099C212yesmaximal, complemented
8#64095C212yesmaximal, complemented
9#64090C212yesmaximal, complemented
10#5645—34nomaximal
11C2×C32×Q8C2211yescharacteristic
12C22×C32×C4C2211yescharacteristic
13C2×#638C2212yes—
14C22×C3⋊Dic3C2211yescharacteristic
15C22×C3⋊Dic3C2211yescharacteristic
16C2×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C4×C3⋊D3C2212yes—
18C32×C4⋊C4C2212yes—
19C32×C4⋊C4C2211yescharacteristic
20C32×C22⋊C4C2212yes—
21#3752C2212yes—
22#3752C2212yes—
23#3752C2212yes—
24#3752C2212yes—
25C4⋊(C3⋊Dic3)C2212yes—
26C4⋊(C3⋊Dic3)C2212yes—
27C4⋊(C3⋊Dic3)C2211yescharacteristic
28#3750C2212yes—
29#3750C2212yes—
30#3750C2212yes—
31C4×C3⋊Dic3C2212yes—
32C3×C22⋊Q8D314yescomplemented
33#946—38no—
34#930—34no—
35C2×Dic3⋊C4—34no—
36#898—38no—
37#894—38no—
38#890—38no—
39#886—38no—
40#67⋊C4—38no—
41C23×C32C2311yescharacteristic
42C22×C3⋊D3C2312yes—
43C32×Q8—41no—
44C2×C32×C4—41no—
45C2×C32×C4C2312yes—
46C2×C32×C4C2311yescharacteristic
47C2×C32×C4C2311yescharacteristic
48C2×C32×C4C2312yes—
49#638—24no—
50C2×C3⋊Dic3—22no—
51C2×C3⋊Dic3—41no—
52C2×C3⋊Dic3C2312yes—
53C2×C3⋊Dic3C2312yes—
54C2×C3⋊Dic3C2312yes—
55C4⋊(C3⋊D3)—41no—
56C4×C3⋊D3—42no—
57#539—91noSylow
58C2×C3×Q8C2×D314yes—
59C22×C3×C4C2×D314yes—
60C2×#67—38no—
61C22×Dic3—34no—
62C22×Dic3—34no—
63C2×D12—34no—
64C2×C4×D3—38no—
65C3×C4⋊C4C2×D318yes—
66C3×C4⋊C4C2×D314yes—
67C3×C22⋊C4C2×D318yes—
68(C2×D3)⋊C4—38no—
69(C2×D3)⋊C4—38no—
70(C2×D3)⋊C4—38no—
71(C2×D3)⋊C4—38no—
72C4⋊Dic3—38no—
73C4⋊Dic3—38no—
74C4⋊Dic3—34no—
75Dic3⋊C4—38no—
76Dic3⋊C4—38no—
77Dic3⋊C4—38no—
78C4×Dic3—38no—
79C22×C32—21no—
80C22×C32—21no—
81C22×C32#4112yes—
82C22×C32C2411yescharacteristic
83C2×C3⋊D3—22no—
84C2×C3⋊D3—22no—
85C2×C3⋊D3—22no—
86C32×C4—22no—
87C32×C4—21no—
88C32×C4—21no—
89C32×C4—22no—
90C3⋊Dic3—22no—
91C3⋊Dic3—22no—
92C3⋊Dic3—22no—
93#126—92no—
94#125—92no—
95#123—92no—
96#123—92no—
97C22⋊Q8—92no—
98C22⋊Q8C3⋊D311yescharacteristic, complemented
99#121—91no—
100C4×D4—92no—
101C2×C4⋊C4—91no—
102C23×C3C22×D314yes—
103C22×D3—38no—
104C3×Q8—44no—
105C2×C3×C4—44no—
106C2×C3×C4C22×D318yes—
107C2×C3×C4C22×D314yes—
108C2×C3×C4C22×D314yes—
109C2×C3×C4C22×D318yes—
110#67—616no—
111C2×Dic3—38no—
112C2×Dic3—38no—
113C2×Dic3—38no—
114C2×Dic3—68no—
115C2×Dic3—124no—
116D12—124no—
117C4×D3—128no—
118C2×C32—22no—
119C2×C32C2×#4111yescharacteristic
120C2×C32H(5,𝔽2)11yescharacteristic
121C2×C322−1+411yescharacteristic
122C3⋊D3—42no—
123C2×Q8C2×C3⋊D311yescharacteristic
124C2×D4—92no—
125C2×D4—91no—
126C22×C4—91no—
127C22×C4—91no—
128C22×C4—92no—
129C22×C4C2×C3⋊D311yescharacteristic
130C4⋊C4—92no—
131C4⋊C4—92no—
132C4⋊C4—92no—
133C4⋊C4—91no—
134C4⋊C4—92no—
135C4⋊C4—92no—
136C4⋊C4C2×C3⋊D312yes—
137C4⋊C4C2×C3⋊D311yescharacteristic
138C22⋊C4—92no—
139C22⋊C4—92no—
140C22⋊C4—92no—
141C22⋊C4—92no—
142C22⋊C4C2×C3⋊D312yes—
143C42—92no—
144C22×C3—24no—
145C22×C3—24no—
146C22×C3#23918yes—
147C22×C3C23×D314yes—
148C2×D3—68no—
149C2×D3—68no—
150C2×D3—68no—
151C3×C4—28no—
152C3×C4—24no—
153C3×C4—24no—
154C3×C4—28no—
155Dic3—68no—
156Dic3—68no—
157Dic3—68no—
158C32#53911yescharacteristic, complemented, Sylow
159C23—92no—
160C23C22×C3⋊D311yescharacteristic
161Q8—41no—
162D4—184no—
163D4—361no—
164C2×C4—41no—
165C2×C4—92no—
166C2×C4—92no—
167C2×C4—92no—
168C2×C4—182no—
169C2×C4—362no—
170C2×C4—361no—
171C2×C4C22×C3⋊D312yes—
172C2×C4C22×C3⋊D311yescharacteristic
173C2×C4C22×C3⋊D312yes—
174C2×C4C22×C3⋊D311yescharacteristic
175C2×C3—28no—
176C2×C3#100414yes—
177C2×C3C2×#23914yes—
178C2×C3#101014yes—
179D3—128no—
180C22—21no—
181C22—21no—
182C22—182no—
183C22—182no—
184C22—182no—
185C22#383212yes—
186C22C23×C3⋊D311yescharacteristic
187C4—21no—
188C4—21no—
189C4—22no—
190C4—22no—
191C4—182no—
192C4—182no—
193C4—182no—
194C3#564514yescomplemented
195C2—22no—
196C2—362no—
197C2#6436111yescharacteristic
198C2C2×#383211yescharacteristic
199C2#6436711yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊Q8C2
prime radical#64139C2
hypercenterC22⋊Q8C3⋊D3
hypocenterC32#539
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C32×C22⋊Q8C2semidirect
2. #64155C2semidirect
4. C2×#3750C2semidirect
5. #64107C2semidirect
6. #64103C2semidirect
7. #64099C2semidirect
8. #64095C2semidirect
9. #64090C2semidirect
32. C3×C22⋊Q8D3semidirect
98. C22⋊Q8C3⋊D3semidirect
158. C32#539semidirect
194. C3#5645semidirect

Maximal subgroups

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

subgroupindex
C32×C22⋊Q82
#641552
#641392
C2×#37502
#641072
#641032
#640992
#640952
#640902
#56453