Finite Group Databasebeta

← #10596146

Subgroups — #10596146

Order 576.

810 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 191 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
1C2×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#3830C211yesmaximal, characteristic
3C32×#124C211yesmaximal, characteristic, complemented
4#64153C211yesmaximal, characteristic, complemented
5#64145C211yesmaximal, characteristic, complemented
6#64092C214yesmaximal, complemented
7#64087C214yesmaximal, complemented
8#64081C211yesmaximal, characteristic, complemented
9#64078C211yesmaximal, characteristic, complemented
10#5675—34nomaximal
11C2×C32×Q8C2211yescharacteristic
12C2×C32×D4C2211yescharacteristic
13C2×#638C2212yes—
14C22×C3⋊Dic3C2212yes—
15Q8×C3⋊D3—24no—
16#3830—24no—
17C2×C4×C3⋊D3C2212yes—
18C2×C4×C3⋊D3C2211yescharacteristic
19C2×#634C2211yescharacteristic
20C2×#634C2211yescharacteristic
21C2×#634C2212yes—
22C32×C22⋊C4C2214yes—
23C32×C42C2211yescharacteristic
24#3757C2214yes—
25#3752C2212yes—
26C4⋊(C3⋊Dic3)C2214yes—
27#3750C2212yes—
28#3750C2214yes—
29C4×C3⋊Dic3C2211yescharacteristic
30C2×D3×Q8—34no—
31C2×#237—34no—
32C3×#124D314yescomplemented
33#944—34no—
34#936—34no—
35#883—316no—
36#878—316no—
37#872—34no—
38#869—34no—
39C23×C32C2312yes—
40C22×C3⋊D3C2311yescharacteristic
41C32×Q8—22no—
42C32×D4—22no—
43C2×C32×C4C2312yes—
44C2×C32×C4C2312yes—
45C2×C32×C4C2311yescharacteristic
46#638—24no—
47C2×C3⋊Dic3—24no—
48C2×C3⋊Dic3C2312yes—
49C2×C3⋊Dic3C2312yes—
50C2×C3⋊Dic3C2312yes—
51C2×C3⋊Dic3C2311yescharacteristic
52C4×C3⋊D3—24no—
53C4×C3⋊D3D414yes—
54#634—22no—
55#634—22no—
56#634—24no—
57#536—91noSylow
58C2×C3×Q8C2×D314yes—
59C2×C3×D4C2×D314yes—
60C2×#67—38no—
61C22×Dic3—38no—
62D3×Q8—616no—
63#237—616no—
64C2×C4×D3—34no—
65C2×C4×D3—38no—
66C2×Dic6—34no—
67C2×Dic6—34no—
68C2×Dic6—38no—
69C3×C22⋊C4C2×D3116yes—
70C3×C42C2×D314yes—
71#217—316no—
72(C2×D3)⋊C4—38no—
73C4⋊Dic3—316no—
74Dic3⋊C4—38no—
75Dic3⋊C4—316no—
76C4×Dic3—34no—
77C22×C32—22no—
78C22×C32—24no—
79C22×C32C2411yescharacteristic
80C2×C3⋊D3—22no—
81C2×C3⋊D3C2×D412yes—
82C32×C4—22no—
83C32×C4—22no—
84C32×C4C2×D412yes—
85C3⋊Dic3—22no—
86C3⋊Dic3—22no—
87C3⋊Dic3—22no—
88C3⋊Dic3C2×D412yes—
89C2×#41—91no—
90C22×Q8—91no—
91C4⋊Q8—91no—
92C4⋊Q8—91no—
93#124—91no—
94#124C3⋊D311yescharacteristic, complemented
95#123—94no—
96C22⋊Q8—94no—
97#117—91no—
98C23×C3C22×D318yes—
99C22×D3—34no—
100C3×Q8—28no—
101C3×D4—28no—
102C2×C3×C4C22×D318yes—
103C2×C3×C4C22×D318yes—
104C2×C3×C4C22×D314yes—
105#67—616no—
106C2×Dic3—34no—
107C2×Dic3—38no—
108C2×Dic3—38no—
109C2×Dic3—38no—
110C2×Dic3—616no—
111C4×D3—316no—
112C4×D3—616no—
113Dic6—616no—
114Dic6—68no—
115Dic6—68no—
116C2×C32—42no—
117C2×C32C22×D411yescharacteristic
118C2×C322−1+412yes—
119C3⋊D3—22no—
120#41—184no—
121C2×Q8—92no—
122C2×Q8—91no—
123C2×Q8—91no—
124C2×Q8—184no—
125C2×Q8C2×C3⋊D311yescharacteristic
126C2×D4—92no—
127C2×D4C2×C3⋊D311yescharacteristic
128C22×C4—91no—
129C22×C4—92no—
130C22×C4—92no—
131C4⋊C4—92no—
132C4⋊C4—94no—
133C4⋊C4—94no—
134C22⋊C4—94no—
135C22⋊C4—92no—
136C22⋊C4C2×C3⋊D314yes—
137C42—91no—
138C42C2×C3⋊D311yescharacteristic
139C22×C3—28no—
140C22×C3—216no—
141C22×C3C23×D314yes—
142C2×D3—38no—
143C2×D3—68no—
144C3×C4—28no—
145C3×C4—28no—
146C3×C4D3×D418yes—
147Dic3—38no—
148Dic3—68no—
149Dic3—68no—
150Dic3—68no—
151C32#53611yescharacteristic, complemented, Sylow
152C23—91no—
153C23C22×C3⋊D312yes—
154Q8—22no—
155Q8—184no—
156Q8—182no—
157Q8—182no—
158D4—22no—
159D4—184no—
160C2×C4—91no—
161C2×C4—94no—
162C2×C4—92no—
163C2×C4—92no—
164C2×C4—92no—
165C2×C4—184no—
166C2×C4—184no—
167C2×C4C22×C3⋊D312yes—
168C2×C4C22×C3⋊D312yes—
169C2×C4C22×C3⋊D311yescharacteristic
170C2×C3—48no—
171C2×C3C2×D3×D414yes—
172C2×C3#101018yes—
173D3—68no—
174C22—22no—
175C22—24no—
176C22—92no—
177C22—182no—
178C22C23×C3⋊D311yescharacteristic
179C4—22no—
180C4—22no—
181C4—92no—
182C4—182no—
183C4—182no—
184C4—182no—
185C4D4×C3⋊D312yes—
186C3#567514yescomplemented
187C2—42no—
188C2—182no—
189C2C2×D4×C3⋊D311yescharacteristic
190C2#6436712yes—
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#124C2
prime radicalC2×#3830C2
hypercenter#124C3⋊D3
hypocenterC32#536
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×Q8×C3⋊D3C2semidirect
3. C32×#124C2semidirect
4. #64153C2semidirect
5. #64145C2semidirect
6. #64092C2semidirect
7. #64087C2semidirect
8. #64081C2semidirect
9. #64078C2semidirect
32. C3×#124D3semidirect
94. #124C3⋊D3semidirect
151. C32#536semidirect
186. C3#5675semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊D32
C2×#38302
C32×#1242
#641532
#641452
#640922
#640872
#640812
#640782
#56753