Finite Group Databasebeta

← #10596200

Subgroups — #10596200

Order 576.

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

normal subgroups only

This group has 618 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
1C32×C4⋊Q8C211yesmaximal, characteristic, complemented
2#64155C212yesmaximal
3#64153C212yesmaximal, complemented
4#64105C212yesmaximal
5#64101C211yesmaximal, characteristic
6C4⋊C4×C3⋊D3C211yesmaximal, characteristic
7#64099C211yesmaximal, characteristic, complemented
8#64098C212yesmaximal, complemented
9#64097C211yesmaximal, characteristic, complemented
10#64081C211yesmaximal, characteristic
11#64079C211yesmaximal, characteristic, complemented
12#5729—34nomaximal
13C2×C32×Q8C2212yes—
14C2×C4×C3⋊D3C2212yes—
15C2×C4×C3⋊D3C2211yescharacteristic
16C2×#634C2212yes—
17C32×C4⋊C4C2211yescharacteristic
18C32×C4⋊C4C2212yes—
19C32×C4⋊C4C2211yescharacteristic
20C32×C42C2211yescharacteristic
21#3752C2211yescharacteristic
22#3752C2211yescharacteristic
23#3752C2212yes—
24C4⋊(C3⋊Dic3)C2211yescharacteristic
25C4⋊(C3⋊Dic3)C2211yescharacteristic
26C4⋊(C3⋊Dic3)C2212yes—
27#3750C2212yes—
28#3750C2212yes—
29#3750C2211yescharacteristic
30#3750C2211yescharacteristic
31#3750C2212yes—
32#3750C2212yes—
33#3750C2212yes—
34C4×C3⋊Dic3C2211yescharacteristic
35C4×C3⋊Dic3C2212yes—
36C3×C4⋊Q8D314yescomplemented
37#946—38no—
38#944—38no—
39#896—38no—
40#892—34no—
41D3×C4⋊C4—34no—
42#890—34no—
43#889—38no—
44#888—34no—
45#872—34no—
46#870—34no—
47C22×C3⋊D3C2311yescharacteristic
48C32×Q8—42no—
49C2×C32×C4C2312yes—
50C2×C32×C4C2311yescharacteristic
51C2×C32×C4C2311yescharacteristic
52C2×C32×C4C2312yes—
53C2×C32×C4C2311yescharacteristic
54C2×C3⋊Dic3C2311yescharacteristic
55C2×C3⋊Dic3C2312yes—
56C2×C3⋊Dic3C2311yescharacteristic
57C2×C3⋊Dic3C2312yes—
58C2×C3⋊Dic3C2311yescharacteristic
59C4×C3⋊D3—42no—
60C4×C3⋊D3Q814yescomplemented
61#634—42no—
62#544—91noSylow
63C2×C3×Q8C2×D318yes—
64C2×C4×D3—34no—
65C2×C4×D3—38no—
66C2×Dic6—38no—
67C3×C4⋊C4C2×D314yes—
68C3×C4⋊C4C2×D318yes—
69C3×C4⋊C4C2×D314yes—
70C3×C42C2×D314yes—
71(C2×D3)⋊C4—34no—
72(C2×D3)⋊C4—34no—
73(C2×D3)⋊C4—38no—
74C4⋊Dic3—34no—
75C4⋊Dic3—34no—
76C4⋊Dic3—38no—
77Dic3⋊C4—38no—
78Dic3⋊C4—38no—
79Dic3⋊C4—34no—
80Dic3⋊C4—34no—
81Dic3⋊C4—38no—
82Dic3⋊C4—38no—
83Dic3⋊C4—38no—
84C4×Dic3—34no—
85C4×Dic3—38no—
86C22×C32C2411yescharacteristic
87C2×C3⋊D3—21no—
88C2×C3⋊D3—21no—
89C2×C3⋊D3C2×Q812yes—
90C32×C4—22no—
91C32×C4—22no—
92C32×C4—21no—
93C32×C4—21no—
94C32×C4C2×Q812yes—
95C3⋊Dic3—21no—
96C3⋊Dic3—22no—
97C3⋊Dic3—21no—
98C3⋊Dic3—22no—
99C3⋊Dic3C2×Q812yes—
100C4⋊Q8—92no—
101C4⋊Q8—91no—
102C4⋊Q8C3⋊D311yescharacteristic, complemented
103#125—92no—
104#125—91no—
105#125—91no—
106C22⋊Q8—92no—
107C22⋊Q8—92no—
108#117—91no—
109#117—91no—
110C2×C4⋊C4—91no—
111C22×D3—34no—
112C3×Q8—48no—
113C2×C3×C4C22×D318yes—
114C2×C3×C4C22×D314yes—
115C2×C3×C4C22×D314yes—
116C2×C3×C4C22×D318yes—
117C2×C3×C4C22×D314yes—
118C2×Dic3—34no—
119C2×Dic3—34no—
120C2×Dic3—38no—
121C2×Dic3—34no—
122C2×Dic3—38no—
123C4×D3—316no—
124C4×D3—128no—
125Dic6—128no—
126C2×C32C22×Q811yescharacteristic
127C2×C32H(5,𝔽2)11yescharacteristic
128C2×C322−1+411yescharacteristic
129C3⋊D3—22no—
130C2×Q8—92no—
131C2×Q8C2×C3⋊D312yes—
132C22×C4—91no—
133C22×C4—92no—
134C4⋊C4—92no—
135C4⋊C4—91no—
136C4⋊C4—91no—
137C4⋊C4—91no—
138C4⋊C4—91no—
139C4⋊C4—92no—
140C4⋊C4—92no—
141C4⋊C4—92no—
142C4⋊C4—92no—
143C4⋊C4—92no—
144C4⋊C4C2×C3⋊D311yescharacteristic
145C4⋊C4C2×C3⋊D311yescharacteristic
146C4⋊C4C2×C3⋊D312yes—
147C22⋊C4—92no—
148C22⋊C4—91no—
149C22⋊C4—91no—
150C42—92no—
151C42—91no—
152C42C2×C3⋊D311yescharacteristic
153C22×C3C23×D314yes—
154C2×D3—38no—
155C2×D3—64no—
156C2×D3—64no—
157C3×C4—28no—
158C3×C4—28no—
159C3×C4—24no—
160C3×C4—24no—
161C3×C4D3×Q818yes—
162Dic3—38no—
163Dic3—64no—
164Dic3—68no—
165Dic3—64no—
166Dic3—68no—
167C32#54411yescharacteristic, complemented, Sylow
168C23—91no—
169Q8—42no—
170Q8—362no—
171C2×C4—91no—
172C2×C4—94no—
173C2×C4—91no—
174C2×C4—92no—
175C2×C4—91no—
176C2×C4—92no—
177C2×C4—362no—
178C2×C4C22×C3⋊D312yes—
179C2×C4C22×C3⋊D311yescharacteristic
180C2×C4C22×C3⋊D311yescharacteristic
181C2×C4C22×C3⋊D312yes—
182C2×C4C22×C3⋊D311yescharacteristic
183C2×C3#100414yes—
184C2×C3C2×D3×Q814yes—
185C2×C3#100714yes—
186D3—68no—
187C22—92no—
188C22—181no—
189C22—181no—
190C22C23×C3⋊D311yescharacteristic
191C4—22no—
192C4—21no—
193C4—22no—
194C4—21no—
195C4—92no—
196C4—182no—
197C4—181no—
198C4—181no—
199C4—182no—
200C4Q8×C3⋊D312yes—
201C3#572914yescomplemented
202C2—182no—
203C2#6436111yescharacteristic
204C2C2×Q8×C3⋊D311yescharacteristic
205C2#6436411yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊Q8C2
prime radicalC22×C3⋊D3C23
hypercenterC4⋊Q8C3⋊D3
hypocenterC32#544
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C32×C4⋊Q8C2semidirect
3. #64153C2semidirect
7. #64099C2semidirect
8. #64098C2semidirect
9. #64097C2semidirect
11. #64079C2semidirect
36. C3×C4⋊Q8D3semidirect
60. C4×C3⋊D3Q8semidirect
102. C4⋊Q8C3⋊D3semidirect
167. C32#544semidirect
201. C3#5729semidirect

Maximal subgroups

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

subgroupindex
C32×C4⋊Q82
#641552
#641532
#641052
#641012
C4⋊C4×C3⋊D32
#640992
#640982
#640972
#640812
#640792
#57293