Finite Group Databasebeta

← #10596010

Subgroups — #10596010

Order 576.

684 conjugacy classes of subgroups; only the 233 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 233 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×D4C211yesmaximal, characteristic
2D4×C3⋊Dic3C211yesmaximal, characteristic
3D4×C3⋊Dic3C211yesmaximal, characteristic
4#64133C212yesmaximal, complemented
5C2×#3750C212yesmaximal, complemented
6#64099C211yesmaximal, characteristic, complemented
7#64097C211yesmaximal, characteristic, complemented
8#64087C212yesmaximal, complemented
9#64087C212yesmaximal, complemented
10#64079C211yesmaximal, characteristic, complemented
11C4×#634C211yesmaximal, characteristic, complemented
12#5539—34nomaximal
13C2×C32×D4C2211yescharacteristic
14C22×C32×C4C2212yes—
15C22×C3⋊Dic3C2212yes—
16C22×C3⋊Dic3C2212yes—
17C2×#634C2211yescharacteristic
18C2×#634C2212yes—
19C32×C4⋊C4C2211yescharacteristic
20C32×C22⋊C4C2212yes—
21C32×C42C2211yescharacteristic
22#3757C2212yes—
23#3757C2212yes—
24C4⋊(C3⋊Dic3)C2211yescharacteristic
25C4⋊(C3⋊Dic3)C2211yescharacteristic
26C4⋊(C3⋊Dic3)C2211yescharacteristic
27C4⋊(C3⋊Dic3)C2212yes—
28#3750C2211yescharacteristic
29#3750C2211yescharacteristic
30#3750C2212yes—
31#3750C2212yes—
32#3750C2212yes—
33#3750C2212yes—
34C4×C3⋊Dic3C2211yescharacteristic
35C4×C3⋊Dic3C2211yescharacteristic
36C3×C4×D4D314yes—
37D4×Dic3—34no—
38D4×Dic3—34no—
39#924—38no—
40C2×Dic3⋊C4—38no—
41#890—34no—
42#888—34no—
43#878—38no—
44#878—38no—
45#870—34no—
46C4×Dic6—34no—
47C23×C32C2312yes—
48C32×D4Q814yescomplemented
49C2×C32×C4—42no—
50C2×C32×C4C2311yescharacteristic
51C2×C32×C4C2311yescharacteristic
52C2×C32×C4C2312yes—
53C2×C32×C4C2311yescharacteristic
54C2×C3⋊Dic3—24no—
55C2×C3⋊Dic3—42no—
56C2×C3⋊Dic3C2311yescharacteristic
57C2×C3⋊Dic3C2311yescharacteristic
58C2×C3⋊Dic3C2311yescharacteristic
59C2×C3⋊Dic3C2312yes—
60C2×C3⋊Dic3C2312yes—
61C2×C3⋊Dic3C2311yescharacteristic
62#634—22no—
63#634—42no—
64#554—91noSylow
65C2×C3×D4C2×D314yes—
66C22×C3×C4C2×D318yes—
67C22×Dic3—38no—
68C22×Dic3—38no—
69C2×Dic6—34no—
70C2×Dic6—38no—
71C3×C4⋊C4C2×D314yes—
72C3×C22⋊C4C2×D318yes—
73C3×C42C2×D314yes—
74#217—38no—
75#217—38no—
76C4⋊Dic3—34no—
77C4⋊Dic3—34no—
78C4⋊Dic3—34no—
79C4⋊Dic3—38no—
80Dic3⋊C4—34no—
81Dic3⋊C4—34no—
82Dic3⋊C4—38no—
83Dic3⋊C4—38no—
84Dic3⋊C4—38no—
85Dic3⋊C4—38no—
86C4×Dic3—34no—
87C4×Dic3—34no—
88C22×C32—22no—
89C22×C32—22no—
90C22×C32C2×Q814yes—
91C22×C32C2411yescharacteristic
92C32×C4—21no—
93C32×C4—22no—
94C32×C4—21no—
95C32×C4C2×Q812yes—
96C3⋊Dic3—21no—
97C3⋊Dic3—22no—
98C3⋊Dic3—21no—
99C3⋊Dic3—22no—
100C3⋊Dic3—21no—
101C3⋊Dic3#4112yes—
102C4⋊Q8—91no—
103#125—91no—
104#125—91no—
105C22⋊Q8—92no—
106C22⋊Q8—92no—
107C22⋊Q8—92no—
108C4×Q8—91no—
109C4×D4—91no—
110C4×D4—91no—
111C4×D4C3⋊D311yescharacteristic
112C2×C4⋊C4—92no—
113C23×C3C22×D318yes—
114C3×D4Dic6116yescomplemented
115C2×C3×C4—48no—
116C2×C3×C4C22×D314yes—
117C2×C3×C4C22×D314yes—
118C2×C3×C4C22×D318yes—
119C2×C3×C4C22×D314yes—
120C2×Dic3—34no—
121C2×Dic3—34no—
122C2×Dic3—34no—
123C2×Dic3—34no—
124C2×Dic3—38no—
125C2×Dic3—38no—
126C2×Dic3—616no—
127C2×Dic3—128no—
128Dic6—68no—
129Dic6—128no—
130C2×C32—24no—
131C2×C32C22×Q811yescharacteristic
132C2×C32C2×#4111yescharacteristic
133C2×C32H(5,𝔽2)11yescharacteristic
134C2×Q8—91no—
135C2×Q8—92no—
136C2×D4C2×C3⋊D311yescharacteristic
137C22×C4—92no—
138C22×C4—92no—
139C22×C4C2×C3⋊D312yes—
140C4⋊C4—91no—
141C4⋊C4—92no—
142C4⋊C4—92no—
143C4⋊C4—92no—
144C4⋊C4—91no—
145C4⋊C4—91no—
146C4⋊C4—91no—
147C4⋊C4—91no—
148C4⋊C4—92no—
149C4⋊C4—92no—
150C4⋊C4C2×C3⋊D311yescharacteristic
151C22⋊C4—92no—
152C22⋊C4—92no—
153C22⋊C4C2×C3⋊D312yes—
154C42—91no—
155C42—91no—
156C42C2×C3⋊D311yescharacteristic
157C22×C3—28no—
158C22×C3—28no—
159C22×C3C2×Dic6116yes—
160C22×C3C23×D314yes—
161C3×C4—28no—
162C3×C4—24no—
163C3×C4—24no—
164C3×C4C2×Dic618yes—
165Dic3—38no—
166Dic3—64no—
167Dic3—68no—
168Dic3—64no—
169Dic3—64no—
170Dic3—68no—
171C32#55411yescharacteristic, complemented, Sylow
172C23C22×C3⋊D312yes—
173Q8—182no—
174Q8—362no—
175D4#63414yescomplemented
176C2×C4—42no—
177C2×C4—91no—
178C2×C4—91no—
179C2×C4—91no—
180C2×C4—91no—
181C2×C4—92no—
182C2×C4—92no—
183C2×C4—184no—
184C2×C4—362no—
185C2×C4C22×C3⋊D311yescharacteristic
186C2×C4C22×C3⋊D311yescharacteristic
187C2×C4C22×C3⋊D312yes—
188C2×C4C22×C3⋊D311yescharacteristic
189C2×C3—216no—
190C2×C3C22×Dic614yes—
191C2×C3#100414yes—
192C2×C3D3×#4114yes—
193C22—22no—
194C22—22no—
195C22C2×#63414yes—
196C22C23×C3⋊D311yescharacteristic
197C4—21no—
198C4—22no—
199C4—21no—
200C4—92no—
201C4—181no—
202C4—182no—
203C4—181no—
204C4—182no—
205C4—181no—
206C4C2×#63412yes—
207C3#553914yescomplemented
208C2—24no—
209C2C22×#63411yescharacteristic
210C2#6436111yescharacteristic
211C2#41×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C4×D4C2
prime radicalC2×C32×D4C22
hypercenterC4×D4C3⋊D3
hypocenterC32#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
4. #64133C2semidirect
5. C2×#3750C2semidirect
6. #64099C2semidirect
7. #64097C2semidirect
8. #64087C2semidirect
9. #64087C2semidirect
10. #64079C2semidirect
11. C4×#634C2semidirect
48. C32×D4Q8semidirect
114. C3×D4Dic6semidirect
171. C32#554semidirect
175. D4#634semidirect
207. C3#5539semidirect

Maximal subgroups

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

subgroupindex
C32×C4×D42
D4×C3⋊Dic32
#641332
C2×#37502
#640992
#640972
#640872
#640792
C4×#6342
#55393