Finite Group Databasebeta

← #10593486

Subgroups — #10593486

Order 576.

302 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 41 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×C22⋊#157C211yesmaximal, characteristic
2C2×#3773C211yesmaximal, characteristic, complemented
3#63974C212yesmaximal
4#63737C212yesmaximal, complemented
5#63737C211yesmaximal, characteristic, complemented
6C23×C3⋊D3C2211yescharacteristic
7C22×#157C2211yescharacteristic
8C22×#157C2211yescharacteristic
9C2×#624C2212yes—
10C2×#624C2212yes—
11C22⋊#157—22no—
12#3773—22no—
13C3×#217—22no—
14#3722—22no—
15D3×C22⋊C4—62no—
16C23×C32D411yescharacteristic
17C22×C3⋊D3—22no—
18C22×C3⋊D3—22no—
19C22×C3⋊D3—21no—
20C22×C3⋊D3—21no—
21C22×C3⋊D3D411yescharacteristic
22C22×C3⋊D3C2311yescharacteristic
23C2×#157—22no—
24C2×#157—21no—
25C2×#157—22no—
26C2×#157D412yes—
27C2×C3×Dic3—22no—
28C2×C3×Dic3—22no—
29#624—22no—
30#624—22no—
31#624—22no—
32#624—22no—
33#397—91nomaximal, Sylow
34C23×D3—62no—
35C2×C4×D3—62no—
36C2×C4×D3—62no—
37C3×C22⋊C4—62no—
38#217—22no—
39(C2×D3)⋊C4—62no—
40(C2×D3)⋊C4—62no—
41C22×C32—22no—
42C22×C32—21no—
43C22×C32C2×D411yescharacteristic
44C2×C3⋊D3—22no—
45C2×C3⋊D3—21no—
46C2×C3⋊D3—41no—
47C2×C3⋊D3—41no—
48C2×C3⋊D3—42no—
49C2×C3⋊D3C2×D411yescharacteristic
50C2×C3⋊D3#4112yes—
51C2×C3⋊D3#4112yes—
52C2×C3⋊D3#4111yescharacteristic
53#157—22no—
54#157—41no—
55C3×Dic3—42no—
56C3×Dic3—42no—
57C2×C4⋊C4—91no—
58C2×C22⋊C4—92no—
59C2×C22⋊C4—91no—
60#95—91no—
61#95—92no—
62C23×C3—22no—
63C22×D3—62no—
64C22×D3—62no—
65C22×D3—64no—
66C22×D3—64no—
67C22×D3—122no—
68C22×D3—122no—
69C22×D3—122no—
70C22×D3—122no—
71C2×C3×C4—62no—
72C2×C3×C4—62no—
73C2×Dic3—22no—
74C2×Dic3—22no—
75C4×D3—122no—
76C4×D3—122no—
77C4×D3—122no—
78C4×D3—122no—
79C2×C32—41no—
80C2×C32#12111yescharacteristic
81C2×C32#12312yes—
82C3⋊D3—41no—
83C3⋊D3#12311yescharacteristic
84C3⋊D3#12411yescharacteristic
85C3⋊D3#12612yes—
86C24—91no—
87C22×C4—92no—
88C22×C4—92no—
89C22×C4—91no—
90C22×C4—91no—
91C4⋊C4—182no—
92C22⋊C4—62no—
93C22⋊C4—182no—
94C22⋊C4—182no—
95C22×C3—24no—
96C22×C3—22no—
97C22×C3—42no—
98C22×C3—42no—
99C2×D3—62no—
100C2×D3—62no—
101C2×D3—62no—
102C2×D3—62no—
103C2×D3—62no—
104C2×D3—62no—
105C2×D3—64no—
106C2×D3—122no—
107C2×D3—122no—
108C2×D3—124no—
109C2×D3—124no—
110C2×D3—124no—
111C2×D3—124no—
112C3×C4—122no—
113C3×C4—122no—
114Dic3—42no—
115Dic3—42no—
116C32#39711yescharacteristic, complemented, Sylow
117C23—91no—
118C23—91no—
119C23—181no—
120C23—182no—
121C23—182no—
122C23—181no—
123C23GO+(4,𝔽2)11yescharacteristic
124C2×C4—62no—
125C2×C4—62no—
126C2×C4—92no—
127C2×C4—182no—
128C2×C4—182no—
129C2×C4—182no—
130C2×C4—182no—
131C2×C4—182no—
132C2×C4—181no—
133C2×C4—182no—
134C2×C3—22no—
135C2×C3—22no—
136C2×C3—22no—
137C2×C3—44no—
138D3—62no—
139D3—62no—
140D3—62no—
141D3—62no—
142D3—124no—
143C22—22no—
144C22—21no—
145C22—92no—
146C22—92no—
147C22—91no—
148C22—91no—
149C22—182no—
150C22—181no—
151C22—361no—
152C22—361no—
153C22—362no—
154C22C2×GO+(4,𝔽2)11yescharacteristic
155C4—122no—
156C4—122no—
157C4—182no—
158C4—361no—
159C3—22no—
160C2—41no—
161C2—91no—
162C2—91no—
163C2—92no—
164C2—361no—
165C2#6423312yes—
166C2#6424211yescharacteristic
subgroupG/H
center, FrattiniC22C2×GO+(4,𝔽2)
commutator, cosocleC22×C3⋊D3C23
FittingC23×C32D4
socleC22×C32C2×D4
prime radicalC23×C3⋊D3C22
hypercenterC23GO+(4,𝔽2)
hypocenterC32#397
LCSG ▷ C22×C3⋊D3 ▷ C32
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#3773C2semidirect
4. #63737C2semidirect
5. #63737C2semidirect
116. C32#397semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊#1572
C2×#37732
#639742
#637372
#3979