Finite Group Databasebeta

← #10593402

Subgroups — #10593402

Order 576.

346 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 45 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×#3797C211yesmaximal, characteristic, complemented
2C2×D32⋊C4C212yesmaximal
3#63921C211yesmaximal, characteristic
4#63781C211yesmaximal, characteristic, complemented
5#63737C212yesmaximal, complemented
6C22×D32C2211yescharacteristic
7C22×#157C2212yes—
8C2×C4×C3⋊D3C2211yescharacteristic
9C2×C3×Dic6—21no—
10C2×#627—21no—
11C2×#624C2211yescharacteristic
12C2×#624C2212yes—
13#3797—24no—
14D32⋊C4—24no—
15C3×(C2×D3)⋊C4—21no—
16(C2×D3)⋊Dic3—21no—
17C2×D3×Q8—61no—
18D3×C22⋊C4—61no—
19C22×C3⋊D3C2311yescharacteristic
20C22×C3×D3—21no—
21C2×D32—21no—
22C2×D32—22no—
23C2×D32—22no—
24C2×D32Q812yescomplemented
25C2×#157—22no—
26C2×#157—22no—
27C2×#157—22no—
28C2×C32×C4D411yescharacteristic
29C2×C3⋊Dic3D411yescharacteristic
30C4×C3⋊D3—22no—
31C2×C3×Dic3—22no—
32C2×C3×Dic3—21no—
33C3×Dic6—24no—
34#627—24no—
35#624—22no—
36#624—22no—
37#624D414yes—
38#393—91nomaximal, Sylow
39C23×D3—61no—
40C2×C3×Q8—61no—
41D3×Q8—68no—
42C2×C4×D3—62no—
43C2×C4×D3—61no—
44C2×C4×D3—61no—
45C2×C4×D3—61no—
46C2×Dic6—21no—
47C2×Dic6—62no—
48C3×C22⋊C4—61no—
49#217—61no—
50(C2×D3)⋊C4—21no—
51(C2×D3)⋊C4—61no—
52C22×C32C2×D411yescharacteristic
53C2×C3⋊D3C2×D412yes—
54C2×C3⋊D3C2×Q811yescharacteristic
55C2×C3⋊D3#4112yes—
56C2×C3⋊D3#4111yescharacteristic
57C2×C3×D3—22no—
58C2×C3×D3—41no—
59C2×C3×D3—41no—
60D32—24no—
61D32—42no—
62#157—42no—
63C32×C4—21no—
64C3⋊Dic3—21no—
65C3×Dic3—24no—
66C3×Dic3—41no—
67C22×Q8—91no—
68C2×C22⋊C4—91no—
69C2×C22⋊C4—92no—
70#95—91no—
71#95—92no—
72C23×C3—61no—
73C22×D3—21no—
74C22×D3—62no—
75C22×D3—61no—
76C22×D3—61no—
77C22×D3—62no—
78C22×D3—122no—
79C22×D3—122no—
80C3×Q8—64no—
81C2×C3×C4—21no—
82C2×C3×C4—21no—
83C2×C3×C4—62no—
84C2×C3×C4—61no—
85C2×Dic3—22no—
86C2×Dic3—21no—
87C2×Dic3—61no—
88C2×Dic3—61no—
89C4×D3—64no—
90C4×D3—64no—
91C4×D3—64no—
92C4×D3—122no—
93C4×D3—122no—
94Dic6—24no—
95Dic6—68no—
96C2×C32#12011yescharacteristic
97C2×C32C22⋊Q811yescharacteristic
98C2×C32#12411yescharacteristic
99C3⋊D3C22⋊Q812yes—
100C3⋊D3#12412yes—
101C3×D3—42no—
102C24—91no—
103C2×Q8—61no—
104C2×Q8—181no—
105C2×Q8—184no—
106C22×C4—92no—
107C22×C4—91no—
108C22×C4—91no—
109C22×C4—92no—
110C22⋊C4—61no—
111C22⋊C4—181no—
112C22⋊C4—184no—
113C22×C3—21no—
114C22×C3—21no—
115C22×C3—62no—
116C22×C3—121no—
117C22×C3—121no—
118C2×D3—22no—
119C2×D3—41no—
120C2×D3—41no—
121C2×D3—62no—
122C2×D3—62no—
123C2×D3—62no—
124C2×D3—61no—
125C2×D3—61no—
126C2×D3—61no—
127C2×D3—61no—
128C2×D3—62no—
129C2×D3—124no—
130C2×D3—124no—
131C3×C4—22no—
132C3×C4—41no—
133C3×C4—64no—
134C3×C4—121no—
135Dic3—24no—
136Dic3—41no—
137Dic3—62no—
138Dic3—121no—
139C32#39311yescharacteristic, complemented, Sylow
140C23—61no—
141C23—92no—
142C23—91no—
143C23—181no—
144C23—182no—
145C23—182no—
146Q8—64no—
147Q8—184no—
148C2×C4—62no—
149C2×C4—61no—
150C2×C4—94no—
151C2×C4—91no—
152C2×C4—182no—
153C2×C4—182no—
154C2×C4—182no—
155C2×C4—182no—
156C2×C4—182no—
157C2×C4—182no—
158C2×C4GO+(4,𝔽2)11yescharacteristic
159C2×C3—21no—
160C2×C3—21no—
161C2×C3—21no—
162C2×C3—21no—
163C2×C3—21no—
164C2×C3—21no—
165C2×C3—122no—
166D3—42no—
167D3—62no—
168D3—62no—
169D3—62no—
170D3—62no—
171C22—62no—
172C22—91no—
173C22—91no—
174C22—92no—
175C22—92no—
176C22—121no—
177C22—121no—
178C22—184no—
179C22—362no—
180C22C2×GO+(4,𝔽2)11yescharacteristic
181C4—21no—
182C4—64no—
183C4—121no—
184C4—181no—
185C4—362no—
186C3—21no—
187C3—21no—
188C2—92no—
189C2—92no—
190C2—122no—
191C2D32⋊Q811yescharacteristic
192C2#6422111yescharacteristic
193C2#6424111yescharacteristic
subgroupG/H
center, FrattiniC22C2×GO+(4,𝔽2)
commutator, cosocleC22×C3⋊D3C23
FittingC2×C32×C4D4
socleC22×C32C2×D4
prime radicalC22×D32C22
hypercenterC2×C4GO+(4,𝔽2)
hypocenterC32#393
LCSG ▷ C22×C3⋊D3 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#3797C2semidirect
4. #63781C2semidirect
5. #63737C2semidirect
24. C2×D32Q8semidirect
139. C32#393semidirect

Maximal subgroups

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

subgroupindex
C2×#37972
C2×D32⋊C42
#639212
#637812
#637372
#3939