Finite Group Databasebeta

← #10591518

Subgroups — #10591518

Order 576.

363 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 68 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
2#64147C211yesmaximal, characteristic
3C2×C3×#214C212yesmaximal, complemented
4#63566C212yesmaximal, complemented
5#63557C211yesmaximal, characteristic, complemented
6#5170—32nomaximal
7C2×C32×D4C2211yescharacteristic
8C2×#638—21no—
9C2×C4×C3⋊D3C2211yescharacteristic
10C2×C3×Dic6C2212yes—
11C2×#627—21no—
12C2×#624—21no—
13#3797—24no—
14#3757—21no—
15C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#214—24no—
17C2×C3×C3⋊C8C2212yes—
18C2×D3×Q8—32no—
19C2×C3×SD16—32no—
20#938—32no—
21#938—61no—
22C2×#214D312yescomplemented
23#835—32no—
24(C2×D3)⋊C8—32no—
25#816—32no—
26C23×C32—21no—
27C22×C3⋊D3D411yescharacteristic
28C32×D4—22no—
29C2×C32×C4C2311yescharacteristic
30#638—42no—
31C2×C3⋊Dic3—21no—
32C2×C3⋊Dic3D411yescharacteristic
33C4×C3⋊D3—22no—
34C2×C3×Dic3—22no—
35C3×Dic6—22no—
36C3×Dic6D414yescomplemented
37#627—24no—
38#624—24no—
39C3×C3⋊C8—22no—
40#448—91noSylow
41C2×C3×Q8—32no—
42C2×C3×D4—21no—
43C2×C3×D4C2×D312yes—
44C2×#67—62no—
45C2×#67—62no—
46D3×Q8—64no—
47D3×Q8—64no—
48C2×C4×D3—32no—
49C2×C4×D3—61no—
50C2×C4×D3—62no—
51C2×Dic6—62no—
52C2×Dic6C2×D312yes—
53C3×SD16—64no—
54C2×C3×C8—32no—
55#217—62no—
56#217—62no—
57#214—24no—
58C4⋊Dic3—32no—
59C4⋊Dic3—61no—
60C2×C3⋊C8C2×D312yes—
61C22×C32—41no—
62C22×C32—41no—
63C22×C32—41no—
64C22×C32C2×D411yescharacteristic
65C2×C3⋊D3—21no—
66C2×C3⋊D3—21no—
67C2×C3⋊D3SD1612yescomplemented
68C32×C4C2×D412yes—
69C3⋊Dic3—21no—
70C3⋊Dic3—41no—
71C3×Dic3—24no—
72C22×Q8—91no—
73C2×SD16—32no—
74#121—91no—
75#103—92no—
76C22⋊C8—91no—
77C23×C3—22no—
78C23×C3—22no—
79C22×D3—32no—
80C22×D3—61no—
81C3×Q8—34no—
82C3×Q8—62no—
83C3×D4—22no—
84C3×D4—22no—
85C3×D4—42no—
86C2×C3×C4—21no—
87C2×C3×C4—62no—
88C2×C3×C4C22×D312yes—
89#67—124no—
90#67—124no—
91C2×Dic3—22no—
92C2×Dic3—32no—
93C2×Dic3—61no—
94C2×Dic3—62no—
95C2×Dic3—62no—
96C4×D3—68no—
97C4×D3—62no—
98C4×D3—62no—
99C4×D3—122no—
100Dic6—22no—
101Dic6—68no—
102Dic6#6714yescomplemented
103C3×C8—62no—
104C3⋊C8—22no—
105C2×C32—81no—
106C2×C32#12011yescharacteristic
107C2×C32C2×SD1611yescharacteristic
108C2×C32#13711yescharacteristic
109C3⋊D3—22no—
110C2×Q8—32no—
111C2×Q8—181no—
112C2×Q8—184no—
113C2×D4—181no—
114C2×D4D3211yescharacteristic
115C22×C4—91no—
116C22×C4—181no—
117SD16—64no—
118C2×C8—32no—
119C4⋊C4—91no—
120C22⋊C4—181no—
121C22×C3—21no—
122C22×C3—42no—
123C22×C3—42no—
124C22×C3—42no—
125C22×C3—42no—
126C22×C3—42no—
127C22×C3—42no—
128C22×C3D3×D412yes—
129C2×D3—34no—
130C2×D3—62no—
131C2×D3—62no—
132C2×D3—62no—
133C2×D3—121no—
134C2×D3—121no—
135C3×C4—22no—
136C3×C4—64no—
137C3×C4D3×D412yes—
138C3×C4C2×#6712yes—
139Dic3—24no—
140Dic3—62no—
141Dic3—121no—
142Dic3—122no—
143Dic3—122no—
144C32#44811yescharacteristic, complemented, Sylow
145C23—21no—
146C23—91no—
147Q8—34no—
148Q8—62no—
149Q8—184no—
150D4—22no—
151D4—362no—
152C2×C4—62no—
153C2×C4—91no—
154C2×C4—184no—
155C2×C4—182no—
156C2×C4—181no—
157C2×C4C2×D3211yescharacteristic
158C8—62no—
159C2×C3—21no—
160C2×C3—21no—
161C2×C3—21no—
162C2×C3—82no—
163C2×C3—82no—
164C2×C3D3×SD1612yes—
165C2×C3#91512yes—
166C2×C3#93712yes—
167D3—64no—
168D3—122no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22—92no—
173C22—181no—
174C22—181no—
175C22#381111yescharacteristic
176C4—64no—
177C4—181no—
178C4—361no—
179C4D3×#6712yes—
180C3—21no—
181C3#517012yescomplemented
182C2—81no—
183C2—182no—
184C2#6393011yescharacteristic
185C2#6393111yescharacteristic
186C2#6398311yescharacteristic
subgroupG/H
centerC22#3811
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×D4C22
socleC22×C32C2×D4
prime radical#64147C2
hypercenterC2×D4D32
hypocenterC32#448
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#3797C2semidirect
3. C2×C3×#214C2semidirect
4. #63566C2semidirect
5. #63557C2semidirect
22. C2×#214D3semidirect
36. C3×Dic6D4semidirect
67. C2×C3⋊D3SD16semidirect
102. Dic6#67semidirect
144. C32#448semidirect
181. C3#5170semidirect

Maximal subgroups

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

subgroupindex
C2×#37972
#641472
C2×C3×#2142
#635662
#635572
#51703