Finite Group Databasebeta

← #72220254

Subgroups — #72220254

Order 2592.

1360 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 1360 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×#10620150C211yesmaximal, characteristic
2C22⋊#66472C212yesmaximal
3#13049411C212yesmaximal, complemented
4#13049409C212yesmaximal, complemented
5C22×C3⋊(C3⋊(C3⋊D3))C2211yescharacteristic
6C2×#66472C2214yes—
7C2×C32×C3⋊Dic3C412yescomplemented
8#10620150C412yescomplemented
9#10620150C2211yescharacteristic
10#10620150C2211yescharacteristic
11C2×#6289—64no—
12C22×C34C2×C411yescharacteristic
13C2×C3⋊(C3⋊(C3⋊D3))—22no—
14C2×C3⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
15C2×C3⋊(C3⋊(C3⋊D3))C2311yescharacteristic
16#66472—24no—
17C32×C3⋊Dic3C2×C412yes—
18C32×C3⋊Dic3C2×C412yes—
19#64286—92nomaximal
20C22×C3⋊(C3⋊D3)—64no—
21C22×C3⋊(C3⋊D3)—128no—
22C2×C3×C3⋊Dic3—24no—
23C2×C32×Dic3—64no—
24#6289—64no—
25#6289—64no—
26#6289—64no—
27#6289—64no—
28C2×C34—21no—
29C2×C34C22×C411yescharacteristic
30C3⋊(C3⋊(C3⋊D3))—21no—
31C3⋊(C3⋊(C3⋊D3))#4112yes—
32C2×C4×C3⋊D3—92no—
33C2×#624—188no—
34C22⋊#157—94no—
35C22⋊#157—188no—
36C4⋊#157—94no—
37C4×#157—94no—
38C22×C33—24no—
39C22×C33—48no—
40C2×C3⋊(C3⋊D3)—64no—
41C2×C3⋊(C3⋊D3)—64no—
42C2×C3⋊(C3⋊D3)—616no—
43C2×C3⋊(C3⋊D3)—128no—
44C2×C3⋊(C3⋊D3)—128no—
45C2×C3⋊(C3⋊D3)—1232no—
46C3×C3⋊Dic3—24no—
47C3×C3⋊Dic3—24no—
48C32×Dic3—64no—
49C32×Dic3—64no—
50C34#11711yescharacteristic, complemented, Sylow
51C22×C3⋊D3—92no—
52C22×C3⋊D3—188no—
53C22×C3⋊D3—188no—
54C22×C3⋊D3—3616no—
55C22×C3⋊D3—368no—
56C2×#157—98no—
57C2×#157—1816no—
58C2×C32×C4—92no—
59C2×C3⋊Dic3#15712yescomplemented
60C4×C3⋊D3—92no—
61C4×C3⋊D3—92no—
62C4×C3⋊D3—92no—
63C4×C3⋊D3—92no—
64C2×C3×Dic3—616no—
65#624—1816no—
66#624—188no—
67#624—188no—
68C2×C33—28no—
69C2×C33—24no—
70C2×C33—48no—
71C2×C33—416no—
72C3⋊(C3⋊D3)—68no—
73C3⋊(C3⋊D3)—68no—
74C3⋊(C3⋊D3)—1216no—
75C3⋊(C3⋊D3)—1216no—
76C2×C4×D3—544no—
77C22×C32—28no—
78C22×C32—28no—
79C22×C32—416no—
80C22×C32—48no—
81C22×C32C2×#15712yes—
82C2×C3⋊D3—92no—
83C2×C3⋊D3—92no—
84C2×C3⋊D3—184no—
85C2×C3⋊D3—188no—
86C2×C3⋊D3—188no—
87C2×C3⋊D3—1832no—
88C2×C3⋊D3—188no—
89C2×C3⋊D3—188no—
90C2×C3⋊D3—3616no—
91C2×C3⋊D3—3616no—
92C2×C3⋊D3—3664no—
93C2×C3⋊D3—3616no—
94C2×C3⋊D3—368no—
95C2×C3⋊D3—368no—
96C2×C3⋊D3—3632no—
97#157—188no—
98#157—3616no—
99C32×C4—92no—
100C32×C4—92no—
101C3⋊Dic3C2×#15712yes—
102C3⋊Dic3C2×#15712yes—
103C3×Dic3—616no—
104C3×Dic3—616no—
105#117—811noSylow
106C33—24no—
107C33—48no—
108C22×D3—544no—
109C22×D3—1088no—
110C2×C3×C4—184no—
111C2×Dic3—64no—
112C4×D3—544no—
113C4×D3—544no—
114C4×D3—544no—
115C4×D3—544no—
116C2×C32—22no—
117C2×C32—28no—
118C2×C32—216no—
119C2×C32—28no—
120C2×C32—432no—
121C2×C32—416no—
122C2×C32—48no—
123C2×C32—416no—
124C2×C32—48no—
125C2×C32C22×#15712yes—
126C3⋊D3—94no—
127C3⋊D3—182no—
128C3⋊D3—1816no—
129C3⋊D3—1816no—
130C3⋊D3—1816no—
131C3⋊D3—3632no—
132C3⋊D3—3632no—
133C3⋊D3—368no—
134C3⋊D3—3616no—
135C3⋊D3—3616no—
136C22×C4—811no—
137C4⋊C4—812no—
138C22⋊C4—812no—
139C42—812no—
140C22×C3—24no—
141C22×C3—48no—
142C2×D3—544no—
143C2×D3—544no—
144C2×D3—5416no—
145C2×D3—1088no—
146C2×D3—1088no—
147C2×D3—10832no—
148C3×C4—184no—
149C3×C4—184no—
150Dic3—64no—
151Dic3—64no—
152C32—28no—
153C32—28no—
154C32—416no—
155C32—48no—
156C32#6428612yescomplemented
157C23—811no—
158C2×C4—92no—
159C2×C4—811no—
160C2×C4—811no—
161C2×C4—812no—
162C2×C4—814no—
163C2×C3—28no—
164C2×C3—24no—
165C2×C3—416no—
166C2×C3—48no—
167D3—548no—
168D3—548no—
169D3—10816no—
170D3—10816no—
171C22—811no—
172C22—811no—
173C22—1622no—
174C22#1062019811yescharacteristic
175C4—92no—
176C4—92no—
177C4—1624no—
178C3—24no—
179C3—48no—
180C2—21no—
181C2—812no—
182C2—1621no—
183C2C2×#1062019811yescharacteristic
subgroupG/H
center, FrattiniC2C2×#10620198
commutator, socleC2×C34C22×C4
FittingC22×C34C2×C4
prime radicalC22×C3⋊(C3⋊(C3⋊D3))C22
cosocleC2×C3⋊(C3⋊(C3⋊D3))C23
hypercenterC22#10620198
hypocenterC34#117
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
3. #13049411C2semidirect
4. #13049409C2semidirect
7. C2×C32×C3⋊Dic3C4semidirect
8. #10620150C4semidirect
50. C34#117semidirect
59. C2×C3⋊Dic3#157semidirect
156. C32#64286semidirect

Maximal subgroups

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

subgroupindex
C2×#106201502
C22⋊#664722
#130494112
#130494092
#642869