Finite Group Databasebeta

← #13154837

Subgroups — #13154837

Order 1728.

316 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 30 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
1#11723451C211yesmaximal, characteristic
2#11723105C211yesmaximal, characteristic
3C3×#63780C211yesmaximal, characteristic
4#10590218—31nomaximal
5C2×C3×#634C2211yescharacteristic
6C2×C32×Dic6C411yescharacteristic
7#88993—21no—
8C3×#3792—21no—
9C4×C3⋊(C3⋊Dic3)C411yescharacteristic
10#88806—21no—
11#88806—21no—
12#64153—31no—
13#63873—62no—
14#63780—31no—
15#63780D311yescharacteristic
16C2×C33×C4C2×C411yescharacteristic
17C2×C3⋊(C3⋊Dic3)—21no—
18C2×C3×C3⋊Dic3D411yescharacteristic
19C2×C3×C3⋊Dic3D411yescharacteristic
20C3×#634—41no—
21C2×C32×Dic3—21no—
22C32×Dic6—41no—
23C3⋊#622—41no—
24C3×#622—41no—
25#4478—91nomaximal
26C2×C32×Q8—31no—
27C2×#634C2×D311yescharacteristic
28C2×C3×Dic6—22no—
29C2×C3×Dic6—62no—
30#3792—21no—
31#3792—61no—
32#3750—61no—
33#3750—61no—
34C4×C3⋊Dic3—31no—
35C4×C3⋊Dic3—62no—
36C4×C3⋊Dic3—122no—
37#3724—64no—
38#3724—64no—
39C22×C33C22⋊C411yescharacteristic
40C33×C4—21no—
41C3⋊(C3⋊Dic3)—22no—
42C3×C3⋊Dic3—21no—
43C3×C3⋊Dic3—21no—
44C32×Dic3—41no—
45#944—91no—
46#944—182no—
47C3×#101—91no—
48#836—91no—
49C32×Q8—121no—
50C2×C32×C4—22no—
51C2×C32×C4—42no—
52C2×C32×C4—61no—
53C2×C32×C4C4×D311yescharacteristic
54C2×C3⋊Dic3—61no—
55C2×C3⋊Dic3—64no—
56C2×C3⋊Dic3—124no—
57C2×C3⋊Dic3D1211yescharacteristic
58C2×C3⋊Dic3#6711yescharacteristic
59#634—41no—
60C2×C3×Dic3—24no—
61C2×C3×Dic3—62no—
62C2×C3×Dic3—62no—
63C3×Dic6—44no—
64C3×Dic6—124no—
65#622—41no—
66#622—121no—
67#356—271noSylow
68C2×C33—21no—
69C2×C33#9911yescharacteristic
70C2×C3×Q8—62no—
71C2×C3×Q8—91no—
72C2×Dic6—62no—
73C2×Dic6#15711yescharacteristic
74C3×M16—181no—
75Dic3⋊C4—181no—
76Dic3⋊C4—181no—
77Dic3⋊C4—184no—
78Dic3⋊C4—184no—
79C4×Dic3—91no—
80C4×Dic3—182no—
81C4×Dic3—362no—
82#208—181no—
83C22×C32—22no—
84C22×C32—42no—
85C22×C32(C2×D3)⋊C411yescharacteristic
86C32×C4—21no—
87C32×C4—24no—
88C32×C4—44no—
89C32×C4—121no—
90C3⋊Dic3—21no—
91C3⋊Dic3—21no—
92C3⋊Dic3—62no—
93C3⋊Dic3—68no—
94C3⋊Dic3—128no—
95C3×Dic3—44no—
96C3×Dic3—122no—
97C3×Dic3—122no—
98C4⋊Q8—271no—
99#101—91no—
100#101—271no—
101C33#35611yescharacteristic, complemented, Sylow
102C3×Q8—124no—
103C3×Q8—361no—
104C2×C3×C4—22no—
105C2×C3×C4—42no—
106C2×C3×C4—64no—
107C2×C3×C4—91no—
108C2×C3×C4—91no—
109C2×C3×C4C2×#15711yescharacteristic
110C2×Dic3—21no—
111C2×Dic3—62no—
112C2×Dic3—62no—
113C2×Dic3—184no—
114C2×Dic3—181no—
115C2×Dic3—364no—
116Dic6—41no—
117Dic6—124no—
118C3×C8—361no—
119C3⋊C8—361no—
120C2×C32—21no—
121C2×C32—24no—
122C2×C32—22no—
123C2×C32—44no—
124C2×C32—42no—
125C2×C32#80611yescharacteristic
126C2×Q8—31no—
127C2×Q8—91no—
128M16—181no—
129M16—541no—
130C4⋊C4—541no—
131C4⋊C4—541no—
132C42—271no—
133C22×C3—22no—
134C22×C3—42no—
135C22×C3C22⋊#15711yescharacteristic
136C3×C4—24no—
137C3×C4—21no—
138C3×C4—44no—
139C3×C4—124no—
140C3×C4—181no—
141C3×C4—181no—
142Dic3—41no—
143Dic3—122no—
144Dic3—122no—
145Dic3—182no—
146Dic3—188no—
147Dic3—368no—
148C32—22no—
149C32—42no—
150C32#447811yescharacteristic, complemented
151Q8—121no—
152Q8—361no—
153C2×C4—61no—
154C2×C4—91no—
155C2×C4—91no—
156C2×C4—541no—
157C2×C4D3×#15711yescharacteristic
158C8—361no—
159C8—1081no—
160C2×C3—24no—
161C2×C3—21no—
162C2×C3—22no—
163C2×C3—44no—
164C2×C3—42no—
165C2×C3#6378611yescharacteristic
166C22(C2×D3)⋊#15711yescharacteristic
167C4—21no—
168C4—121no—
169C4—181no—
170C4—181no—
171C4—542no—
172C3—22no—
173C3—42no—
174C3#1059021811yescharacteristic, complemented
175C2—21no—
176C2#1172272711yescharacteristic
subgroupG/H
centerC2#11722727
commutator, FittingC2×C33×C4C2×C4
Frattini, hypercenterC2×C4D3×#157
socleC2×C33#99
prime radicalC22×C33C22⋊C4
cosocleC2×C3×#634C22
hypocenterC33#356
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
101. C33#356semidirect
150. C32#4478semidirect
174. C3#10590218semidirect

Maximal subgroups

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

subgroupindex
#117234512
#117231052
C3×#637802
#105902183
#44789