Finite Group Databasebeta

← #13154838

Subgroups — #13154838

Order 1728.

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

normal subgroups only

This group has 332 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#11723443C211yesmaximal, characteristic
2#11723105C211yesmaximal, characteristic
3C3×#63780C211yesmaximal, characteristic
4#10590144—31nomaximal
5C2×#6308C411yescharacteristic
6C2×C3×#634C2211yescharacteristic
7#88993—21no—
8C3×#3792—21no—
9C32×C4×Dic3C411yescharacteristic
10Dic3⋊(C3⋊Dic3)—21no—
11Dic3⋊(C3⋊Dic3)—21no—
12#64078—31no—
13#63866—62no—
14#63780—31no—
15#63780D311yescharacteristic
16C2×C33×C4C2×C411yescharacteristic
17C2×C3⋊(C3⋊Dic3)—21no—
18#6308—41no—
19C2×C3×C3⋊Dic3D411yescharacteristic
20C2×C3×C3⋊Dic3D411yescharacteristic
21C3×#634—41no—
22C2×C32×Dic3—21no—
23C3⋊#622—41no—
24C3×#622—41no—
25#4478—91nomaximal
26C2×#634—31no—
27C2×#634—62no—
28C2×#634—122no—
29C2×#634C2×D311yescharacteristic
30C2×C3×Dic6—62no—
31#3792—21no—
32#3792—61no—
33C32×C42—31no—
34C4⋊(C3⋊Dic3)—61no—
35C4⋊(C3⋊Dic3)—61no—
36C3×C4×Dic3—22no—
37Dic3⋊Dic3—64no—
38Dic3⋊Dic3—64no—
39C22×C33C22⋊C411yescharacteristic
40C33×C4—21no—
41C3⋊(C3⋊Dic3)—41no—
42C3×C3⋊Dic3—21no—
43C3×C3⋊Dic3—21no—
44C32×Dic3—22no—
45#944—91no—
46#869—182no—
47C3×#101—91no—
48#836—91no—
49C2×C32×C4—22no—
50C2×C32×C4—42no—
51C2×C32×C4—61no—
52C2×C32×C4C4×D311yescharacteristic
53C2×C3⋊Dic3—61no—
54C2×C3⋊Dic3—64no—
55C2×C3⋊Dic3—124no—
56C2×C3⋊Dic3D1211yescharacteristic
57C2×C3⋊Dic3#6711yescharacteristic
58#634—41no—
59#634—124no—
60#634—128no—
61#634—121no—
62C2×C3×Dic3—24no—
63C2×C3×Dic3—62no—
64C2×C3×Dic3—62no—
65C3×Dic6—124no—
66#622—41no—
67#622—121no—
68#356—271noSylow
69C2×C33—21no—
70C2×C33#9911yescharacteristic
71C2×C3×Q8—91no—
72C2×Dic6—62no—
73C2×Dic6—91no—
74C2×Dic6—182no—
75C2×Dic6—362no—
76C3×M16—181no—
77C3×C42—62no—
78C4⋊Dic3—184no—
79C4⋊Dic3—184no—
80Dic3⋊C4—181no—
81Dic3⋊C4—181no—
82C4×Dic3#15711yescharacteristic
83#208—181no—
84C22×C32—22no—
85C22×C32—42no—
86C22×C32(C2×D3)⋊C411yescharacteristic
87C32×C4—21no—
88C32×C4—24no—
89C32×C4—44no—
90C32×C4—62no—
91C3⋊Dic3—21no—
92C3⋊Dic3—21no—
93C3⋊Dic3—124no—
94C3⋊Dic3—128no—
95C3⋊Dic3—121no—
96C3×Dic3—28no—
97C3×Dic3—122no—
98C3×Dic3—122no—
99C4⋊Q8—271no—
100#101—91no—
101#101—271no—
102C33#35611yescharacteristic, complemented, Sylow
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—124no—
117Dic6—364no—
118Dic6—368no—
119Dic6—361no—
120C3×C8—361no—
121C3⋊C8—361no—
122C2×C32—21no—
123C2×C32—24no—
124C2×C32—22no—
125C2×C32—44no—
126C2×C32—42no—
127C2×C32#80611yescharacteristic
128C2×Q8—91no—
129C2×Q8—271no—
130M16—181no—
131M16—541no—
132C4⋊C4—541no—
133C4⋊C4—541no—
134C42—31no—
135C22×C3—22no—
136C22×C3—42no—
137C22×C3C22⋊#15711yescharacteristic
138C3×C4—24no—
139C3×C4—21no—
140C3×C4—44no—
141C3×C4—68no—
142C3×C4—181no—
143C3×C4—181no—
144Dic3—22no—
145Dic3—122no—
146Dic3—122no—
147Dic3—364no—
148Dic3—368no—
149Dic3—361no—
150C32—22no—
151C32—42no—
152C32#447811yescharacteristic, complemented
153Q8—361no—
154Q8—1081no—
155C2×C4—61no—
156C2×C4—91no—
157C2×C4—91no—
158C2×C4—541no—
159C2×C4D3×#15711yescharacteristic
160C8—361no—
161C8—1081no—
162C2×C3—24no—
163C2×C3—21no—
164C2×C3—22no—
165C2×C3—44no—
166C2×C3—42no—
167C2×C3#6377411yescharacteristic
168C22(C2×D3)⋊#15711yescharacteristic
169C4—21no—
170C4—62no—
171C4—181no—
172C4—181no—
173C4—1081no—
174C3—22no—
175C3—42no—
176C3#1059014411yescharacteristic, complemented
177C2—21no—
178C2#1172272811yescharacteristic
subgroupG/H
centerC2#11722728
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
102. C33#356semidirect
152. C32#4478semidirect
176. C3#10590144semidirect

Maximal subgroups

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

subgroupindex
#117234432
#117231052
C3×#637802
#105901443
#44789