Finite Group Databasebeta

← #13157225

Subgroups — #13157225

Order 1728.

488 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 69 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×#88993C212yesmaximal
2C2×C3×#3750C211yesmaximal, characteristic
3#10590208—31nomaximal
4C22×C33×C4C411yescharacteristic
5C22×C3×C3⋊Dic3C411yescharacteristic
6C22×C3×C3⋊Dic3C2211yescharacteristic
7#88993—22no—
8#88993C414yescomplemented
9C2×C3⋊#622—22no—
10C3×#3750—22no—
11C2×#3792—32no—
12C2×#3750D311yescharacteristic
13C2×C3×Dic3⋊C4—62no—
14C23×C33C2×C411yescharacteristic
15C2×C33×C4—21no—
16C2×C33×C4—41no—
17C2×C3×C3⋊Dic3—21no—
18C2×C3×C3⋊Dic3—41no—
19C2×C3×C3⋊Dic3C2×C412yes—
20C2×C3×C3⋊Dic3D411yescharacteristic
21C2×C3×C3⋊Dic3D412yes—
22C2×C3×C3⋊Dic3Q811yescharacteristic
23C3⋊#622—24no—
24#4530—91nomaximal
25C22×C32×C4—22no—
26C22×C32×C4—42no—
27C22×C32×C4Dic311yescharacteristic
28C22×C3⋊Dic3Dic311yescharacteristic
29C22×C3⋊Dic3C2×D311yescharacteristic
30C22×C3×Dic3—62no—
31C22×C3×Dic3—62no—
32#3792—34no—
33#3792—62no—
34C2×#622—62no—
35#3750—22no—
36C3×Dic3⋊C4—68no—
37C22×C33—21no—
38C22×C33—21no—
39C22×C33C4211yescharacteristic
40C22×C33C22⋊C411yescharacteristic
41C22×C33C4⋊C411yescharacteristic
42C33×C4—41no—
43C3×C3⋊Dic3—41no—
44C3×C3⋊Dic3C22⋊C412yes—
45C3×C3⋊Dic3C4⋊C412yes—
46C2×C3×C4⋊C4—91no—
47C2×Dic3⋊C4—62no—
48C2×#208—92no—
49C23×C32—22no—
50C23×C32—42no—
51C23×C32C2×Dic311yescharacteristic
52C2×C32×C4—21no—
53C2×C32×C4—24no—
54C2×C32×C4—44no—
55C2×C32×C4—44no—
56C2×C32×C4—48no—
57C2×C32×C4—41no—
58C2×C3⋊Dic3—21no—
59C2×C3⋊Dic3—41no—
60C2×C3⋊Dic3Dic611yescharacteristic
61C2×C3⋊Dic3C4×D312yes—
62C2×C3⋊Dic3D1211yescharacteristic
63C2×C3⋊Dic3#6712yes—
64C2×C3×Dic3—64no—
65C2×C3×Dic3—64no—
66C2×C3×Dic3—62no—
67C2×C3×Dic3—62no—
68C2×C3×Dic3—64no—
69C2×C3×Dic3—124no—
70#622—64no—
71#343—271noSylow
72C2×C33—21no—
73C2×C33—21no—
74C2×C33#9511yescharacteristic
75C2×C33#10011yescharacteristic
76C2×C33#10111yescharacteristic
77C22×C3×C4—22no—
78C22×C3×C4—42no—
79C22×C3×C4—91no—
80C22×C3×C4—91no—
81C22×C3×C4#15711yescharacteristic
82C22×Dic3—62no—
83C22×Dic3—62no—
84C3×C4⋊C4—182no—
85Dic3⋊C4—68no—
86#208—94no—
87#208—182no—
88C2×C3⋊C8—182no—
89C22×C32—21no—
90C22×C32—21no—
91C22×C32—24no—
92C22×C32—24no—
93C22×C32—22no—
94C22×C32—22no—
95C22×C32—22no—
96C22×C32—44no—
97C22×C32—44no—
98C22×C32—42no—
99C22×C32—42no—
100C22×C32—42no—
101C22×C32C4×Dic311yescharacteristic
102C22×C32C4⋊Dic311yescharacteristic
103C22×C32#21711yescharacteristic
104C32×C4—44no—
105C32×C4—48no—
106C32×C4—41no—
107C3⋊Dic3—41no—
108C3⋊Dic3Dic3⋊C412yes—
109C3⋊Dic3(C2×D3)⋊C412yes—
110C3×Dic3—64no—
111C3×Dic3—64no—
112C3×Dic3—124no—
113C2×M16—272no—
114C2×C4⋊C4—91no—
115C33#34311yescharacteristic, complemented, Sylow
116C23×C3—22no—
117C23×C3—42no—
118C23×C3C2×#15711yescharacteristic
119C2×C3×C4—24no—
120C2×C3×C4—21no—
121C2×C3×C4—44no—
122C2×C3×C4—44no—
123C2×C3×C4—48no—
124C2×C3×C4—41no—
125C2×C3×C4—92no—
126C2×C3×C4—91no—
127C2×C3×C4—91no—
128C2×C3×C4—92no—
129C2×C3×C4—181no—
130C2×C3×C4—361no—
131C2×Dic3—62no—
132C2×Dic3—64no—
133C2×Dic3—62no—
134C2×Dic3—64no—
135C2×Dic3—64no—
136C2×Dic3—124no—
137C3⋊C8—184no—
138C2×C32—21no—
139C2×C32—21no—
140C2×C32—24no—
141C2×C32—24no—
142C2×C32—22no—
143C2×C32—22no—
144C2×C32—22no—
145C2×C32—44no—
146C2×C32—44no—
147C2×C32—42no—
148C2×C32—42no—
149C2×C32—42no—
150C2×C32C3⋊#9511yescharacteristic
151C2×C32#83311yescharacteristic
152C2×C32#83611yescharacteristic
153C22×C4—91no—
154C22×C4—91no—
155C22×C4C3⋊#15711yescharacteristic
156M16—274no—
157M16—542no—
158C2×C8—542no—
159C4⋊C4—182no—
160C22×C3—24no—
161C22×C3—24no—
162C22×C3—21no—
163C22×C3—21no—
164C22×C3—22no—
165C22×C3—22no—
166C22×C3—22no—
167C22×C3—44no—
168C22×C3—44no—
169C22×C3—42no—
170C22×C3—42no—
171C22×C3—42no—
172C22×C3C4×#15711yescharacteristic
173C22×C3C4⋊#15711yescharacteristic
174C22×C3C22⋊#15711yescharacteristic
175C3×C4—44no—
176C3×C4—48no—
177C3×C4—41no—
178C3×C4—92no—
179C3×C4—92no—
180C3×C4—361no—
181Dic3—64no—
182Dic3—64no—
183Dic3—124no—
184C32—22no—
185C32—42no—
186C32#453011yescharacteristic, complemented
187C23C2×C3⋊#15711yescharacteristic
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—92no—
191C2×C4—91no—
192C2×C4—91no—
193C2×C4—92no—
194C2×C4—181no—
195C2×C4—361no—
196C8—544no—
197C2×C3—24no—
198C2×C3—24no—
199C2×C3—21no—
200C2×C3—21no—
201C2×C3—22no—
202C2×C3—22no—
203C2×C3—22no—
204C2×C3—44no—
205C2×C3—44no—
206C2×C3—42no—
207C2×C3—42no—
208C2×C3—42no—
209C2×C3#6378011yescharacteristic
210C2×C3#6378111yescharacteristic
211C2×C3#6378811yescharacteristic
212C22—21no—
213C22—21no—
214C22C4×C3⋊#15711yescharacteristic
215C22C4⋊(C3⋊#157)11yescharacteristic
216C22#8899411yescharacteristic
217C4—41no—
218C4—92no—
219C4—92no—
220C4—361no—
221C3—22no—
222C3—42no—
223C3#1059020811yescharacteristic, complemented
224C2—21no—
225C2—21no—
226C2#1172310511yescharacteristic
227C2#1172310611yescharacteristic
228C2#1172311311yescharacteristic
subgroupG/H
centerC22#88994
commutatorC22×C33C42
FrattiniC23C2×C3⋊#157
FittingC22×C33×C4C4
socleC22×C33C22⋊C4
prime radicalC23×C33C2×C4
cosocleC22×C3×C3⋊Dic3C22
hypercenterC22×C4C3⋊#157
hypocenterC33#343
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
8. #88993C4semidirect
115. C33#343semidirect
186. C32#4530semidirect
223. C3#10590208semidirect

Maximal subgroups

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

subgroupindex
C2×#889932
C2×C3×#37502
#105902083
#45309