Finite Group Databasebeta

← #13173418

Subgroups — #13173418

Order 1728.

540 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 83 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, complemented
2C3×D4×C3⋊Dic3C211yesmaximal, characteristic
3#11723112C212yesmaximal, complemented
4C4⋊(C3⋊#622)C211yesmaximal, characteristic, complemented
5C4×C3⋊#622C211yesmaximal, characteristic, complemented
6#10593821—31nomaximal
7C2×C33×D4C411yescharacteristic
8C22×C3×C3⋊Dic3C2212yes—
9#88993—24no—
10C2×C3⋊#622C2211yescharacteristic
11C2×C3⋊#622C2212yes—
12C2×C3⋊#622C2211yescharacteristic
13C3×#3757C412yes—
14C3×C4⋊(C3⋊Dic3)C411yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×#3792—32no—
17D4×C3⋊Dic3D311yescharacteristic
18C3×D4×Dic3—62no—
19C22⋊#622—32no—
20C4⋊#622—31no—
21C4×#622—31no—
22C23×C33C2×C412yes—
23C33×D4—22no—
24C2×C33×C4C2×C411yescharacteristic
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3—22no—
27C2×C3×C3⋊Dic3C2×C411yescharacteristic
28C2×C3×C3⋊Dic3C2×C412yes—
29C2×C3×C3⋊Dic3C2311yescharacteristic
30C3⋊#622—22no—
31C3⋊#622—21no—
32C3⋊#622D412yescomplemented
33#5012—91nomaximal
34C2×C32×D4—22no—
35C2×C32×D4—42no—
36C2×C32×D4Dic311yescharacteristic
37C22×C3⋊Dic3C2×D312yes—
38C22×C3×Dic3—64no—
39#3792—64no—
40C2×#622—31no—
41C2×#622—31no—
42C2×#622—32no—
43#3757Dic312yes—
44C4⋊(C3⋊Dic3)Dic311yescharacteristic
45C4×C3⋊Dic3C2×D311yescharacteristic
46C3×#217—64no—
47C3×C4⋊Dic3—62no—
48C3×C4×Dic3—62no—
49C22×C33—22no—
50C22×C33—22no—
51C22×C33—22no—
52C22×C33C22×C411yescharacteristic
53C33×C4M1612yescomplemented
54C3×C3⋊Dic3—22no—
55C3×C3⋊Dic3—21no—
56C3×C3⋊Dic3C2×D411yescharacteristic
57C3×C3⋊Dic3#4111yescharacteristic
58C3×C4×D4—91no—
59D4×Dic3—62no—
60C2×#208—92no—
61#830—92no—
62C4⋊(C3⋊C8)—91no—
63C4×C3⋊C8—91no—
64C23×C32—24no—
65C23×C32—44no—
66C23×C32C2×Dic312yes—
67C32×D4—22no—
68C32×D4—28no—
69C32×D4—48no—
70C2×C32×C4—22no—
71C2×C32×C4—42no—
72C2×C32×C4C2×Dic311yescharacteristic
73C2×C3⋊Dic3—22no—
74C2×C3⋊Dic3—22no—
75C2×C3⋊Dic3C2×Dic311yescharacteristic
76C2×C3⋊Dic3C2×Dic312yes—
77C2×C3⋊Dic3C22×D311yescharacteristic
78C2×C3×Dic3—62no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—64no—
81C2×C3×Dic3—124no—
82C2×C3×Dic3—124no—
83#622—32no—
84#622—62no—
85#622—61no—
86#436—271noSylow
87C2×C33—42no—
88C2×C33C4×D411yescharacteristic
89C2×C33C2×M1611yescharacteristic
90C2×C33#13111yescharacteristic
91C2×C3×D4—22no—
92C2×C3×D4—42no—
93C2×C3×D4#15711yescharacteristic
94C22×C3×C4—92no—
95C22×Dic3—64no—
96C3×C4⋊C4—91no—
97C3×C22⋊C4—92no—
98C3×C42—91no—
99#217—64no—
100C4⋊Dic3—62no—
101C4×Dic3—62no—
102#208—184no—
103C2×C3⋊C8—91no—
104C2×C3⋊C8—91no—
105C2×C3⋊C8—92no—
106C22×C32—22no—
107C22×C32—28no—
108C22×C32—22no—
109C22×C32—22no—
110C22×C32—22no—
111C22×C32—48no—
112C22×C32—42no—
113C22×C32—44no—
114C22×C32—44no—
115C22×C32—84no—
116C22×C32—84no—
117C22×C32C22×Dic311yescharacteristic
118C32×C4—24no—
119C32×C4—44no—
120C32×C4#20812yescomplemented
121C3⋊Dic3—22no—
122C3⋊Dic3—21no—
123C3⋊Dic3D3×D411yescharacteristic
124C3⋊Dic3#23711yescharacteristic
125C3×Dic3—62no—
126C3×Dic3—62no—
127C3×Dic3—122no—
128C3×Dic3—124no—
129C2×M16—272no—
130C4×D4—91no—
131C4⋊C8—271no—
132C22⋊C8—272no—
133C4×C8—271no—
134C33#43611yescharacteristic, complemented, Sylow
135C23×C3—24no—
136C23×C3—44no—
137C23×C3C2×#15712yes—
138C3×D4—28no—
139C3×D4—22no—
140C3×D4—48no—
141C2×C3×C4—22no—
142C2×C3×C4—42no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—92no—
146C2×C3×C4—182no—
147C2×C3×C4—182no—
148C2×C3×C4C2×#15711yescharacteristic
149C2×Dic3—62no—
150C2×Dic3—62no—
151C2×Dic3—64no—
152C2×Dic3—124no—
153C2×Dic3—124no—
154C3⋊C8—92no—
155C3⋊C8—181no—
156C3⋊C8—182no—
157C2×C32—22no—
158C2×C32—22no—
159C2×C32—22no—
160C2×C32—42no—
161C2×C32—42no—
162C2×C32—42no—
163C2×C32—42no—
164C2×C32—48no—
165C2×C32—88no—
166C2×C32C2×#20811yescharacteristic
167C2×C32D4×Dic311yescharacteristic
168C2×C32#94811yescharacteristic
169C2×D4C3⋊#15711yescharacteristic
170C22×C4—92no—
171M16—544no—
172C2×C8—271no—
173C2×C8—271no—
174C2×C8—272no—
175C4⋊C4—91no—
176C22⋊C4—92no—
177C42—91no—
178C22×C3—28no—
179C22×C3—22no—
180C22×C3—22no—
181C22×C3—22no—
182C22×C3—22no—
183C22×C3—44no—
184C22×C3—44no—
185C22×C3—48no—
186C22×C3—42no—
187C22×C3—84no—
188C22×C3—84no—
189C22×C3C22×#15711yescharacteristic
190C3×C4—24no—
191C3×C4—44no—
192C3×C4—91no—
193C3×C4—91no—
194C3×C4—181no—
195C3×C4—182no—
196C3×C4#379212yescomplemented
197Dic3—62no—
198Dic3—62no—
199Dic3—124no—
200Dic3—122no—
201C32—22no—
202C32—42no—
203C32#501211yescharacteristic, complemented
204C23C2×C3⋊#15712yes—
205D4—22no—
206C2×C4—91no—
207C2×C4—91no—
208C2×C4—92no—
209C2×C4—182no—
210C2×C4—182no—
211C2×C4C2×C3⋊#15711yescharacteristic
212C8—272no—
213C8—542no—
214C8—541no—
215C2×C3—22no—
216C2×C3—22no—
217C2×C3—22no—
218C2×C3—48no—
219C2×C3—42no—
220C2×C3—42no—
221C2×C3—42no—
222C2×C3—42no—
223C2×C3—88no—
224C2×C3#6428711yescharacteristic
225C2×C3D4×#15711yescharacteristic
226C2×C3C2×#379211yescharacteristic
227C22—22no—
228C22—22no—
229C22—22no—
230C22C22×C3⋊#15711yescharacteristic
231C4—91no—
232C4—91no—
233C4—181no—
234C4—182no—
235C4#8899312yescomplemented
236C3—22no—
237C3—42no—
238C3#1059382111yescharacteristic, complemented
239C2—42no—
240C2#1172451511yescharacteristic
241C2D4×C3⋊#15711yescharacteristic
242C2C2×#8899311yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
Fitting, prime radicalC2×C33×D4C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×D4C3⋊#157
hypocenterC33#436
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88993C2semidirect
3. #11723112C2semidirect
4. C4⋊(C3⋊#622)C2semidirect
5. C4×C3⋊#622C2semidirect
32. C3⋊#622D4semidirect
53. C33×C4M16semidirect
120. C32×C4#208semidirect
134. C33#436semidirect
196. C3×C4#3792semidirect
203. C32#5012semidirect
235. C4#88993semidirect
238. C3#10593821semidirect

Maximal subgroups

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

subgroupindex
C2×#889932
C3×D4×C3⋊Dic32
#117231122
C4⋊(C3⋊#622)2
C4×C3⋊#6222
#105938213
#50129