Finite Group Databasebeta

← #13154832

Subgroups — #13154832

Order 1728.

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

normal subgroups only

This group has 496 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#11723517C211yesmaximal, characteristic
2#11723099C211yesmaximal, characteristic, complemented
3C3×#63774C211yesmaximal, characteristic, complemented
4#10590142—31nomaximal
5C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic, complemented
6C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
7C2×#6292—21no—
8C2×#6292—21no—
9#88994—21no—
10C3×C22⋊#157—21no—
11C32×C4×Dic3C411yescharacteristic, complemented
12#64083—31no—
13#63911—62no—
14#63774—31no—
15#63774D311yescharacteristic, complemented
16C22×C3⋊(C3⋊D3)—21no—
17C22×C3×C3⋊D3D411yescharacteristic
18C22×C3×C3⋊D3D411yescharacteristic
19C2×C3⋊#157—41no—
20C2×C3×#157—41no—
21C2×C33×C4C2×C411yescharacteristic
22C4⋊(C3⋊(C3⋊D3))—41no—
23C3×C4⋊(C3⋊D3)—41no—
24C2×C32×Dic3—21no—
25#6292—42no—
26#6292—42no—
27#4474—91nomaximal
28C2×C4⋊(C3⋊D3)—31no—
29C2×C4⋊(C3⋊D3)—62no—
30C2×C4⋊(C3⋊D3)—61no—
31C2×C4⋊(C3⋊D3)—61no—
32C2×C4⋊(C3⋊D3)—122no—
33C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
34C2×C3×D12—62no—
35C2×#626—64no—
36C2×#626—64no—
37C22⋊#157—21no—
38C22⋊#157—61no—
39C32×C42—31no—
40C3×C4×Dic3—22no—
41C22×C33C22⋊C411yescharacteristic
42C2×C3⋊(C3⋊D3)—41no—
43C2×C3⋊(C3⋊D3)—41no—
44C2×C3⋊(C3⋊D3)—41no—
45C2×C3×C3⋊D3—21no—
46C2×C3×C3⋊D3—21no—
47C2×C3×C3⋊D3—41no—
48C2×C3×C3⋊D3—41no—
49C3⋊#157—81no—
50C3×#157—81no—
51C33×C4—21no—
52C32×Dic3—22no—
53#940—91no—
54#874—182no—
55C3×#99—91no—
56#834—91no—
57C22×C3⋊D3—61no—
58C22×C3⋊D3—64no—
59C22×C3⋊D3—124no—
60C22×C3⋊D3D1211yescharacteristic
61C22×C3⋊D3#6711yescharacteristic
62C22×C3×D3—62no—
63C22×C3×D3—62no—
64C2×#157—41no—
65C2×#157—121no—
66C2×C32×C4—22no—
67C2×C32×C4—42no—
68C2×C32×C4—61no—
69C2×C32×C4C4×D311yescharacteristic
70C4⋊(C3⋊D3)—41no—
71C4⋊(C3⋊D3)—121no—
72C4⋊(C3⋊D3)—124no—
73C4⋊(C3⋊D3)—128no—
74C4⋊(C3⋊D3)—122no—
75C4⋊(C3⋊D3)—122no—
76C2×C3×Dic3—24no—
77C3×D12—124no—
78#626—128no—
79#626—128no—
80#353—271noSylow
81C2×C33—21no—
82C2×C33#9911yescharacteristic
83C3⋊(C3⋊D3)—81no—
84C3×C3⋊D3—41no—
85C3×C3⋊D3—41no—
86C2×C3×D4—91no—
87C2×#67—181no—
88C2×#67—181no—
89C2×D12—62no—
90C2×D12—91no—
91C2×D12—182no—
92C2×D12—184no—
93C2×D12—184no—
94C2×D12—362no—
95C3×C22⋊C4—181no—
96C3×C42—62no—
97#217—181no—
98C4×Dic3#15711yescharacteristic, complemented
99C22×C32—22no—
100C22×C32—42no—
101C22×C32(C2×D3)⋊C411yescharacteristic
102C2×C3⋊D3—21no—
103C2×C3⋊D3—21no—
104C2×C3⋊D3—41no—
105C2×C3⋊D3—41no—
106C2×C3⋊D3—124no—
107C2×C3⋊D3—121no—
108C2×C3⋊D3—121no—
109C2×C3⋊D3—128no—
110C2×C3⋊D3—121no—
111C2×C3⋊D3—124no—
112C2×C3⋊D3—124no—
113C2×C3⋊D3—244no—
114C2×C3⋊D3—244no—
115C2×C3×D3—122no—
116C2×C3×D3—122no—
117C2×C3×D3—124no—
118C2×C3×D3—124no—
119#157—81no—
120#157—241no—
121C32×C4—21no—
122C32×C4—24no—
123C32×C4—44no—
124C32×C4—62no—
125C3×Dic3—28no—
126#127—271no—
127#99—91no—
128#99—271no—
129C33#35311yescharacteristic, complemented, Sylow
130C23×C3—91no—
131C23×C3—91no—
132C22×D3—62no—
133C22×D3—62no—
134C22×D3—184no—
135C22×D3—181no—
136C22×D3—364no—
137C3×D4—361no—
138C2×C3×C4—22no—
139C2×C3×C4—42no—
140C2×C3×C4—64no—
141C2×C3×C4—361no—
142C2×C3×C4C2×#15711yescharacteristic
143#67—362no—
144#67—362no—
145C2×Dic3—21no—
146C2×Dic3—361no—
147D12—124no—
148D12—361no—
149D12—364no—
150D12—368no—
151D12—368no—
152D12—368no—
153C2×C32—21no—
154C2×C32—24no—
155C2×C32—22no—
156C2×C32—44no—
157C2×C32—42no—
158C2×C32#80611yescharacteristic
159C3⋊D3—41no—
160C3⋊D3—41no—
161C3⋊D3—244no—
162C3⋊D3—248no—
163C3⋊D3—241no—
164C3×D3—242no—
165C3×D3—242no—
166C2×D4—91no—
167C2×D4—271no—
168C2×D4—541no—
169C2×D4—541no—
170C22⋊C4—181no—
171C22⋊C4—541no—
172C42—31no—
173C22×C3—22no—
174C22×C3—42no—
175C22×C3—181no—
176C22×C3—181no—
177C22×C3—361no—
178C22×C3—361no—
179C22×C3C22⋊#15711yescharacteristic
180C2×D3—122no—
181C2×D3—122no—
182C2×D3—124no—
183C2×D3—124no—
184C2×D3—364no—
185C2×D3—364no—
186C2×D3—364no—
187C2×D3—361no—
188C2×D3—361no—
189C2×D3—368no—
190C2×D3—361no—
191C2×D3—724no—
192C2×D3—724no—
193C3×C4—24no—
194C3×C4—21no—
195C3×C4—44no—
196C3×C4—68no—
197C3×C4—721no—
198Dic3—22no—
199Dic3—721no—
200C32—22no—
201C32—42no—
202C32#447411yescharacteristic, complemented
203C23—91no—
204C23—91no—
205C23—541no—
206D4—361no—
207D4—1081no—
208D4—1082no—
209D4—1082no—
210C2×C4—61no—
211C2×C4—361no—
212C2×C4—1081no—
213C2×C4D3×#15711yescharacteristic
214C2×C3—24no—
215C2×C3—21no—
216C2×C3—22no—
217C2×C3—44no—
218C2×C3—42no—
219C2×C3—361no—
220C2×C3—361no—
221C2×C3#6377411yescharacteristic
222D3—242no—
223D3—242no—
224D3—724no—
225D3—728no—
226D3—721no—
227C22—181no—
228C22—181no—
229C22—361no—
230C22—361no—
231C22—1081no—
232C22—1081no—
233C22—1081no—
234C22(C2×D3)⋊#15711yescharacteristic
235C4—21no—
236C4—62no—
237C4—721no—
238C4—2161no—
239C3—22no—
240C3—42no—
241C3#1059014211yescharacteristic, complemented
242C2—21no—
243C2—361no—
244C2—361no—
245C2—2161no—
246C2#1172272811yescharacteristic
subgroupG/H
centerC2#11722728
commutator, FittingC2×C33×C4C2×C4
Frattini, hypercenterC2×C4D3×#157
socleC2×C33#99
prime radical#11723517C2
cosocleC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#353
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
2. #11723099C2semidirect
3. C3×#63774C2semidirect
5. C2×C4⋊(C3⋊(C3⋊D3))C4semidirect
11. C32×C4×Dic3C4semidirect
15. #63774D3semidirect
98. C4×Dic3#157semidirect
129. C33#353semidirect
202. C32#4474semidirect
241. C3#10590142semidirect

Maximal subgroups

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

subgroupindex
#117235172
#117230992
C3×#637742
#105901423
#44749