Finite Group Databasebeta

← #13154846

Subgroups — #13154846

Order 1728.

448 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 50 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×Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic
2#11723112C211yesmaximal, characteristic
3C3×C22⋊#622C211yesmaximal, characteristic
4#10590202—31nomaximal
5C22×C3⋊(C3⋊Dic3)C411yescharacteristic
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C22×C32×Dic3C411yescharacteristic
8C2×C3⋊#622—21no—
9C2×C3×#622—21no—
10Dic3⋊(C3⋊Dic3)—21no—
11Dic3⋊(C3⋊Dic3)—21no—
12C2×C4⋊(C3⋊Dic3)—31no—
13C2×Dic3⋊Dic3—62no—
14C22⋊#622—31no—
15C22⋊#622D311yescharacteristic
16C23×C33C2×C411yescharacteristic
17C2×C3⋊(C3⋊Dic3)—21no—
18C2×C3⋊(C3⋊Dic3)—41no—
19C2×C3×C3⋊Dic3—41no—
20C2×C3×C3⋊Dic3D411yescharacteristic
21C2×C3×C3⋊Dic3D411yescharacteristic
22C2×C32×Dic3—21no—
23C2×C32×Dic3—21no—
24C2×C32×Dic3C812yescomplemented
25C3⋊#622—41no—
26C3×#622—41no—
27#4469—91nomaximal
28C22×C32×C4—31no—
29C22×C3⋊Dic3—31no—
30C22×C3⋊Dic3—62no—
31C22×C3⋊Dic3—122no—
32C22×C3⋊Dic3C2×D311yescharacteristic
33C22×C3×Dic3—22no—
34C22×C3×Dic3—62no—
35C2×#622—21no—
36C2×#622—61no—
37C4⋊(C3⋊Dic3)—61no—
38C4⋊(C3⋊Dic3)—61no—
39Dic3⋊Dic3—64no—
40Dic3⋊Dic3—64no—
41C22×C33—21no—
42C22×C33—21no—
43C22×C33C22⋊C411yescharacteristic
44C22×C33C2×C811yescharacteristic
45C22×C33M1611yescharacteristic
46C3⋊(C3⋊Dic3)—41no—
47C3×C3⋊Dic3—21no—
48C3×C3⋊Dic3—21no—
49C32×Dic3—22no—
50C2×C4⋊Dic3—182no—
51C2×Dic3⋊C4—91no—
52C3×C22⋊C8—91no—
53#830—91no—
54C23×C32—22no—
55C23×C32—42no—
56C23×C32C4×D311yescharacteristic
57C2×C32×C4—32no—
58C2×C32×C4—61no—
59C2×C32×C4—61no—
60C2×C3⋊Dic3—41no—
61C2×C3⋊Dic3—61no—
62C2×C3⋊Dic3—64no—
63C2×C3⋊Dic3—124no—
64C2×C3⋊Dic3—124no—
65C2×C3⋊Dic3—128no—
66C2×C3⋊Dic3—121no—
67C2×C3⋊Dic3D1211yescharacteristic
68C2×C3⋊Dic3#6711yescharacteristic
69C2×C3×Dic3—24no—
70C2×C3×Dic3—24no—
71C2×C3×Dic3—24no—
72C2×C3×Dic3—62no—
73C2×C3×Dic3—62no—
74C2×C3×Dic3—124no—
75#622—41no—
76#622—121no—
77#324—271noSylow
78C2×C33—21no—
79C2×C33—21no—
80C2×C33C22⋊C811yescharacteristic
81C2×C33#9911yescharacteristic
82C2×C33#10111yescharacteristic
83C22×C3×C4—62no—
84C22×C3×C4—91no—
85C22×Dic3—62no—
86C22×Dic3—91no—
87C22×Dic3—182no—
88C22×Dic3—362no—
89C22×Dic3#15711yescharacteristic
90C2×C3×C8—181no—
91C4⋊Dic3—184no—
92C4⋊Dic3—184no—
93Dic3⋊C4—181no—
94Dic3⋊C4—181no—
95C2×C3⋊C8—181no—
96C22×C32—21no—
97C22×C32—21no—
98C22×C32—24no—
99C22×C32—24no—
100C22×C32—22no—
101C22×C32—22no—
102C22×C32—22no—
103C22×C32—44no—
104C22×C32—44no—
105C22×C32—42no—
106C22×C32—42no—
107C22×C32—42no—
108C22×C32D3×C811yescharacteristic
109C22×C32ΓL(1,𝔽25)11yescharacteristic
110C22×C32(C2×D3)⋊C411yescharacteristic
111C32×C4—62no—
112C3⋊Dic3—21no—
113C3⋊Dic3—21no—
114C3⋊Dic3—124no—
115C3⋊Dic3—128no—
116C3⋊Dic3—121no—
117C3×Dic3—28no—
118C3×Dic3—122no—
119C3×Dic3—122no—
120C2×C4⋊C4—271no—
121C22⋊C8—91no—
122C22⋊C8—271no—
123C33#32411yescharacteristic, complemented, Sylow
124C23×C3—22no—
125C23×C3—42no—
126C23×C3C2×#15711yescharacteristic
127C2×C3×C4—64no—
128C2×C3×C4—64no—
129C2×C3×C4—64no—
130C2×C3×C4—91no—
131C2×C3×C4—91no—
132C2×C3×C4—361no—
133C2×Dic3—21no—
134C2×Dic3—21no—
135C2×Dic3—62no—
136C2×Dic3—62no—
137C2×Dic3—124no—
138C2×Dic3—184no—
139C2×Dic3—181no—
140C2×Dic3—364no—
141C2×Dic3—364no—
142C2×Dic3—368no—
143C2×Dic3—361no—
144C2×Dic3#62212yescomplemented
145C3×C8—361no—
146C3⋊C8—361no—
147C2×C32—21no—
148C2×C32—21no—
149C2×C32—24no—
150C2×C32—24no—
151C2×C32—22no—
152C2×C32—22no—
153C2×C32—22no—
154C2×C32—44no—
155C2×C32—44no—
156C2×C32—42no—
157C2×C32—42no—
158C2×C32—42no—
159C2×C32#80611yescharacteristic
160C2×C32(C2×D3)⋊C811yescharacteristic
161C2×C32#82411yescharacteristic
162C22×C4—31no—
163C22×C4—91no—
164C22×C4—271no—
165C2×C8—181no—
166C2×C8—541no—
167C4⋊C4—541no—
168C4⋊C4—541no—
169C22×C3—24no—
170C22×C3—24no—
171C22×C3—21no—
172C22×C3—21no—
173C22×C3—22no—
174C22×C3—22no—
175C22×C3—22no—
176C22×C3—44no—
177C22×C3—44no—
178C22×C3—42no—
179C22×C3—42no—
180C22×C3—42no—
181C22×C3C2×#62211yescharacteristic
182C22×C3#379211yescharacteristic
183C22×C3C22⋊#15711yescharacteristic
184C3×C4—68no—
185C3×C4—181no—
186C3×C4—181no—
187Dic3—22no—
188Dic3—122no—
189Dic3—122no—
190Dic3—364no—
191Dic3—368no—
192Dic3—361no—
193C32—22no—
194C32—42no—
195C32#446911yescharacteristic, complemented
196C23D3×#15711yescharacteristic
197C2×C4—32no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—91no—
201C2×C4—91no—
202C2×C4—361no—
203C2×C4—541no—
204C2×C4—1081no—
205C8—361no—
206C8—1081no—
207C2×C3—24no—
208C2×C3—24no—
209C2×C3—21no—
210C2×C3—21no—
211C2×C3—22no—
212C2×C3—22no—
213C2×C3—22no—
214C2×C3—44no—
215C2×C3—44no—
216C2×C3—42no—
217C2×C3—42no—
218C2×C3—42no—
219C2×C3#6377411yescharacteristic
220C2×C3#6378011yescharacteristic
221C2×C3C22⋊#62211yescharacteristic
222C22—21no—
223C22—21no—
224C22(C2×D3)⋊#15711yescharacteristic
225C22D3×#62211yescharacteristic
226C22#8892511yescharacteristic
227C4—62no—
228C4—181no—
229C4—181no—
230C4—1081no—
231C3—22no—
232C3—42no—
233C3#1059020211yescharacteristic, complemented
234C2—21no—
235C2—21no—
236C2(C2×D3)⋊#62211yescharacteristic
237C2#1172272811yescharacteristic
238C2#1172273011yescharacteristic
subgroupG/H
centerC22(C2×D3)⋊#157
commutatorC22×C33C2×C8
Frattini, hypercenterC23D3×#157
Fitting, prime radicalC23×C33C2×C4
socleC22×C33C22⋊C4
cosocleC22×C3×C3⋊Dic3C22
hypocenterC33#324
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
24. C2×C32×Dic3C8semidirect
123. C33#324semidirect
144. C2×Dic3#622semidirect
195. C32#4469semidirect
233. C3#10590202semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3⋊(C3⋊Dic3)2
#117231122
C3×C22⋊#6222
#105902023
#44699