Finite Group Databasebeta

← #13154845

Subgroups — #13154845

Order 1728.

448 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 48 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×#88806C211yesmaximal, characteristic
2#11723112C211yesmaximal, characteristic
3C3×C22⋊#622C211yesmaximal, characteristic
4#10590203—31nomaximal
5C22×C3⋊(C3⋊Dic3)C411yescharacteristic
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C22×C32×Dic3C411yescharacteristic
8C2×C3⋊#622—21no—
9C2×C3×#622—21no—
10#88806—21no—
11#88806—21no—
12C2×#3750—31no—
13C2×#3724—62no—
14C22⋊#622—31no—
15C22⋊#622D311yescharacteristic
16C23×C33C2×C411yescharacteristic
17C2×C3⋊(C3⋊Dic3)—21no—
18C2×C3⋊(C3⋊Dic3)—21no—
19C2×C3⋊(C3⋊Dic3)C812yescomplemented
20C2×C3×C3⋊Dic3—41no—
21C2×C3×C3⋊Dic3D411yescharacteristic
22C2×C3×C3⋊Dic3D411yescharacteristic
23C2×C32×Dic3—21no—
24C2×C32×Dic3—41no—
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—
37#3750—61no—
38#3750—61no—
39#3724—64no—
40#3724—64no—
41C22×C33—21no—
42C22×C33—21no—
43C22×C33C22⋊C411yescharacteristic
44C22×C33C2×C811yescharacteristic
45C22×C33M1611yescharacteristic
46C3⋊(C3⋊Dic3)—22no—
47C3×C3⋊Dic3—21no—
48C3×C3⋊Dic3—21no—
49C32×Dic3—41no—
50C2×Dic3⋊C4—91no—
51C2×Dic3⋊C4—182no—
52C3×C22⋊C8—91no—
53#830—91no—
54C23×C32—22no—
55C23×C32—42no—
56C23×C32C4×D311yescharacteristic
57C2×C32×C4—61no—
58C2×C32×C4—121no—
59C2×C3⋊Dic3—32no—
60C2×C3⋊Dic3—41no—
61C2×C3⋊Dic3—64no—
62C2×C3⋊Dic3—61no—
63C2×C3⋊Dic3—61no—
64C2×C3⋊Dic3—64no—
65C2×C3⋊Dic3—64no—
66C2×C3⋊Dic3—124no—
67C2×C3⋊Dic3—124no—
68C2×C3⋊Dic3—124no—
69C2×C3⋊Dic3D1211yescharacteristic
70C2×C3⋊Dic3#6711yescharacteristic
71C2×C3×Dic3—24no—
72C2×C3×Dic3—44no—
73C2×C3×Dic3—62no—
74C2×C3×Dic3—62no—
75C2×C3×Dic3—124no—
76#622—41no—
77#622—121no—
78#324—271noSylow
79C2×C33—21no—
80C2×C33—21no—
81C2×C33C22⋊C811yescharacteristic
82C2×C33#9911yescharacteristic
83C2×C33#10111yescharacteristic
84C22×C3×C4—62no—
85C22×C3×C4—91no—
86C22×Dic3—62no—
87C22×Dic3—91no—
88C22×Dic3—182no—
89C22×Dic3—362no—
90C22×Dic3#15711yescharacteristic
91C2×C3×C8—181no—
92Dic3⋊C4—181no—
93Dic3⋊C4—181no—
94Dic3⋊C4—184no—
95Dic3⋊C4—184no—
96C2×C3⋊C8—181no—
97C22×C32—21no—
98C22×C32—21no—
99C22×C32—24no—
100C22×C32—24no—
101C22×C32—22no—
102C22×C32—22no—
103C22×C32—22no—
104C22×C32—44no—
105C22×C32—44no—
106C22×C32—42no—
107C22×C32—42no—
108C22×C32—42no—
109C22×C32D3×C811yescharacteristic
110C22×C32ΓL(1,𝔽25)11yescharacteristic
111C22×C32(C2×D3)⋊C411yescharacteristic
112C32×C4—121no—
113C3⋊Dic3—21no—
114C3⋊Dic3—21no—
115C3⋊Dic3—62no—
116C3⋊Dic3—68no—
117C3⋊Dic3—128no—
118C3×Dic3—44no—
119C3×Dic3—122no—
120C3×Dic3—122no—
121C2×C4⋊C4—271no—
122C22⋊C8—91no—
123C22⋊C8—271no—
124C33#32411yescharacteristic, complemented, Sylow
125C23×C3—22no—
126C23×C3—42no—
127C23×C3C2×#15711yescharacteristic
128C2×C3×C4—64no—
129C2×C3×C4—91no—
130C2×C3×C4—91no—
131C2×C3×C4—124no—
132C2×C3×C4—361no—
133C2×Dic3—21no—
134C2×Dic3—41no—
135C2×Dic3—62no—
136C2×Dic3—62no—
137C2×Dic3—92no—
138C2×Dic3—124no—
139C2×Dic3—184no—
140C2×Dic3—184no—
141C2×Dic3—184no—
142C2×Dic3—181no—
143C2×Dic3—181no—
144C2×Dic3—364no—
145C2×Dic3—364no—
146C2×Dic3—364no—
147C3×C8—361no—
148C3⋊C8—361no—
149C2×C32—21no—
150C2×C32—21no—
151C2×C32—24no—
152C2×C32—24no—
153C2×C32—22no—
154C2×C32—22no—
155C2×C32—22no—
156C2×C32—44no—
157C2×C32—44no—
158C2×C32—42no—
159C2×C32—42no—
160C2×C32—42no—
161C2×C32#80611yescharacteristic
162C2×C32(C2×D3)⋊C811yescharacteristic
163C2×C32#82411yescharacteristic
164C22×C4—31no—
165C22×C4—91no—
166C22×C4—271no—
167C2×C8—181no—
168C2×C8—541no—
169C4⋊C4—541no—
170C4⋊C4—541no—
171C22×C3—24no—
172C22×C3—24no—
173C22×C3—21no—
174C22×C3—21no—
175C22×C3—22no—
176C22×C3—22no—
177C22×C3—22no—
178C22×C3—44no—
179C22×C3—44no—
180C22×C3—42no—
181C22×C3—42no—
182C22×C3—42no—
183C22×C3#378711yescharacteristic
184C22×C3#378811yescharacteristic
185C22×C3C22⋊#15711yescharacteristic
186C3×C4—124no—
187C3×C4—181no—
188C3×C4—181no—
189Dic3—41no—
190Dic3—122no—
191Dic3—122no—
192Dic3—188no—
193Dic3—182no—
194Dic3—368no—
195C32—22no—
196C32—42no—
197C32#446911yescharacteristic, complemented
198C23D3×#15711yescharacteristic
199C2×C4—61no—
200C2×C4—91no—
201C2×C4—91no—
202C2×C4—121no—
203C2×C4—272no—
204C2×C4—361no—
205C2×C4—541no—
206C2×C4—541no—
207C8—361no—
208C8—1081no—
209C2×C3—24no—
210C2×C3—24no—
211C2×C3—21no—
212C2×C3—21no—
213C2×C3—22no—
214C2×C3—22no—
215C2×C3—22no—
216C2×C3—44no—
217C2×C3—44no—
218C2×C3—42no—
219C2×C3—42no—
220C2×C3—42no—
221C2×C3#6377811yescharacteristic
222C2×C3#6378011yescharacteristic
223C2×C3#6378611yescharacteristic
224C22—21no—
225C22—21no—
226C22(C2×D3)⋊#15711yescharacteristic
227C22#8892411yescharacteristic
228C22#8892611yescharacteristic
229C4—121no—
230C4—181no—
231C4—181no—
232C4—542no—
233C3—22no—
234C3—42no—
235C3#1059020311yescharacteristic, complemented
236C2—21no—
237C2—21no—
238C2#1172272611yescharacteristic
239C2#1172272711yescharacteristic
240C2#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
19. C2×C3⋊(C3⋊Dic3)C8semidirect
124. C33#324semidirect
197. C32#4469semidirect
235. C3#10590203semidirect

Maximal subgroups

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

subgroupindex
C2×#888062
#117231122
C3×C22⋊#6222
#105902033
#44699