Finite Group Databasebeta

← #13155054

Subgroups — #13155054

Order 1728.

336 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 29 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#11723718C211yesmaximal, characteristic
2#11723111C211yesmaximal, characteristic
3C3×#63581C211yesmaximal, characteristic
4#10590081—31nomaximal
5C2×C3×#638C2211yescharacteristic
6C2×#6296C411yescharacteristic, complemented
7#88994—21no—
8C3×#3757—21no—
9C32×#217—21no—
10#88814—21no—
11#88813C411yescharacteristic, complemented
12#63959—61no—
13#63953—31no—
14#63786—31no—
15C3×#806—61no—
16#63581D311yescharacteristic
17C23×C33D411yescharacteristic
18C22×C3×C3⋊D3D411yescharacteristic
19C2×C3⋊#157—41no—
20C3×#638—41no—
21C2×C3×C3⋊Dic3—21no—
22C2×C3×C3⋊Dic3—21no—
23C2×C3×C3⋊Dic3C2×C411yescharacteristic
24C2×C32×Dic3—41no—
25#6296—41no—
26#4473—91nomaximal
27C2×#638C2×D311yescharacteristic
28C2×C3×#67—61no—
29C2×C3×#67—61no—
30C2×#627—31no—
31C2×#627—61no—
32C22⋊#157—61no—
33C32×C22⋊C4—61no—
34#3757—21no—
35C3×#217—21no—
36C3×#217—21no—
37C3×#217—41no—
38C3×#217—61no—
39C3×#217—61no—
40C3×#217—121no—
41#3722—61no—
42(C2×D3)⋊Dic3—61no—
43(C2×D3)⋊Dic3—61no—
44Dic32—31no—
45Dic32—61no—
46C22×C33—21no—
47C22×C33—41no—
48C22×C33C22⋊C411yescharacteristic
49C2×C3×C3⋊D3—21no—
50C2×C3×C3⋊D3—41no—
51C3⋊#157—81no—
52C3×C3⋊Dic3—21no—
53C3×C3⋊Dic3—41no—
54C3×C3⋊Dic3—41no—
55C32×Dic3—81no—
56#936—91no—
57#885—181no—
58C3×#99—91no—
59#834—91no—
60#806—61no—
61C23×C32—21no—
62C23×C32—21no—
63C23×C32—41no—
64C23×C32—41no—
65C23×C32#6711yescharacteristic
66C22×C3⋊D3D1211yescharacteristic
67C22×C3×D3—61no—
68C22×C3×D3—61no—
69C2×#157—121no—
70C2×C32×C4—121no—
71#638—41no—
72C2×C3⋊Dic3—21no—
73C2×C3⋊Dic3—21no—
74C2×C3⋊Dic3C4×D311yescharacteristic
75C3×#67—122no—
76C3×#67—121no—
77C3×#67—121no—
78C2×C3×Dic3—41no—
79C2×C3×Dic3—41no—
80C2×C3×Dic3—42no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—61no—
83C2×C3×Dic3—61no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—121no—
88C2×C3×Dic3—121no—
89#627—122no—
90#627—121no—
91#354—271noSylow
92C2×C33—21no—
93C2×C33—41no—
94C2×C33#9911yescharacteristic
95C3×C3⋊D3—41no—
96C2×C3×D4—91no—
97C2×#67—61no—
98C2×#67—61no—
99C2×Dic6—91no—
100C2×Dic6—181no—
101C3×C22⋊C4—61no—
102C3×C22⋊C4—61no—
103C3×C22⋊C4—121no—
104C3×C22⋊C4—181no—
105#217—21no—
106#217—61no—
107#217—61no—
108#217—121no—
109#217—181no—
110#217—181no—
111(C2×D3)⋊C4—181no—
112(C2×D3)⋊C4—181no—
113C4×Dic3—91no—
114C4×Dic3—181no—
115C22×C32—21no—
116C22×C32—21no—
117C22×C32—21no—
118C22×C32—21no—
119C22×C32—21no—
120C22×C32—41no—
121C22×C32—41no—
122C22×C32—41no—
123C22×C32—41no—
124C22×C32—41no—
125C22×C32—41no—
126C22×C32—41no—
127C22×C32—41no—
128C22×C32—42no—
129C22×C32—81no—
130C22×C32—81no—
131C22×C32—81no—
132C22×C32—82no—
133C22×C32(C2×D3)⋊C411yescharacteristic
134C2×C3⋊D3—21no—
135C2×C3⋊D3—41no—
136C2×C3×D3—121no—
137C2×C3×D3—121no—
138C2×C3×D3—121no—
139C2×C3×D3—121no—
140C2×C3×D3—241no—
141#157—241no—
142C32×C4—241no—
143C3⋊Dic3—21no—
144C3⋊Dic3—41no—
145C3⋊Dic3—41no—
146C3×Dic3—62no—
147C3×Dic3—81no—
148C3×Dic3—82no—
149C3×Dic3—81no—
150C3×Dic3—121no—
151C3×Dic3—121no—
152C3×Dic3—121no—
153C3×Dic3—121no—
154C3×Dic3—121no—
155C3×Dic3—241no—
156C3×Dic3—241no—
157#124—271no—
158#99—91no—
159#99—271no—
160C33#35411yescharacteristic, complemented, Sylow
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—41no—
164C23×C3—41no—
165C23×C3—91no—
166C23×C3GO+(4,𝔽2)11yescharacteristic
167C22×D3—61no—
168C22×D3—61no—
169C3×D4—361no—
170C2×C3×C4—91no—
171C2×C3×C4—121no—
172C2×C3×C4—121no—
173C2×C3×C4—122no—
174C2×C3×C4—181no—
175C2×C3×C4—181no—
176#67—122no—
177#67—121no—
178#67—121no—
179C2×Dic3—41no—
180C2×Dic3—61no—
181C2×Dic3—61no—
182C2×Dic3—61no—
183C2×Dic3—61no—
184C2×Dic3—61no—
185C2×Dic3—61no—
186C2×Dic3—121no—
187C2×Dic3—121no—
188C2×Dic3—361no—
189Dic6—362no—
190Dic6—361no—
191C2×C32—21no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32—21no—
195C2×C32—21no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32—41no—
199C2×C32—41no—
200C2×C32—41no—
201C2×C32—41no—
202C2×C32—41no—
203C2×C32—42no—
204C2×C32—41no—
205C2×C32—81no—
206C2×C32—81no—
207C2×C32—81no—
208C2×C32—82no—
209C2×C32#80611yescharacteristic
210C3⋊D3—41no—
211C3×D3—241no—
212C3×D3—241no—
213C2×Q8—271no—
214C2×D4—91no—
215C22⋊C4—61no—
216C22⋊C4—181no—
217C22⋊C4—541no—
218C22⋊C4—541no—
219C42—271no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—41no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—42no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—81no—
235C22×C3—81no—
236C22×C3—81no—
237C22×C3—82no—
238C22×C3—181no—
239C22×C3—361no—
240C22×C3D32⋊C411yescharacteristic
241C2×D3—121no—
242C2×D3—121no—
243C2×D3—121no—
244C2×D3—121no—
245C2×D3—241no—
246C3×C4—181no—
247C3×C4—241no—
248C3×C4—242no—
249C3×C4—241no—
250C3×C4—361no—
251C3×C4—361no—
252Dic3—62no—
253Dic3—81no—
254Dic3—121no—
255Dic3—121no—
256Dic3—121no—
257Dic3—121no—
258Dic3—121no—
259Dic3—241no—
260Dic3—241no—
261Dic3—721no—
262C32—21no—
263C32—21no—
264C32—41no—
265C32—41no—
266C32#447311yescharacteristic, complemented
267C23—91no—
268C23D32⋊D311yescharacteristic
269Q8—1081no—
270D4—361no—
271C2×C4—91no—
272C2×C4—121no—
273C2×C4—181no—
274C2×C4—181no—
275C2×C4—1081no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—41no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—41no—
285C2×C3—41no—
286C2×C3—42no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—81no—
291C2×C3—81no—
292C2×C3—81no—
293C2×C3—82no—
294C2×C3—361no—
295C2×C3#6373811yescharacteristic
296D3—241no—
297D3—241no—
298C22—21no—
299C22—41no—
300C22—181no—
301C22—361no—
302C22#8893211yescharacteristic
303C4—181no—
304C4—241no—
305C4—361no—
306C4—361no—
307C4—2161no—
308C3—21no—
309C3—21no—
310C3—41no—
311C3—41no—
312C3#1059008111yescharacteristic, complemented
313C2—21no—
314C2—41no—
315C2—361no—
316C2#1172277011yescharacteristic
subgroupG/H
centerC2#11722770
commutatorC2×C3×C3⋊Dic3C2×C4
Frattini, hypercenterC23D32⋊D3
FittingC23×C33D4
socleC2×C33#99
prime radical, cosocleC2×C3×#638C22
hypocenterC33#354
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
6. C2×#6296C4semidirect
11. #88813C4semidirect
160. C33#354semidirect
266. C32#4473semidirect
312. C3#10590081semidirect

Maximal subgroups

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

subgroupindex
#117237182
#117231112
C3×#635812
#105900813
#44739