Finite Group Databasebeta

← #13155051

Subgroups — #13155051

Order 1728.

334 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 31 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
1C3×#63975C211yesmaximal, characteristic
2#11723113C211yesmaximal, characteristic
3#11722083C211yesmaximal, characteristic
4#10590079—31nomaximal
5C2×C3×#638C2211yescharacteristic
6C2×C3×#624C411yescharacteristic, complemented
7#88993—21no—
8C3×#3757—21no—
9C32×#217—21no—
10C3×Dic3⋊Dic3—21no—
11C3×#3722C411yescharacteristic, complemented
12C3×#883—61no—
13#63975D311yescharacteristic
14#63788—31no—
15#63581—31no—
16#63580—61no—
17C23×C33D411yescharacteristic
18C22×C3×C3⋊D3C2×C411yescharacteristic
19C3×#638—41no—
20C2×C3×C3⋊Dic3—41no—
21C2×C3×C3⋊Dic3D411yescharacteristic
22C2×C32×Dic3—21no—
23C2×C32×Dic3—21no—
24C3×#624—41no—
25C3⋊#622—41no—
26#4539—91nomaximal
27C2×#638C2×D311yescharacteristic
28C2×C3×#67—61no—
29C2×C3×#67—61no—
30C2×C3×C4×D3—61no—
31C2×#624Dic311yescharacteristic, complemented
32#3792—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—
41C3×(C2×D3)⋊C4—61no—
42C3×C4⋊Dic3—61no—
43C3×Dic3⋊C4—61no—
44Dic3⋊Dic3—21no—
45#3722Dic311yescharacteristic, complemented
46C22×C33—21no—
47C22×C33—41no—
48C22×C33C22⋊C411yescharacteristic
49C2×C3×C3⋊D3—21no—
50C2×C3×C3⋊D3—41no—
51C3×C3⋊Dic3—21no—
52C3×C3⋊Dic3—81no—
53C32×Dic3—41no—
54C32×Dic3—41no—
55C3×#123—91no—
56#883—61no—
57#834—91no—
58#833—91no—
59#806—181no—
60C23×C32—21no—
61C23×C32—21no—
62C23×C32—41no—
63C23×C32—41no—
64C23×C32#6711yescharacteristic
65C22×C3⋊D3C2×Dic311yescharacteristic
66C22×C3×D3—61no—
67C22×C3×D3—61no—
68C2×C32×C4—61no—
69C2×C32×C4—61no—
70#638—41no—
71C2×C3⋊Dic3—41no—
72C2×C3⋊Dic3#6711yescharacteristic
73C3×#67—122no—
74C3×#67—121no—
75C3×#67—121no—
76C2×C3×Dic3—21no—
77C2×C3×Dic3—21no—
78C2×C3×Dic3—21no—
79C2×C3×Dic3—21no—
80C2×C3×Dic3—41no—
81C2×C3×Dic3—41no—
82C2×C3×Dic3—61no—
83C2×C3×Dic3—61no—
84C2×C3×Dic3—121no—
85C2×C3×Dic3—121no—
86C2×C3×Dic3—122no—
87C3×C4×D3—122no—
88#624—41no—
89#622—121no—
90#352—271noSylow
91C2×C33—21no—
92C2×C33—41no—
93C2×C33#9911yescharacteristic
94C3×C3⋊D3—41no—
95C2×C3×D4—91no—
96C22×C3×C4—91no—
97C2×#67—61no—
98C2×#67—61no—
99C2×C4×D3—61no—
100C3×C4⋊C4—181no—
101C3×C22⋊C4—61no—
102C3×C22⋊C4—61no—
103C3×C22⋊C4—91no—
104C3×C22⋊C4—121no—
105C3×C22⋊C4—181no—
106#217—21no—
107#217—61no—
108#217—61no—
109#217—121no—
110(C2×D3)⋊C4—61no—
111C4⋊Dic3—61no—
112Dic3⋊C4—61no—
113#208—181no—
114C22×C32—21no—
115C22×C32—21no—
116C22×C32—21no—
117C22×C32—21no—
118C22×C32—21no—
119C22×C32—41no—
120C22×C32—41no—
121C22×C32—41no—
122C22×C32—41no—
123C22×C32—41no—
124C22×C32—41no—
125C22×C32—41no—
126C22×C32—41no—
127C22×C32—42no—
128C22×C32—81no—
129C22×C32—81no—
130C22×C32—81no—
131C22×C32—82no—
132C22×C32#21711yescharacteristic
133C2×C3⋊D3—21no—
134C2×C3⋊D3—41no—
135C2×C3×D3—62no—
136C2×C3×D3—121no—
137C2×C3×D3—121no—
138C2×C3×D3—121no—
139C2×C3×D3—241no—
140C32×C4—121no—
141C32×C4—121no—
142C3⋊Dic3—21no—
143C3⋊Dic3—81no—
144C3×Dic3—41no—
145C3×Dic3—41no—
146C3×Dic3—41no—
147C3×Dic3—41no—
148C3×Dic3—81no—
149C3×Dic3—81no—
150C3×Dic3—121no—
151C3×Dic3—121no—
152C3×Dic3—241no—
153C3×Dic3—242no—
154C3×Dic3—241no—
155#123—91no—
156#100—271no—
157#99—271no—
158C33#35211yescharacteristic, complemented, Sylow
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—41no—
162C23×C3—41no—
163C23×C3—91no—
164C23×C3GO+(4,𝔽2)11yescharacteristic
165C22×D3—61no—
166C22×D3—61no—
167C3×D4—361no—
168C2×C3×C4—61no—
169C2×C3×C4—61no—
170C2×C3×C4—61no—
171C2×C3×C4—61no—
172C2×C3×C4—91no—
173C2×C3×C4—121no—
174C2×C3×C4—121no—
175C2×C3×C4—361no—
176C2×C3×C4—361no—
177#67—122no—
178#67—121no—
179#67—121no—
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—61no—
183C2×Dic3—61no—
184C2×Dic3—121no—
185C2×Dic3—121no—
186C2×Dic3—122no—
187C4×D3—122no—
188C3⋊C8—361no—
189C2×C32—21no—
190C2×C32—21no—
191C2×C32—21no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32—41no—
195C2×C32—41no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32—41no—
199C2×C32—41no—
200C2×C32—41no—
201C2×C32—42no—
202C2×C32—41no—
203C2×C32—81no—
204C2×C32—81no—
205C2×C32—81no—
206C2×C32—82no—
207C2×C32#83411yescharacteristic
208C3⋊D3—41no—
209C3×D3—122no—
210C3×D3—241no—
211C2×D4—91no—
212C22×C4—91no—
213M16—541no—
214C4⋊C4—181no—
215C22⋊C4—61no—
216C22⋊C4—91no—
217C22⋊C4—181no—
218C22×C3—21no—
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—41no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—42no—
230C22×C3—41no—
231C22×C3—41no—
232C22×C3—81no—
233C22×C3—81no—
234C22×C3—81no—
235C22×C3—82no—
236C22×C3—181no—
237C22×C3—361no—
238C22×C3D32⋊C411yescharacteristic
239C2×D3—62no—
240C2×D3—121no—
241C2×D3—121no—
242C2×D3—121no—
243C2×D3—241no—
244C3×C4—121no—
245C3×C4—121no—
246C3×C4—121no—
247C3×C4—121no—
248C3×C4—181no—
249C3×C4—241no—
250C3×C4—241no—
251C3×C4—721no—
252Dic3—41no—
253Dic3—41no—
254Dic3—121no—
255Dic3—121no—
256Dic3—241no—
257Dic3—242no—
258Dic3—241no—
259C32—21no—
260C32—21no—
261C32—41no—
262C32—41no—
263C32#453911yescharacteristic, complemented
264C23—91no—
265C23D32⋊D311yescharacteristic
266D4—361no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—91no—
270C2×C4—361no—
271C2×C4—361no—
272C8—1081no—
273C2×C3—21no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—41no—
279C2×C3—41no—
280C2×C3—41no—
281C2×C3—41no—
282C2×C3—41no—
283C2×C3—42no—
284C2×C3—41no—
285C2×C3—41no—
286C2×C3—41no—
287C2×C3—81no—
288C2×C3—81no—
289C2×C3—81no—
290C2×C3—82no—
291C2×C3—361no—
292C2×C3#6373811yescharacteristic
293D3—122no—
294D3—241no—
295C22—21no—
296C22—41no—
297C22—181no—
298C22—361no—
299C22D32⋊Dic311yescharacteristic
300C4—121no—
301C4—121no—
302C4—181no—
303C4—721no—
304C3—21no—
305C3—21no—
306C3—41no—
307C3—41no—
308C3#1059007911yescharacteristic, complemented
309C2—21no—
310C2—41no—
311C2—361no—
312C2#1172277111yescharacteristic
subgroupG/H
centerC2#11722771
commutatorC22×C3×C3⋊D3C2×C4
Frattini, hypercenterC23D32⋊D3
FittingC23×C33D4
socleC2×C33#99
prime radical, cosocleC2×C3×#638C22
hypocenterC33#352
LCSG ▷ C22×C3×C3⋊D3 ▷ 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×C3×#624C4semidirect
11. C3×#3722C4semidirect
31. C2×#624Dic3semidirect
45. #3722Dic3semidirect
158. C33#352semidirect
263. C32#4539semidirect
308. C3#10590079semidirect

Maximal subgroups

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

subgroupindex
C3×#639752
#117231132
#117220832
#105900793
#45399