Finite Group Databasebeta

← C3×#10590055 #13154905

Subgroups — C3×#10590055 #13154905

Order 1728.

358 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 66 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×#64286C211yesmaximal, characteristic
2C2×C3×#3712C211yesmaximal, characteristic
3C3×#63818C211yesmaximal, characteristic
4#10590055C311yesmaximal, characteristic, direct product factor
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6C3×C22⋊#157C412yes—
7C3×C4⋊#157—21no—
8C3×C4×#157—21no—
9C32×#208—21no—
10C2×C32×C3⋊C8—21no—
11C3×#3712—21no—
12C3×#3712—21no—
13C3×#3712C412yescomplemented
14C3×#3710C412yescomplemented
15#64286C2×C311yescharacteristic
16C3×D3×M16—61no—
17C2×C3×ΓL(1,𝔽25)—61no—
18C2×#3712C2×C311yescharacteristic
19#63818C2×C311yescharacteristic
20C22×C3×C3⋊D3C2×C411yescharacteristic
21C2×C3×#157—22no—
22C2×C33×C4D411yescharacteristic
23C2×C3×C3⋊Dic3Q811yescharacteristic
24C3×C4×C3⋊D3C2×C411yescharacteristic
25C3×C4×C3⋊D3C2×C411yescharacteristic
26C3×C4×C3⋊D3D411yescharacteristic
27C3×C4×C3⋊D3D411yescharacteristic
28C32×C3⋊C8—22no—
29C32×C3⋊C8—22no—
30C3×#344—91nomaximal
31C2×C4×C3⋊D3C22×C311yescharacteristic
32C2×C3×C4×D3—61no—
33C2×C3×C4×D3—61no—
34C22⋊#157C3×C412yes—
35C4⋊#157—21no—
36C4×#157—21no—
37C32×M16—61no—
38C2×C32×C8—61no—
39C3×#208—21no—
40C3×#208—21no—
41C3×#208—41no—
42C2×C3×C3⋊C8—21no—
43C2×C3×C3⋊C8—21no—
44C2×C3×C3⋊C8—41no—
45C3×ΓL(1,𝔽25)—62no—
46C3×ΓL(1,𝔽25)—62no—
47C3×ΓL(1,𝔽25)—62no—
48C3×D3×C8—62no—
49#3712—21no—
50#3712—21no—
51#3712C3×C412yescomplemented
52#3710C3×C412yescomplemented
53C22×C33C4⋊C411yescharacteristic
54C2×C3×C3⋊D3—41no—
55C2×C3×C3⋊D3C4211yescharacteristic
56C2×C3×C3⋊D3C22⋊C411yescharacteristic
57C3×#157—42no—
58C33×C4C22⋊C411yescharacteristic
59C33×C4C22⋊C411yescharacteristic
60C3×C3⋊Dic3C4⋊C411yescharacteristic
61C3×C3⋊Dic3C4⋊C411yescharacteristic
62C2×C3×M16—91no—
63C2×C3×M16—91no—
64C3×#117—91no—
65D3×M16—61no—
66C2×ΓL(1,𝔽25)—61no—
67C22×C3⋊D3C2×C3×C411yescharacteristic
68C22×C3×D3—61no—
69C22×C3×D3—61no—
70C2×#157—22no—
71C2×C32×C4—21no—
72C2×C32×C4—21no—
73C2×C32×C4—41no—
74C2×C32×C4—41no—
75C2×C32×C4C3×D411yescharacteristic
76C2×C3⋊Dic3C3×Q811yescharacteristic
77C4×C3⋊D3C2×C3×C411yescharacteristic
78C4×C3⋊D3C2×C3×C411yescharacteristic
79C4×C3⋊D3C3×D411yescharacteristic
80C4×C3⋊D3C3×D411yescharacteristic
81C2×C3×Dic3—61no—
82C2×C3×Dic3—61no—
83C3×C4×D3—61no—
84C3×C4×D3—61no—
85C3×C4×D3—61no—
86C3×C4×D3—61no—
87C3×C4×D3—61no—
88C3×C4×D3—61no—
89C3×C4×D3—61no—
90C3×C4×D3—61no—
91C32×C8—62no—
92C32×C8—62no—
93C3×C3⋊C8—22no—
94C3×C3⋊C8—22no—
95C3×C3⋊C8—22no—
96C3×C3⋊C8—22no—
97C3×C3⋊C8—42no—
98C3×C3⋊C8—42no—
99#344—91noSylow
100C2×C33—21no—
101C2×C33#9511yescharacteristic
102C3×C3⋊D3—21no—
103C3×C3⋊D3—21no—
104C22×C3×C4—91no—
105C2×C4×D3—61no—
106C2×C4×D3—61no—
107C3×M16—61no—
108C3×M16—61no—
109C3×M16—92no—
110C3×M16—121no—
111C3×M16—181no—
112C3×M16—181no—
113C2×C3×C8—61no—
114C2×C3×C8—61no—
115C2×C3×C8—92no—
116C2×C3×C8—121no—
117C3×C4⋊C4—181no—
118C3×C22⋊C4—92no—
119C3×C42—181no—
120#208—21no—
121C2×C3⋊C8—21no—
122ΓL(1,𝔽25)—62no—
123ΓL(1,𝔽25)—62no—
124ΓL(1,𝔽25)—62no—
125D3×C8—62no—
126C22×C32—21no—
127C22×C32—21no—
128C22×C32—41no—
129C22×C32—41no—
130C22×C32C3×C4⋊C411yescharacteristic
131C2×C3⋊D3—41no—
132C2×C3⋊D3C3×C4211yescharacteristic
133C2×C3⋊D3C3×C22⋊C411yescharacteristic
134C2×C3×D3—61no—
135C2×C3×D3—61no—
136C2×C3×D3—61no—
137C2×C3×D3—61no—
138C2×C3×D3—122no—
139C2×C3×D3—122no—
140#157—42no—
141C32×C4—21no—
142C32×C4—21no—
143C32×C4—21no—
144C32×C4—21no—
145C32×C4—41no—
146C32×C4—41no—
147C32×C4—41no—
148C32×C4—41no—
149C32×C4C3×C22⋊C411yescharacteristic
150C32×C4C3×C22⋊C411yescharacteristic
151C3⋊Dic3C3×C4⋊C411yescharacteristic
152C3⋊Dic3C3×C4⋊C411yescharacteristic
153C3×Dic3—61no—
154C3×Dic3—61no—
155C3×Dic3—61no—
156C3×Dic3—61no—
157C2×M16—91no—
158C2×M16—91no—
159#117—91no—
160C33#34411yescharacteristic, complemented, Sylow
161C23×C3—91no—
162C22×D3—61no—
163C22×D3—61no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—91no—
169C2×C3×C4—91no—
170C2×C3×C4—91no—
171C2×C3×C4—91no—
172C2×C3×C4—91no—
173C2×C3×C4—182no—
174C2×C3×C4GO+(4,𝔽2)11yescharacteristic
175C2×Dic3—61no—
176C2×Dic3—61no—
177C4×D3—61no—
178C4×D3—61no—
179C4×D3—61no—
180C4×D3—61no—
181C4×D3—61no—
182C4×D3—61no—
183C4×D3—61no—
184C4×D3—61no—
185C3×C8—62no—
186C3×C8—62no—
187C3×C8—62no—
188C3×C8—62no—
189C3×C8—122no—
190C3×C8—122no—
191C3⋊C8—22no—
192C3⋊C8—22no—
193C2×C32—22no—
194C2×C32—21no—
195C2×C32—21no—
196C2×C32—21no—
197C2×C32—41no—
198C2×C32—41no—
199C2×C32—41no—
200C2×C32—42no—
201C2×C32—81no—
202C2×C32C3×#9511yescharacteristic
203C3⋊D3—21no—
204C3⋊D3—21no—
205C3×D3—62no—
206C3×D3—121no—
207C3×D3—121no—
208C3×D3—121no—
209C22×C4—91no—
210M16—61no—
211M16—92no—
212M16—181no—
213M16—181no—
214C2×C8—61no—
215C2×C8—92no—
216C4⋊C4—181no—
217C22⋊C4—92no—
218C42—181no—
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—91no—
224C22×C3—91no—
225C22×C3—361no—
226C22×C3#377311yescharacteristic
227C2×D3—61no—
228C2×D3—61no—
229C2×D3—61no—
230C2×D3—61no—
231C2×D3—122no—
232C2×D3—122no—
233C3×C4—21no—
234C3×C4—21no—
235C3×C4—21no—
236C3×C4—21no—
237C3×C4—41no—
238C3×C4—41no—
239C3×C4—41no—
240C3×C4—41no—
241C3×C4—91no—
242C3×C4—91no—
243C3×C4—362no—
244C3×C4D32⋊C411yescharacteristic
245C3×C4D32⋊C411yescharacteristic
246Dic3—61no—
247Dic3—61no—
248Dic3—61no—
249Dic3—61no—
250C32—21no—
251C32—21no—
252C32—41no—
253C32—41no—
254C32C3×#34411yescharacteristic, complemented
255C23—91no—
256C2×C4—91no—
257C2×C4—91no—
258C2×C4—91no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4—182no—
262C2×C4C3×GO+(4,𝔽2)11yescharacteristic
263C8—62no—
264C8—62no—
265C2×C3—22no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—41no—
270C2×C3—42no—
271C2×C3—41no—
272C2×C3—41no—
273C2×C3—81no—
274C2×C3—181no—
275C2×C3—181no—
276C2×C3#6373711yescharacteristic
277D3—62no—
278D3—121no—
279D3—121no—
280D3—121no—
281C22—91no—
282C22—91no—
283C22—361no—
284C22C3×#377311yescharacteristic
285C4—91no—
286C4—91no—
287C4—362no—
288C4C3×D32⋊C411yescharacteristic
289C4C3×D32⋊C411yescharacteristic
290C3—21no—
291C3—21no—
292C3—41no—
293C3—41no—
294C3#1059005511yescharacteristic, direct product factor
295C2—21no—
296C2—181no—
297C2—181no—
298C2C3×#6373711yescharacteristic
subgroupG/H
centerC3×C4D32⋊C4
commutatorC2×C3⋊D3C3×C42
FrattiniC2×C4C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33#95
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C4×C3⋊D3C22×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#344
LCSG ▷ C2×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
4. #10590055C3direct (N × H)
13. C3×#3712C4semidirect
14. C3×#3710C4semidirect
51. #3712C3×C4semidirect
52. #3710C3×C4semidirect
160. C33#344semidirect
254. C32C3×#344semidirect
294. C3#10590055direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642862
C2×C3×#37122
C3×#638182
#105900553
C3×#3449