Finite Group Databasebeta

← #13155009

Subgroups — #13155009

Order 1728.

358 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 59 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#11724514C211yesmaximal, characteristic
2C2×#88809C211yesmaximal, characteristic
3C3×#63818C211yesmaximal, characteristic
4#10590055—31nomaximal
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6#88994C412yes—
7C4⋊(C3⋊#157)—21no—
8C4×C3⋊#157—21no—
9C2×C3×C3⋊(C3⋊C8)—21no—
10C32×#208—21no—
11#88809—21no—
12#88809C412yescomplemented
13C3×#3712—21no—
14C3×#3710C412yescomplemented
15#64286—31no—
16C3×D3×M16—61no—
17C2×#3712—31no—
18C2×#3711—61no—
19#63818D311yescharacteristic
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
28C3×C3⋊(C3⋊C8)—22no—
29C32×C3⋊C8—22no—
30#4556—91nomaximal
31C2×C4×C3⋊D3C2×D311yescharacteristic
32C2×C3×C4×D3—61no—
33C2×C3×C4×D3—61no—
34C22⋊#157—32no—
35C4⋊#157—61no—
36C4×#157—61no—
37C32×M16—61no—
38C2×C3⋊(C3⋊C8)—21no—
39C3×#208—21no—
40C3×#208—21no—
41C3×#208—41no—
42C2×C3×C3⋊C8—61no—
43C2×C3×C3⋊C8—61no—
44C2×C3×C3⋊C8—121no—
45C3×ΓL(1,𝔽25)—62no—
46C3×D3×C8—62no—
47#3712—21no—
48#3712—32no—
49#3712—61no—
50#3711—62no—
51#3711—62no—
52#3710Dic312yescomplemented
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—
63#926—91no—
64C2×#208—91no—
65D3×M16—61no—
66C2×ΓL(1,𝔽25)—181no—
67C22×C3⋊D3C4×D311yescharacteristic
68C22×C3×D3—61no—
69C22×C3×D3—61no—
70C2×#157—62no—
71C2×C32×C4—21no—
72C2×C32×C4—21no—
73C2×C32×C4—41no—
74C2×C32×C4—41no—
75C2×C32×C4#6711yescharacteristic
76C2×C3⋊Dic3Dic611yescharacteristic
77C4×C3⋊D3C4×D311yescharacteristic
78C4×C3⋊D3D1211yescharacteristic
79C4×C3⋊D3C2×Dic311yescharacteristic
80C4×C3⋊D3#6711yescharacteristic
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—
92C3⋊(C3⋊C8)—22no—
93C3×C3⋊C8—22no—
94C3×C3⋊C8—22no—
95C3×C3⋊C8—42no—
96C3×C3⋊C8—62no—
97C3×C3⋊C8—62no—
98C3×C3⋊C8—122no—
99#344—271noSylow
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—121no—
110C3×M16—181no—
111C2×C3×C8—92no—
112C2×C3×C8—181no—
113#217—92no—
114C4⋊Dic3—181no—
115C4×Dic3—181no—
116#208—21no—
117#208—92no—
118#208—181no—
119C2×C3⋊C8—61no—
120C2×C3⋊C8—61no—
121C2×C3⋊C8—121no—
122ΓL(1,𝔽25)—62no—
123ΓL(1,𝔽25)—182no—
124ΓL(1,𝔽25)—182no—
125D3×C8—62no—
126C22×C32—21no—
127C22×C32—21no—
128C22×C32—41no—
129C22×C32—41no—
130C22×C32Dic3⋊C411yescharacteristic
131C2×C3⋊D3—41no—
132C2×C3⋊D3C4×Dic311yescharacteristic
133C2×C3⋊D3(C2×D3)⋊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—122no—
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×C4(C2×D3)⋊C411yescharacteristic
150C32×C4#21711yescharacteristic
151C3⋊Dic3Dic3⋊C411yescharacteristic
152C3⋊Dic3C4⋊Dic311yescharacteristic
153C3×Dic3—61no—
154C3×Dic3—61no—
155C3×Dic3—61no—
156C3×Dic3—61no—
157C2×M16—91no—
158C2×M16—271no—
159#117—271no—
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×C4GO+(4,𝔽2)11yescharacteristic
174C2×Dic3—61no—
175C2×Dic3—61no—
176C2×Dic3—182no—
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—122no—
188C3×C8—182no—
189C3⋊C8—22no—
190C3⋊C8—62no—
191C3⋊C8—62no—
192C3⋊C8—122no—
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—181no—
212M16—272no—
213M16—541no—
214C2×C8—92no—
215C2×C8—181no—
216C4⋊C4—541no—
217C22⋊C4—272no—
218C42—541no—
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×C4D32⋊C411yescharacteristic
244C3×C4D32⋊C411yescharacteristic
245Dic3—61no—
246Dic3—61no—
247Dic3—61no—
248Dic3—61no—
249Dic3—362no—
250C32—21no—
251C32—21no—
252C32—41no—
253C32—41no—
254C32#455611yescharacteristic, complemented
255C23—91no—
256C2×C4—91no—
257C2×C4—91no—
258C2×C4—91no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4—542no—
262C2×C4D32⋊D311yescharacteristic
263C8—62no—
264C8—182no—
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—
284C22#8893411yescharacteristic
285C4—91no—
286C4—91no—
287C4—1082no—
288C4#8893211yescharacteristic
289C4D32⋊Dic311yescharacteristic
290C3—21no—
291C3—21no—
292C3—41no—
293C3—41no—
294C3#1059005511yescharacteristic, complemented
295C2—21no—
296C2—181no—
297C2—181no—
298C2#1172276911yescharacteristic
subgroupG/H
centerC4D32⋊Dic3
commutatorC2×C3×C3⋊D3C42
Frattini, hypercenterC2×C4D32⋊D3
FittingC2×C33×C4D4
socleC2×C33#95
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C4×C3⋊D3C22
hypocenterC33#344
LCSG ▷ C2×C3×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
12. #88809C4semidirect
14. C3×#3710C4semidirect
52. #3710Dic3semidirect
160. C33#344semidirect
254. C32#4556semidirect
294. C3#10590055semidirect

Maximal subgroups

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

subgroupindex
#117245142
C2×#888092
C3×#638182
#105900553
#45569