Finite Group Databasebeta

← #13155010

Subgroups — #13155010

Order 1728.

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

normal subgroups only

This group has 370 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#11724511C211yesmaximal, characteristic
2#11723656C211yesmaximal, characteristic
3C3×#63849C211yesmaximal, characteristic
4#10590056—31nomaximal
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6#88993—21no—
7#88989—21no—
8#88988C412yescomplemented
9C3×#3748—21no—
10C32×C4⋊Dic3—21no—
11#88809C412yescomplemented
12#88808—21no—
13C3×#3722C412yes—
14C3×Dic32—21no—
15#64283—31no—
16C3×#892—61no—
17#63849D311yescharacteristic
18#63818—31no—
19D3×#208—61no—
20C22×C3×C3⋊D3C2×C411yescharacteristic
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3D411yescharacteristic
23C3×C4×C3⋊D3C2×C411yescharacteristic
24C3×C4×C3⋊D3C2×C411yescharacteristic
25C3×C4×C3⋊D3D411yescharacteristic
26C3×C4×C3⋊D3Q811yescharacteristic
27C2×C32×Dic3—22no—
28C3⋊#622—22no—
29C3×C3⋊(C3⋊C8)—22no—
30#4533—91nomaximal
31C2×C4×C3⋊D3C2×D311yescharacteristic
32C2×C3×C4×D3—61no—
33C2×C3×C4×D3—61no—
34#3792—61no—
35#3788—61no—
36#3787—32no—
37C32×C4⋊C4—61no—
38#3748—21no—
39C3×(C2×D3)⋊C4—62no—
40C3×C4⋊Dic3—21no—
41C3×C4⋊Dic3—21no—
42C3×C4⋊Dic3—41no—
43C3×C4×Dic3—62no—
44C3×#208—61no—
45C3×#208—61no—
46C3×#208—121no—
47#3722Dic312yes—
48Dic32—21no—
49#3712—32no—
50#3711—62no—
51#3710—61no—
52D3×C3⋊C8—62no—
53C22×C33C22⋊C411yescharacteristic
54C2×C3×C3⋊D3—41no—
55C2×C3×C3⋊D3C4211yescharacteristic
56C2×C3×C3⋊D3C4⋊C411yescharacteristic
57C33×C4C22⋊C411yescharacteristic
58C33×C4C4⋊C411yescharacteristic
59C3×C3⋊Dic3C22⋊C411yescharacteristic
60C3×C3⋊Dic3C4⋊C411yescharacteristic
61C32×Dic3—42no—
62C3×#117—91no—
63C2×#208—91no—
64C2×#208—91no—
65D3×M16—181no—
66#892—61no—
67C22×C3⋊D3C2×Dic311yescharacteristic
68C22×C3×D3—61no—
69C22×C3×D3—61no—
70C2×C32×C4—21no—
71C2×C32×C4—21no—
72C2×C32×C4—41no—
73C2×C32×C4—41no—
74C2×C32×C4—62no—
75C2×C32×C4#6711yescharacteristic
76C2×C3⋊Dic3#6711yescharacteristic
77C4×C3⋊D3Dic611yescharacteristic
78C4×C3⋊D3C4×D311yescharacteristic
79C4×C3⋊D3C4×D311yescharacteristic
80C4×C3⋊D3D1211yescharacteristic
81C2×C3×Dic3—22no—
82C2×C3×Dic3—22no—
83C2×C3×Dic3—42no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C3×C4×D3—61no—
87C3×C4×D3—61no—
88C3×C4×D3—61no—
89C3×C4×D3—61no—
90C3×C4×D3—61no—
91C3×C4×D3—61no—
92C3×C4×D3—61no—
93C3×C4×D3—61no—
94#622—62no—
95C3⋊(C3⋊C8)—22no—
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—181no—
108C3×C4⋊C4—61no—
109C3×C4⋊C4—61no—
110C3×C4⋊C4—121no—
111C3×C22⋊C4—92no—
112C3×C42—181no—
113(C2×D3)⋊C4—62no—
114C4⋊Dic3—21no—
115C4×Dic3—62no—
116#208—61no—
117#208—61no—
118#208—92no—
119#208—121no—
120#208—181no—
121#208—181no—
122C2×C3⋊C8—92no—
123C2×C3⋊C8—181no—
124ΓL(1,𝔽25)—182no—
125D3×C8—182no—
126C22×C32—21no—
127C22×C32—21no—
128C22×C32—41no—
129C22×C32—41no—
130C22×C32#21711yescharacteristic
131C2×C3⋊D3—41no—
132C2×C3⋊D3C4×Dic311yescharacteristic
133C2×C3⋊D3C4⋊Dic311yescharacteristic
134C2×C3×D3—61no—
135C2×C3×D3—61no—
136C2×C3×D3—61no—
137C2×C3×D3—61no—
138C2×C3×D3—122no—
139C2×C3×D3—121no—
140C2×C3×D3—121no—
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—122no—
150C32×C4Dic3⋊C411yescharacteristic
151C32×C4(C2×D3)⋊C411yescharacteristic
152C3⋊Dic3Dic3⋊C411yescharacteristic
153C3⋊Dic3(C2×D3)⋊C411yescharacteristic
154C3×Dic3—42no—
155C3×Dic3—42no—
156C3×Dic3—61no—
157C3×Dic3—61no—
158C3×Dic3—61no—
159C3×Dic3—61no—
160C3×Dic3—82no—
161C2×M16—271no—
162C2×M16—271no—
163#117—91no—
164C33#34411yescharacteristic, complemented, Sylow
165C23×C3—91no—
166C22×D3—61no—
167C22×D3—61no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—62no—
173C2×C3×C4—62no—
174C2×C3×C4—91no—
175C2×C3×C4—91no—
176C2×C3×C4—91no—
177C2×C3×C4—91no—
178C2×C3×C4—91no—
179C2×C3×C4—122no—
180C2×C3×C4GO+(4,𝔽2)11yescharacteristic
181C2×Dic3—22no—
182C2×Dic3—61no—
183C2×Dic3—61no—
184C4×D3—61no—
185C4×D3—61no—
186C4×D3—61no—
187C4×D3—61no—
188C4×D3—61no—
189C4×D3—61no—
190C4×D3—61no—
191C4×D3—61no—
192C3×C8—182no—
193C3⋊C8—62no—
194C3⋊C8—62no—
195C3⋊C8—122no—
196C3⋊C8—182no—
197C2×C32—21no—
198C2×C32—21no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32—41no—
205C2×C32—41no—
206C2×C32—41no—
207C2×C32—81no—
208C2×C32C3⋊#9511yescharacteristic
209C3⋊D3—21no—
210C3⋊D3—21no—
211C3×D3—62no—
212C3×D3—121no—
213C3×D3—121no—
214C3×D3—121no—
215C22×C4—91no—
216M16—181no—
217M16—272no—
218M16—541no—
219M16—541no—
220C2×C8—272no—
221C2×C8—541no—
222C4⋊C4—61no—
223C22⋊C4—92no—
224C42—181no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—91no—
230C22×C3—91no—
231C22×C3—361no—
232C22×C3D32⋊C411yescharacteristic
233C2×D3—61no—
234C2×D3—61no—
235C2×D3—61no—
236C2×D3—61no—
237C2×D3—122no—
238C2×D3—121no—
239C2×D3—121no—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—41no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C4—41no—
248C3×C4—91no—
249C3×C4—91no—
250C3×C4—122no—
251C3×C4—122no—
252C3×C4—242no—
253C3×C4D32⋊C411yescharacteristic
254C3×C4#377311yescharacteristic
255Dic3—42no—
256Dic3—61no—
257Dic3—61no—
258Dic3—61no—
259Dic3—61no—
260C32—21no—
261C32—21no—
262C32—41no—
263C32—41no—
264C32#453311yescharacteristic, complemented
265C23—91no—
266C2×C4—62no—
267C2×C4—91no—
268C2×C4—91no—
269C2×C4—91no—
270C2×C4—91no—
271C2×C4—91no—
272C2×C4D32⋊D311yescharacteristic
273C8—182no—
274C8—542no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—41no—
281C2×C3—41no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—41no—
285C2×C3—81no—
286C2×C3—181no—
287C2×C3—181no—
288C2×C3#6373711yescharacteristic
289D3—62no—
290D3—121no—
291D3—121no—
292D3—121no—
293C22—91no—
294C22—91no—
295C22—361no—
296C22D32⋊Dic311yescharacteristic
297C4—91no—
298C4—91no—
299C4—122no—
300C4#8893211yescharacteristic
301C4#8893411yescharacteristic
302C3—21no—
303C3—21no—
304C3—41no—
305C3—41no—
306C3#1059005611yescharacteristic, complemented
307C2—21no—
308C2—181no—
309C2—181no—
310C2#1172276911yescharacteristic
subgroupG/H
centerC2#11722769
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 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
8. #88988C4semidirect
11. #88809C4semidirect
164. C33#344semidirect
264. C32#4533semidirect
306. C3#10590056semidirect

Maximal subgroups

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

subgroupindex
#117245112
#117236562
C3×#638492
#105900563
#45339