Finite Group Databasebeta

← #13155000

Subgroups — #13155000

Order 1728.

366 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 63 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
2C2×#88808C211yesmaximal, characteristic
3C3×#63818C211yesmaximal, characteristic
4#10590050—31nomaximal
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6#88993C412yes—
7#88989—21no—
8#88988—21no—
9C2×C3×C3⋊(C3⋊C8)—21no—
10C32×#208—21no—
11#88808—21no—
12#88808C412yes—
13C3×#3712C412yes—
14C3×#3710—21no—
15#64283—31no—
16C3×D3×M16—61no—
17#63818D311yescharacteristic
18C2×#3710—31no—
19C2×D3×C3⋊C8—61no—
20C22×C3×C3⋊D3Q811yescharacteristic
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3C2×C411yescharacteristic
23C3×C4×C3⋊D3C2×C411yescharacteristic
24C3×C4×C3⋊D3C2×C411yescharacteristic
25C3×C4×C3⋊D3D411yescharacteristic
26C3×C4×C3⋊D3D411yescharacteristic
27C3⋊#622—22no—
28C3×C3⋊(C3⋊C8)—22no—
29C32×C3⋊C8—22no—
30#4558—91nomaximal
31C2×C4×C3⋊D3C2×D311yescharacteristic
32C2×C3×C4×D3—61no—
33C2×C3×C4×D3—61no—
34#3792—32no—
35#3788—61no—
36#3787—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#3712Dic312yes—
48#3710—21no—
49#3710—32no—
50#3710—61no—
51D3×C3⋊C8—62no—
52D3×C3⋊C8—62no—
53C22×C33C4⋊C411yescharacteristic
54C2×C3×C3⋊D3—22no—
55C2×C3×C3⋊D3C4⋊C411yescharacteristic
56C2×C3×C3⋊D3C4⋊C411yescharacteristic
57C33×C4C22⋊C411yescharacteristic
58C33×C4C22⋊C411yescharacteristic
59C3×C3⋊Dic3C4211yescharacteristic
60C3×C3⋊Dic3C22⋊C411yescharacteristic
61C2×C3×M16—91no—
62C2×#208—91no—
63C22×C3⋊C8—91no—
64D3×M16—61no—
65C2×D3×C8—181no—
66C22×C3⋊D3Dic611yescharacteristic
67C22×C3×D3—61no—
68C22×C3×D3—61no—
69C2×C32×C4—21no—
70C2×C32×C4—21no—
71C2×C32×C4—41no—
72C2×C32×C4—41no—
73C2×C32×C4#6711yescharacteristic
74C2×C3⋊Dic3C4×D311yescharacteristic
75C4×C3⋊D3C4×D311yescharacteristic
76C4×C3⋊D3D1211yescharacteristic
77C4×C3⋊D3C2×Dic311yescharacteristic
78C4×C3⋊D3#6711yescharacteristic
79C2×C3×Dic3—61no—
80C2×C3×Dic3—61no—
81C3×C4×D3—61no—
82C3×C4×D3—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—
89#622—62no—
90C32×C8—62no—
91C3⋊(C3⋊C8)—22no—
92C3×C3⋊C8—22no—
93C3×C3⋊C8—22no—
94C3×C3⋊C8—42no—
95C3×C3⋊C8—62no—
96C3×C3⋊C8—62no—
97C3×C3⋊C8—122no—
98#341—271noSylow
99C2×C33—21no—
100C2×C33#9511yescharacteristic
101C3×C3⋊D3—21no—
102C3×C3⋊D3#10812yes—
103C22×C3×C4—91no—
104C2×C4×D3—61no—
105C2×C4×D3—61no—
106C3×M16—61no—
107C3×M16—61no—
108C3×M16—92no—
109C3×M16—121no—
110C2×C3×C8—181no—
111C2×C3×C8—181no—
112#208—21no—
113#208—92no—
114#208—181no—
115C2×C3⋊C8—61no—
116C2×C3⋊C8—61no—
117C2×C3⋊C8—92no—
118C2×C3⋊C8—121no—
119C2×C3⋊C8—181no—
120C2×C3⋊C8—181no—
121ΓL(1,𝔽25)—62no—
122D3×C8—62no—
123D3×C8—182no—
124D3×C8—182no—
125C22×C32—21no—
126C22×C32—21no—
127C22×C32—41no—
128C22×C32—41no—
129C22×C32Dic3⋊C411yescharacteristic
130C2×C3⋊D3—22no—
131C2×C3⋊D3Dic3⋊C411yescharacteristic
132C2×C3⋊D3C4⋊Dic311yescharacteristic
133C2×C3×D3—61no—
134C2×C3×D3—61no—
135C2×C3×D3—61no—
136C2×C3×D3—61no—
137C2×C3×D3—64no—
138C2×C3×D3—122no—
139C32×C4—21no—
140C32×C4—21no—
141C32×C4—21no—
142C32×C4—21no—
143C32×C4—41no—
144C32×C4—41no—
145C32×C4—41no—
146C32×C4—41no—
147C32×C4(C2×D3)⋊C411yescharacteristic
148C32×C4#21711yescharacteristic
149C3⋊Dic3C4×Dic311yescharacteristic
150C3⋊Dic3(C2×D3)⋊C411yescharacteristic
151C3×Dic3—61no—
152C3×Dic3—61no—
153C3×Dic3—61no—
154C3×Dic3—61no—
155C2×M16—91no—
156C2×M16—271no—
157C22×C8—271no—
158C33#34111yescharacteristic, complemented, Sylow
159C23×C3—91no—
160C22×D3—61no—
161C22×D3—61no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—41no—
165C2×C3×C4—41no—
166C2×C3×C4—91no—
167C2×C3×C4—91no—
168C2×C3×C4—91no—
169C2×C3×C4—91no—
170C2×C3×C4—91no—
171C2×C3×C4GO+(4,𝔽2)11yescharacteristic
172C2×Dic3—61no—
173C2×Dic3—61no—
174C4×D3—61no—
175C4×D3—61no—
176C4×D3—61no—
177C4×D3—61no—
178C4×D3—61no—
179C4×D3—61no—
180C4×D3—61no—
181C4×D3—61no—
182C3×C8—62no—
183C3×C8—62no—
184C3×C8—122no—
185C3×C8—182no—
186C3⋊C8—22no—
187C3⋊C8—62no—
188C3⋊C8—62no—
189C3⋊C8—122no—
190C3⋊C8—182no—
191C2×C32—22no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32—21no—
195C2×C32—41no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32—42no—
199C2×C32—81no—
200C2×C32C3⋊#9511yescharacteristic
201C3⋊D3—21no—
202C3⋊D3#82212yes—
203C3×D3—62no—
204C3×D3—62no—
205C3×D3—62no—
206C3×D3—121no—
207C22×C4—91no—
208M16—61no—
209M16—92no—
210M16—272no—
211M16—541no—
212C2×C8—181no—
213C2×C8—181no—
214C2×C8—272no—
215C2×C8—541no—
216C2×C8—541no—
217C22×C3—21no—
218C22×C3—21no—
219C22×C3—41no—
220C22×C3—41no—
221C22×C3—91no—
222C22×C3—91no—
223C22×C3—182no—
224C22×C3#377311yescharacteristic
225C2×D3—61no—
226C2×D3—61no—
227C2×D3—61no—
228C2×D3—61no—
229C2×D3—64no—
230C2×D3—122no—
231C3×C4—21no—
232C3×C4—21no—
233C3×C4—21no—
234C3×C4—21no—
235C3×C4—41no—
236C3×C4—41no—
237C3×C4—41no—
238C3×C4—41no—
239C3×C4—91no—
240C3×C4—91no—
241C3×C4D32⋊C411yescharacteristic
242C3×C4D32⋊C411yescharacteristic
243Dic3—61no—
244Dic3—61no—
245Dic3—61no—
246Dic3—61no—
247C32—21no—
248C32—21no—
249C32—41no—
250C32—41no—
251C32#455811yescharacteristic, complemented
252C23—91no—
253C2×C4—91no—
254C2×C4—91no—
255C2×C4—91no—
256C2×C4—91no—
257C2×C4—91no—
258C2×C4D32⋊D311yescharacteristic
259C8—62no—
260C8—182no—
261C8—542no—
262C2×C3—22no—
263C2×C3—21no—
264C2×C3—21no—
265C2×C3—21no—
266C2×C3—41no—
267C2×C3—42no—
268C2×C3—41no—
269C2×C3—41no—
270C2×C3—81no—
271C2×C3—92no—
272C2×C3—181no—
273C2×C3#6373711yescharacteristic
274D3—62no—
275D3—62no—
276D3—62no—
277D3—121no—
278C22—91no—
279C22—91no—
280C22—182no—
281C22#8893411yescharacteristic
282C4—91no—
283C4—91no—
284C4#8893211yescharacteristic
285C4D32⋊Dic311yescharacteristic
286C3—21no—
287C3—21no—
288C3—41no—
289C3—41no—
290C3#1059005011yescharacteristic, complemented
291C2—21no—
292C2—92no—
293C2—181no—
294C2#1172276911yescharacteristic
subgroupG/H
centerC4D32⋊Dic3
commutatorC3×C3⋊Dic3C42
Frattini, hypercenterC2×C4D32⋊D3
FittingC2×C33×C4D4
socleC2×C33#95
prime radicalC22×C3×C3⋊D3Q8
cosocleC2×C3×C4×C3⋊D3C22
hypocenterC33#341
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
158. C33#341semidirect
251. C32#4558semidirect
290. C3#10590050semidirect

Maximal subgroups

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

subgroupindex
#117245112
C2×#888082
C3×#638182
#105900503
#45589