Finite Group Databasebeta

← #13139353

Subgroups — #13139353

Order 1728.

462 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×C4×C3⋊Dic9C211yesmaximal, characteristic
2C9×#830C211yesmaximal, characteristic
3C3×#63389C211yesmaximal, characteristic
4#11720273C212yesmaximal, complemented
5#11720271C212yesmaximal, complemented
6#10591061—31nomaximal
7#10588398—31nomaximal
8C22×C3×C4×C9C2211yescharacteristic
9C22×C3⋊Dic9C411yescharacteristic
10C22×C3⋊Dic9C411yescharacteristic
11C4×C3⋊Dic9C411yescharacteristic
12C4×C3⋊Dic9C411yescharacteristic
13C4×C3⋊Dic9C2211yescharacteristic
14C4×C3⋊Dic9C2211yescharacteristic
15C2×C9×C3⋊C8C2212yes—
16C2×C3×C9⋊C8C2212yes—
17C2×C4×C3⋊Dic3—31no—
18C3×#830—31no—
19C3×#830D311yescharacteristic
20#63572—32no—
21#63553—32no—
22C2×C4×Dic9—31no—
23C2×C4×Dic9—61no—
24C9×C22⋊C8—31no—
25#63389D311yescharacteristic
26Dic9⋊C8—32no—
27C8×Dic9—32no—
28C23×C3×C9C2×C411yescharacteristic
29C2×C3×C4×C9—22no—
30C2×C3×C4×C9C2×C411yescharacteristic
31C2×C3×C4×C9C2311yescharacteristic
32C2×C3⋊Dic9—22no—
33C2×C3⋊Dic9C814yescomplemented
34C2×C3⋊Dic9C2×C411yescharacteristic
35C2×C3⋊Dic9C2×C411yescharacteristic
36C2×C3⋊Dic9C2×C411yescharacteristic
37C2×C3⋊Dic9C2×C411yescharacteristic
38C9×C3⋊C8—22no—
39C3×C9⋊C8—22no—
40#4718—91no—
41#4718—91no—
42C22×C32×C4C2×D311yescharacteristic
43C22×C3⋊Dic3—31no—
44C22×C3⋊Dic3—31no—
45C4×C3⋊Dic3—31no—
46C4×C3⋊Dic3—31no—
47C4×C3⋊Dic3—31no—
48C4×C3⋊Dic3—31no—
49C2×C3×C3⋊C8—32no—
50C2×C3×C3⋊C8C2×D312yes—
51C22×C4×C9—21no—
52C22×C4×C9C2×D311yescharacteristic
53C22×Dic9—31no—
54C22×Dic9—31no—
55C22×Dic9—61no—
56C22×Dic9—61no—
57C2×C8×C9—32no—
58C4×Dic9—31no—
59C4×Dic9—31no—
60C4×Dic9—31no—
61C4×Dic9—31no—
62C4×Dic9—61no—
63C4×Dic9—61no—
64C4×Dic9—61no—
65C4×Dic9—61no—
66C2×C9⋊C8C2×D312yes—
67C22×C3×C9—21no—
68C22×C3×C9—21no—
69C22×C3×C9C2×C812yes—
70C22×C3×C9C22×C411yescharacteristic
71C3×C4×C9—21no—
72C3×C4×C9#4112yes—
73C3⋊Dic9—21no—
74C3⋊Dic9C2×C812yes—
75C3⋊Dic9C2×C812yes—
76C3⋊Dic9M1612yes—
77C2×C4×Dic3—91no—
78C2×C4×Dic3—91no—
79C2×C4×Dic3—181no—
80C3×C22⋊C8—31no—
81C3×C22⋊C8—91no—
82#830—31no—
83#830D911yescharacteristic
84Dic3⋊C8—92no—
85Dic3⋊C8—92no—
86C8×Dic3—92no—
87C8×Dic3—92no—
88C23×C32C4×D311yescharacteristic
89C2×C32×C4—22no—
90C2×C32×C4C4×D311yescharacteristic
91C2×C32×C4C22×D311yescharacteristic
92C2×C3⋊Dic3—34no—
93C2×C3⋊Dic3—31no—
94C2×C3⋊Dic3—31no—
95C2×C3⋊Dic3—31no—
96C2×C3⋊Dic3—31no—
97C2×C3⋊Dic3—62no—
98C23×C9—21no—
99C23×C9C4×D311yescharacteristic
100C3×C3⋊C8—22no—
101C3×C3⋊C8—62no—
102C2×C4×C9—24no—
103C2×C4×C9—22no—
104C2×C4×C9—21no—
105C2×C4×C9—21no—
106C2×C4×C9C4×D311yescharacteristic
107C2×C4×C9C22×D311yescharacteristic
108C2×Dic9—34no—
109C2×Dic9—31no—
110C2×Dic9—31no—
111C2×Dic9—31no—
112C2×Dic9—31no—
113C2×Dic9—61no—
114C2×Dic9—61no—
115C2×Dic9—61no—
116C2×Dic9—61no—
117C2×Dic9—64no—
118C2×Dic9—62no—
119C2×Dic9—64no—
120C8×C9—62no—
121C9⋊C8—22no—
122#431—271noSylow
123C2×C3×C9—22no—
124C2×C3×C9#11711yescharacteristic
125C2×C3×C9C22×C811yescharacteristic
126C2×C3×C9C2×M1611yescharacteristic
127C22×C3×C4—21no—
128C22×C3×C4C2×D911yescharacteristic
129C22×C3×C4D3211yescharacteristic
130C22×Dic3—91no—
131C22×Dic3—91no—
132C22×Dic3—91no—
133C22×Dic3—91no—
134C22×Dic3—181no—
135C22×Dic3—181no—
136C2×C3×C8—32no—
137C2×C3×C8—92no—
138C4×Dic3—91no—
139C4×Dic3—91no—
140C4×Dic3—91no—
141C4×Dic3—91no—
142C4×Dic3—91no—
143C4×Dic3—91no—
144C4×Dic3—91no—
145C4×Dic3—91no—
146C4×Dic3—181no—
147C4×Dic3—181no—
148C4×Dic3—181no—
149C4×Dic3—181no—
150C2×C3⋊C8—32no—
151C2×C3⋊C8C2×D912yes—
152C22×C32—21no—
153C22×C32—21no—
154C22×C32D3×C812yes—
155C22×C32C2×C4×D311yescharacteristic
156C32×C4—21no—
157C32×C4#23712yes—
158C3⋊Dic3—32no—
159C3⋊Dic3—32no—
160C3⋊Dic3—32no—
161C3⋊Dic3—61no—
162C22×C9—22no—
163C22×C9—22no—
164C22×C9—21no—
165C22×C9—21no—
166C22×C9—22no—
167C22×C9—21no—
168C22×C9D3×C812yes—
169C22×C9C2×C4×D311yescharacteristic
170C4×C9—22no—
171C4×C9—21no—
172C4×C9—22no—
173C4×C9#23712yes—
174Dic9—32no—
175Dic9—32no—
176Dic9—32no—
177Dic9—62no—
178Dic9—62no—
179Dic9—62no—
180Dic9—62no—
181Dic9—61no—
182C2×C42—271no—
183C4⋊C8—272no—
184C22⋊C8—31no—
185C22⋊C8—91no—
186C4×C8—272no—
187C3×C9#43111yescharacteristic, complemented, Sylow
188C23×C3—21no—
189C23×C3C4×D911yescharacteristic
190C23×C3#62411yescharacteristic
191C2×C3×C4—24no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4C4×D911yescharacteristic
197C2×C3×C4C22×D911yescharacteristic
198C2×C3×C4#62411yescharacteristic
199C2×C3×C4C2×D3211yescharacteristic
200C2×Dic3—94no—
201C2×Dic3—91no—
202C2×Dic3—91no—
203C2×Dic3—91no—
204C2×Dic3—94no—
205C2×Dic3—91no—
206C2×Dic3—91no—
207C2×Dic3—91no—
208C2×Dic3—91no—
209C2×Dic3—91no—
210C2×Dic3—181no—
211C2×Dic3—181no—
212C2×Dic3—181no—
213C2×Dic3—181no—
214C2×Dic3—184no—
215C2×Dic3—182no—
216C2×Dic3—182no—
217C2×Dic3—184no—
218C3×C8—62no—
219C3×C8—182no—
220C3⋊C8—22no—
221C3⋊C8—62no—
222C2×C32—22no—
223C2×C32#87711yescharacteristic
224C2×C32C2×D3×C811yescharacteristic
225C2×C32D3×M1611yescharacteristic
226C2×C9—24no—
227C2×C9—22no—
228C2×C9—21no—
229C2×C9—21no—
230C2×C9—21no—
231C2×C9#87711yescharacteristic
232C2×C9C2×D3×C811yescharacteristic
233C2×C9D3×M1611yescharacteristic
234C22×C4—271no—
235C22×C4—271no—
236C22×C4D3×D911yescharacteristic
237C2×C8—32no—
238C2×C8—92no—
239C42—271no—
240C42—271no—
241C42—271no—
242C42—271no—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—22no—
251C22×C3C8×D912yes—
252C22×C3C2×C4×D911yescharacteristic
253C22×C3#371012yes—
254C22×C3C2×#62411yescharacteristic
255C3×C4—22no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—22no—
259C3×C4#369912yes—
260C3×C4#380512yes—
261Dic3—92no—
262Dic3—92no—
263Dic3—92no—
264Dic3—92no—
265Dic3—92no—
266Dic3—92no—
267Dic3—182no—
268Dic3—182no—
269Dic3—182no—
270Dic3—182no—
271Dic3—181no—
272Dic3—181no—
273C32#471811yescharacteristic
274C9—21no—
275C9#471811yescharacteristic, complemented
276C23(C3⋊D9)⋊C411yescharacteristic
277C2×C4—22no—
278C2×C4—271no—
279C2×C4—271no—
280C2×C4—271no—
281C2×C4—274no—
282C2×C4—271no—
283C2×C4—542no—
284C2×C4(C3⋊D9)⋊C411yescharacteristic
285C2×C4C2×D3×D911yescharacteristic
286C8—62no—
287C8—182no—
288C2×C3—24no—
289C2×C3—22no—
290C2×C3—22no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3#6343911yescharacteristic
295C2×C3C2×C8×D911yescharacteristic
296C2×C3M16×D911yescharacteristic
297C2×C3C2×#371011yescharacteristic
298C2×C3#6381811yescharacteristic
299C2×C3#6395711yescharacteristic
300C22—21no—
301C22—21no—
302C22(C3⋊D9)⋊C812yes—
303C22C2×(C3⋊D9)⋊C411yescharacteristic
304C4—21no—
305C4—272no—
306C4—272no—
307C4—272no—
308C4—541no—
309C4#8866312yes—
310C3—21no—
311C3#1058839811yescharacteristic, complemented
312C3#1059106111yescharacteristic
313C2—22no—
314C2C2×(C3⋊D9)⋊C811yescharacteristic
315C2#1172086311yescharacteristic
316C2#1172124811yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutatorC2×C3×C9C22×C8
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×C4×D3
prime radicalC23×C32C4×D3
cosocleC2×C3×C4×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#431
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11720273C2semidirect
5. #11720271C2semidirect
33. C2×C3⋊Dic9C8semidirect
187. C3×C9#431semidirect
275. C9#4718semidirect
311. C3#10588398semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊Dic92
C9×#8302
C3×#633892
#117202732
#117202712
#105910613
#105883983