Finite Group Databasebeta

← #13137658

Subgroups — #13137658

Order 1728.

372 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 101 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×#63449C211yesmaximal, characteristic
2#11720913C211yesmaximal, characteristic
3C2×#88418C211yesmaximal, characteristic
4#11720527C211yesmaximal, characteristic
5C9×#807C211yesmaximal, characteristic
6C3⋊C8×Dic9C211yesmaximal, characteristic
7#11720268C211yesmaximal, characteristic
8#10590481—31nomaximal
9#10588553—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C3×Dic18C2211yescharacteristic
12C3×C9×C4⋊C4C2211yescharacteristic
13C2×C3⋊(C9⋊C8)C2211yescharacteristic
14C9×C4⋊Dic3C2211yescharacteristic
15C3×Dic9⋊C4—21no—
16C3×C4×Dic9—21no—
17C3×C4×Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19#88442—21no—
20Dic3×Dic9—21no—
21#88418C414yescomplemented
22C3×#887—31no—
23#63844—31no—
24C2×#3719—31no—
25#63637—31no—
26C3×#807D311yescharacteristic
27#63565—31no—
28Dic3×C3⋊C8—31no—
29C2×Dic36—31no—
30#63449—31no—
31#63449D311yescharacteristic
32C9×#107—31no—
33C8×Dic9—31no—
34#63368—31no—
35#63368—31no—
36#63368—61no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40#6226—21no—
41#6226C2×C412yes—
42C2×C9×Dic3—21no—
43C2×C3×Dic9—21no—
44C2×C3×Dic9D411yescharacteristic
45C2×C3×Dic9D411yescharacteristic
46C3×Dic18—21no—
47C3×Dic18C2×C412yes—
48C3⋊(C9⋊C8)—21no—
49C9×C3⋊C8C2×C412yes—
50C4×#216—91no—
51#4873—91no—
52C2×#634—31no—
53C2×C3×Dic6—31no—
54C32×C4⋊C4C2×D311yescharacteristic
55C2×C3⋊(C3⋊C8)—31no—
56C3×C4⋊Dic3C2×D311yescharacteristic
57C3×Dic3⋊C4—61no—
58C3×C4×Dic3—31no—
59C3×C4×Dic3—61no—
60C2×C3×C3⋊C8C2×D311yescharacteristic
61#3724—61no—
62Dic32—61no—
63#3719—34no—
64C2×Dic18—31no—
65C2×Dic18—61no—
66C2×Dic18C2×D311yescharacteristic
67C2×C8×C9—31no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4—31no—
70C9×C4⋊C4C2×D311yescharacteristic
71Dic9⋊C4—21no—
72Dic9⋊C4—61no—
73C4×Dic9—21no—
74C4×Dic9—61no—
75C4×Dic9C2×D311yescharacteristic
76C2×C9⋊C8—31no—
77C2×C9⋊C8—61no—
78Dic36—34no—
79C22×C3×C9C2×D411yescharacteristic
80C3×C4×C9—41no—
81C3×C4×C9C22×C411yescharacteristic
82C3×C4×C9#4111yescharacteristic
83C3⋊Dic9—22no—
84C9×Dic3—41no—
85C3×Dic9—22no—
86C3×Dic9—21no—
87C3×Dic9Q1612yescomplemented
88C3×C4×Q8—91no—
89C2×#216—91no—
90C2×Dic12—91no—
91#887—31no—
92#887—91no—
93C4×Dic6—91no—
94C3×#107—31no—
95#835—91no—
96C8×Dic3—91no—
97#810—91no—
98#810—91no—
99#810—91no—
100#810—181no—
101#807D911yescharacteristic
102C4×C3⋊C8—91no—
103C2×C32×C4—21no—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—61no—
106#634—32no—
107#634—61no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—31no—
110C2×C3×Dic3—31no—
111C2×C3×Dic3—61no—
112C3×Dic6—32no—
113C3×Dic6—61no—
114C3⋊(C3⋊C8)—61no—
115C3×C3⋊C8C4×D312yes—
116C2×C4×C9—21no—
117C2×C4×C9—21no—
118C2×C4×C9—41no—
119C2×C4×C9—61no—
120C2×C4×C9C22×D311yescharacteristic
121C2×Dic9—21no—
122C2×Dic9—61no—
123C2×Dic9—121no—
124C2×Dic9#6711yescharacteristic
125C2×Dic9#6711yescharacteristic
126Dic18—21no—
127Dic18—32no—
128Dic18—62no—
129Dic18—61no—
130Dic18—121no—
131Dic18C4×D312yes—
132C8×C9—32no—
133C9⋊C8—61no—
134C9⋊C8—121no—
135C4×Q16—271noSylow
136C2×C3×C9C4×D411yescharacteristic
137C2×C3×C9C2×Q1611yescharacteristic
138C2×C3×C9#13511yescharacteristic
139C2×C3×Q8—91no—
140C2×Dic6—31no—
141C2×Dic6—91no—
142C2×Dic6—91no—
143C2×Dic6—181no—
144C2×C3×C8—31no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—31no—
147C3×C4⋊C4—181no—
148C3×C4⋊C4C2×D911yescharacteristic
149C3×C4⋊C4D3211yescharacteristic
150C3×C42—91no—
151C3×C42—181no—
152#216—94no—
153C4⋊Dic3C2×D911yescharacteristic
154Dic3⋊C4—61no—
155Dic3⋊C4—181no—
156Dic3⋊C4—181no—
157C4×Dic3—31no—
158C4×Dic3—61no—
159C4×Dic3—181no—
160C4×Dic3—181no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—181no—
164C2×C3⋊C8C2×D911yescharacteristic
165Dic12—94no—
166C22×C32D3×D411yescharacteristic
167C32×C4—41no—
168C32×C4C2×C4×D311yescharacteristic
169C32×C4#23911yescharacteristic
170C3⋊Dic3—62no—
171C3×Dic3—32no—
172C3×Dic3—41no—
173C3×Dic3—62no—
174C3×Dic3—61no—
175C22×C9—21no—
176C22×C9C2×#6711yescharacteristic
177C4×C9—21no—
178C4×C9—21no—
179C4×C9—41no—
180C4×C9—81no—
181C4×C9—121no—
182C4×C9C2×C4×D311yescharacteristic
183C4×C9#23511yescharacteristic
184Dic9—22no—
185Dic9—21no—
186Dic9—62no—
187Dic9—122no—
188Dic9#21612yescomplemented
189C2×Q16—271no—
190C4×Q8—91no—
191C4×Q8—271no—
192#107—31no—
193#103—271no—
194#103—271no—
195C4×C8—271no—
196C3×C9C4×Q1611yescharacteristic, complemented, Sylow
197C3×Q8—92no—
198C3×Q8—181no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—41no—
203C2×C3×C4—61no—
204C2×C3×C4—91no—
205C2×C3×C4—91no—
206C2×C3×C4—181no—
207C2×C3×C4C22×D911yescharacteristic
208C2×C3×C4C2×D3211yescharacteristic
209C2×Dic3—21no—
210C2×Dic3—31no—
211C2×Dic3—31no—
212C2×Dic3—61no—
213C2×Dic3—181no—
214C2×Dic3—181no—
215C2×Dic3—361no—
216Dic6—32no—
217Dic6—61no—
218Dic6—92no—
219Dic6—92no—
220Dic6—182no—
221Dic6—181no—
222Dic6—181no—
223Dic6—361no—
224C3×C8—32no—
225C3⋊C8—181no—
226C3⋊C8—181no—
227C3⋊C8—361no—
228C3⋊C8C4×D912yes—
229C2×C32#89311yescharacteristic
230C2×C32#91211yescharacteristic
231C2×C32D3×Q1611yescharacteristic
232C2×C9—21no—
233C2×C9—21no—
234C2×C9—21no—
235C2×C9C4×#6711yescharacteristic
236C2×C9C2×#21611yescharacteristic
237C2×C9#95011yescharacteristic
238C2×Q8—91no—
239C2×Q8—271no—
240Q16—274no—
241C2×C8—31no—
242C2×C8—271no—
243C4⋊C4—31no—
244C4⋊C4—181no—
245C4⋊C4—541no—
246C4⋊C4D3×D911yescharacteristic
247C42—91no—
248C42—181no—
249C42—541no—
250C22×C3—21no—
251C22×C3D4×D911yescharacteristic
252C22×C3D3×#6711yescharacteristic
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—81no—
258C3×C4—92no—
259C3×C4—121no—
260C3×C4—182no—
261C3×C4—181no—
262C3×C4C2×C4×D911yescharacteristic
263C3×C4#370111yescharacteristic
264C3×C4#379911yescharacteristic
265C3×C4C4×D3211yescharacteristic
266Dic3—32no—
267Dic3—41no—
268Dic3—62no—
269Dic3—61no—
270Dic3—182no—
271Dic3—182no—
272Dic3—362no—
273C32#487311yescharacteristic
274C9—21no—
275C9C4×#21611yescharacteristic, complemented
276Q8—92no—
277Q8—181no—
278Q8—272no—
279Q8—541no—
280C2×C4—21no—
281C2×C4—61no—
282C2×C4—91no—
283C2×C4—91no—
284C2×C4—181no—
285C2×C4—541no—
286C2×C4C2×D3×D911yescharacteristic
287C8—32no—
288C8—541no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3#6345511yescharacteristic
293C2×C3#6347411yescharacteristic
294C2×C3Q16×D911yescharacteristic
295C2×C3#6390411yescharacteristic
296C2×C3#6393311yescharacteristic
297C2×C3D3×#21611yescharacteristic
298C22D9×#6711yescharacteristic
299C4—41no—
300C4—92no—
301C4—121no—
302C4—182no—
303C4—181no—
304C4—542no—
305C4#8863511yescharacteristic
306C4C4×D3×D911yescharacteristic
307C3—21no—
308C3#1058855311yescharacteristic, complemented
309C3#1059048111yescharacteristic
310C2#1172100211yescharacteristic
311C2#1172117711yescharacteristic
312C2D9×#21611yescharacteristic
subgroupG/H
centerC22D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socle, prime radicalC22×C32D3×D4
cosocleC2×C3×C4×C9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9C4×Q16
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
21. #88418C4semidirect
87. C3×Dic9Q16semidirect
188. Dic9#216semidirect
196. C3×C9C4×Q16semidirect
275. C9C4×#216semidirect
308. C3#10588553semidirect

Maximal subgroups

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

subgroupindex
C3×#634492
#117209132
C2×#884182
#117205272
C9×#8072
C3⋊C8×Dic92
#117202682
#105904813
#105885533