Finite Group Databasebeta

← #13137494

Subgroups — #13137494

Order 1728.

336 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C9×#888C211yesmaximal, characteristic
2#11720932C211yesmaximal, characteristic
3#11720527C211yesmaximal, characteristic
4C3×#63368C211yesmaximal, characteristic
5#11720269C211yesmaximal, characteristic
6#11720268C211yesmaximal, characteristic
7Dic3×C9⋊C8C211yesmaximal, characteristic
8#10590560—31nomaximal
9#10588766—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C9×Dic6—21no—
12C2×C3×Dic18C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C9×Dic3⋊C4—21no—
17C4×C9×Dic3C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20Dic3⋊Dic9—21no—
21C3×#888D311yescharacteristic
22#63866—31no—
23#63637—31no—
24C3×#810—31no—
25#63567—31no—
26#63565—31no—
27Dic3×C3⋊C8—31no—
28C9×C4⋊Q8—31no—
29#63431—31no—
30#63368—31no—
31#63368—31no—
32#63368—31no—
33#63368—61no—
34#63368D311yescharacteristic
35C4×C9⋊C8—31no—
36C2×C3×C4×C9—21no—
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9—21no—
39#6226—21no—
40#6226—21no—
41C2×C9×Dic3—21no—
42C2×C9×Dic3D411yescharacteristic
43C2×C9×Dic3D411yescharacteristic
44C2×C3×Dic9—21no—
45C9×Dic6—42no—
46C3×Dic18—21no—
47C3×Dic18—21no—
48C3⋊(C9⋊C8)—21no—
49C3×C9⋊C8—21no—
50#5093—91no—
51#4792—91no—
52C2×#634—31no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—31no—
55C32×C4⋊C4C2×D311yescharacteristic
56C2×C3⋊(C3⋊C8)—31no—
57C3×C4⋊Dic3C2×D311yescharacteristic
58C3×Dic3⋊C4—21no—
59C3×C4×Dic3C2×D311yescharacteristic
60C2×C3×C3⋊C8—31no—
61Dic3⋊Dic3—61no—
62Dic3⋊Dic3—61no—
63C2×Q8×C9—61no—
64C2×Dic18—31no—
65C2×Dic18—61no—
66C2×Dic18C2×D311yescharacteristic
67C9×C4⋊C4—21no—
68C9×C4⋊C4—31no—
69C9×C4⋊C4—61no—
70C9×C4⋊C4C2×D311yescharacteristic
71C42×C9—31no—
72C4⋊Dic9—61no—
73C4⋊Dic9—61no—
74C2×C9⋊C8—31no—
75C2×C9⋊C8—61no—
76C2×C9⋊C8C2×D311yescharacteristic
77C22×C3×C9C2×D411yescharacteristic
78C3×C4×C9—41no—
79C3×C4×C9#4111yescharacteristic
80C3×C4×C9#4111yescharacteristic
81C3⋊Dic9—41no—
82C9×Dic3—41no—
83C9×Dic3SD1612yes—
84C9×Dic3Q1612yes—
85C3×Dic9—41no—
86C3×C4⋊Q8—31no—
87#944—91no—
88#888D911yescharacteristic
89#869—91no—
90C3×#103—91no—
91#835—91no—
92#816—91no—
93C8×Dic3—91no—
94#810—31no—
95#810—91no—
96#810—91no—
97#810—91no—
98#810—91no—
99#810—181no—
100C4×C3⋊C8—91no—
101C2×C32×C4—21no—
102C2×C32×C4C22×D311yescharacteristic
103C2×C3⋊Dic3—61no—
104#634—61no—
105#634—61no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3#6711yescharacteristic
109C2×C3×Dic3#6711yescharacteristic
110C3×Dic6—42no—
111C3×Dic6—61no—
112C3×Dic6—61no—
113C3⋊(C3⋊C8)—61no—
114C3×C3⋊C8—61no—
115Q8×C9—122no—
116C2×C4×C9—21no—
117C2×C4×C9—21no—
118C2×C4×C9—31no—
119C2×C4×C9—31no—
120C2×C4×C9—41no—
121C2×C4×C9—61no—
122C2×C4×C9C22×D311yescharacteristic
123C2×Dic9—21no—
124C2×Dic9—61no—
125C2×Dic9—121no—
126Dic18—21no—
127Dic18—21no—
128Dic18—62no—
129Dic18—61no—
130Dic18—61no—
131Dic18—121no—
132C9⋊C8—21no—
133C9⋊C8—61no—
134C9⋊C8—121no—
135#487—271noSylow
136C2×C3×C9#12411yescharacteristic
137C2×C3×C9C2×SD1611yescharacteristic
138C2×C3×C9C2×Q1611yescharacteristic
139C2×C3×Q8—61no—
140C2×C3×Q8—91no—
141C2×Dic6—21no—
142C2×Dic6—31no—
143C2×Dic6—91no—
144C2×Dic6—91no—
145C2×Dic6—181no—
146C2×C3×C8—91no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—31no—
149C3×C4⋊C4—61no—
150C3×C4⋊C4C2×D911yescharacteristic
151C3×C4⋊C4D3211yescharacteristic
152C3×C42—31no—
153C4⋊Dic3—181no—
154C4⋊Dic3—181no—
155C4⋊Dic3C2×D911yescharacteristic
156Dic3⋊C4—21no—
157Dic3⋊C4—181no—
158Dic3⋊C4—181no—
159C4×Dic3C2×D911yescharacteristic
160C2×C3⋊C8—31no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—181no—
164C22×C32C2×#6711yescharacteristic
165C32×C4—41no—
166C32×C4#23911yescharacteristic
167C32×C4#23911yescharacteristic
168C3⋊Dic3—121no—
169C3×Dic3—41no—
170C3×Dic3—121no—
171C3×Dic3#21412yes—
172C3×Dic3#21612yes—
173C22×C9—21no—
174C22×C9D3×D411yescharacteristic
175C4×C9—21no—
176C4×C9—21no—
177C4×C9—32no—
178C4×C9—32no—
179C4×C9—41no—
180C4×C9—81no—
181C4×C9—121no—
182C4×C9#23511yescharacteristic
183C4×C9#23711yescharacteristic
184Dic9—41no—
185Dic9—122no—
186Dic9—121no—
187C4⋊Q8—31no—
188C4⋊Q8—271no—
189#103—91no—
190#103—271no—
191#103—271no—
192#103—271no—
193C4×C8—271no—
194C3×C9#48711yescharacteristic, complemented, Sylow
195C3×Q8—122no—
196C3×Q8—181no—
197C3×Q8—181no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—41no—
204C2×C3×C4—61no—
205C2×C3×C4—181no—
206C2×C3×C4C22×D911yescharacteristic
207C2×C3×C4C2×D3211yescharacteristic
208C2×Dic3—21no—
209C2×Dic3—61no—
210C2×Dic3—181no—
211C2×Dic3—181no—
212C2×Dic3—361no—
213C2×Dic3#61111yescharacteristic
214C2×Dic3#61111yescharacteristic
215Dic6—42no—
216Dic6—61no—
217Dic6—61no—
218Dic6—182no—
219Dic6—181no—
220Dic6—181no—
221Dic6—181no—
222Dic6—181no—
223Dic6—361no—
224C3×C8—181no—
225C3⋊C8—61no—
226C3⋊C8—181no—
227C3⋊C8—181no—
228C3⋊C8—361no—
229C2×C32C2×#21411yescharacteristic
230C2×C32C2×#21611yescharacteristic
231C2×C32#94711yescharacteristic
232C2×C9—21no—
233C2×C9—21no—
234C2×C9—21no—
235C2×C9#88511yescharacteristic
236C2×C9D3×SD1611yescharacteristic
237C2×C9D3×Q1611yescharacteristic
238C2×Q8—61no—
239C2×Q8—91no—
240C2×Q8—271no—
241C2×C8—91no—
242C2×C8—271no—
243C4⋊C4—31no—
244C4⋊C4—61no—
245C4⋊C4—541no—
246C4⋊C4—541no—
247C4⋊C4D3×D911yescharacteristic
248C42—31no—
249C22×C3—21no—
250C22×C3C2×#61111yescharacteristic
251C22×C3D3×#6711yescharacteristic
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—32no—
255C3×C4—32no—
256C3×C4—41no—
257C3×C4—41no—
258C3×C4—81no—
259C3×C4—121no—
260C3×C4—361no—
261C3×C4#370111yescharacteristic
262C3×C4#370111yescharacteristic
263C3×C4#379611yescharacteristic
264C3×C4#379911yescharacteristic
265Dic3—41no—
266Dic3—121no—
267Dic3—362no—
268Dic3—361no—
269Dic3—361no—
270Dic3#367212yes—
271Dic3#367412yes—
272C32#509311yescharacteristic
273C9—21no—
274C9#479211yescharacteristic, complemented
275Q8—122no—
276Q8—181no—
277Q8—181no—
278Q8—541no—
279Q8—541no—
280C2×C4—21no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—61no—
284C2×C4—181no—
285C2×C4—541no—
286C2×C4C2×D3×D911yescharacteristic
287C8—181no—
288C8—541no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3C2×#367211yescharacteristic
293C2×C3C2×#367411yescharacteristic
294C2×C3#6350911yescharacteristic
295C2×C3#6390711yescharacteristic
296C2×C3D3×#21411yescharacteristic
297C2×C3D3×#21611yescharacteristic
298C22D3×#61111yescharacteristic
299C4—32no—
300C4—32no—
301C4—41no—
302C4—121no—
303C4—361no—
304C4—1081no—
305C4#8863411yescharacteristic
306C4#8863511yescharacteristic
307C3—21no—
308C3#1058876611yescharacteristic, complemented
309C3#1059056011yescharacteristic
310C2#1172100311yescharacteristic
311C2D3×#367211yescharacteristic
312C2D3×#367411yescharacteristic
subgroupG/H
centerC22D3×#611
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socle, prime radicalC22×C32C2×#67
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#487
LCSG ▷ C2×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
194. C3×C9#487semidirect
274. C9#4792semidirect
308. C3#10588766semidirect

Maximal subgroups

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

subgroupindex
C9×#8882
#117209322
#117205272
C3×#633682
#117202692
#117202682
Dic3×C9⋊C82
#105905603
#105887663