Finite Group Databasebeta

← #13137576

Subgroups — #13137576

Order 1728.

324 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 81 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×#63459C211yesmaximal, characteristic
2#11720907C211yesmaximal, characteristic
3#11720525C211yesmaximal, characteristic, complemented
4C9×#808C211yesmaximal, characteristic, complemented
5(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic
6#11720276C211yesmaximal, characteristic, complemented
7#11720268C211yesmaximal, characteristic, complemented
8#10590447—31nomaximal
9#10588548—31nomaximal
10C2×C3×C4×D9C2211yescharacteristic
11C2×C3×Dic18C2211yescharacteristic
12C3×C9×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C3×(C2×D9)⋊C4—21no—
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9—21no—
18C2×C9×C3⋊C8C2211yescharacteristic
19(C2×D9)⋊Dic3—21no—
20Dic3×Dic9—21no—
21C3×#897—31no—
22#63841—31no—
23#63634—31no—
24C3×#808D311yescharacteristic
25#63566—31no—
26#63565—31no—
27(C2×D3)⋊(C3⋊C8)—31no—
28#63459D311yescharacteristic
29#63454—31no—
30C9×#106—31no—
31(C2×D9)⋊C8—31no—
32#63376—31no—
33#63368—31no—
34#63366—31no—
35#63366—61no—
36C22×C3×D9D411yescharacteristic
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C9×Dic3—21no—
41C2×C3×Dic9—21no—
42C2×C3×Dic9D411yescharacteristic
43C3×C4×D9—21no—
44C3×C4×D9—21no—
45C3×Dic18—21no—
46C3×Dic18—21no—
47C3⋊(C9⋊C8)—21no—
48C9×C3⋊C8—21no—
49#5045—91no—
50#4868—91no—
51C2×C3×C4×D3—31no—
52C2×C3×Dic6—31no—
53C32×C4⋊C4C2×D311yescharacteristic
54C4⋊(C3⋊Dic3)—31no—
55C2×C3⋊(C3⋊C8)—31no—
56C3×(C2×D3)⋊C4—61no—
57C3×C4⋊Dic3—61no—
58C3×C4⋊Dic3C2×D311yescharacteristic
59C2×C3×C3⋊C8C2×D311yescharacteristic
60(C2×D3)⋊Dic3—61no—
61Dic32—61no—
62C2×C4×D9C2×D311yescharacteristic
63C2×Dic18C2×D311yescharacteristic
64C2×C8×C9—31no—
65C9×C4⋊C4—21no—
66C9×C4⋊C4—31no—
67C9×C4⋊C4C2×D311yescharacteristic
68(C2×D9)⋊C4—21no—
69(C2×D9)⋊C4—61no—
70C4⋊Dic9—21no—
71C4⋊Dic9—31no—
72C4⋊Dic9—61no—
73C4×Dic9—61no—
74C2×C9⋊C8—31no—
75C2×C9⋊C8—61no—
76C22×C3×C9C2×D411yescharacteristic
77C2×C3×D9—21no—
78C2×C3×D9—21no—
79C2×C3×D9—21no—
80C3×C4×C9—41no—
81C3×C4×C9#4111yescharacteristic
82C3×C4×C9#4111yescharacteristic
83C3⋊Dic9—41no—
84C9×Dic3—41no—
85C3×Dic9—21no—
86C3×Dic9—41no—
87C3×C22⋊Q8—91no—
88#926—91no—
89#897—31no—
90#892—91no—
91C3×#106—31no—
92#835—91no—
93#835—91no—
94#830—91no—
95(C2×D3)⋊C8—91no—
96#816—91no—
97#810—91no—
98#808D911yescharacteristic, complemented
99#807—91no—
100#807—91no—
101#807—181no—
102C22×C3×D3—31no—
103C2×C32×C4—21no—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—61no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—31no—
108C2×C3×Dic3—61no—
109C3×C4×D3—61no—
110C3×C4×D3—61no—
111C3×Dic6—61no—
112C3×Dic6—61no—
113C22×D9#6711yescharacteristic
114C3⋊(C3⋊C8)—61no—
115C3×C3⋊C8—21no—
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
125C4×D9—21no—
126C4×D9—21no—
127Dic18—21no—
128Dic18—21no—
129C8×C9—61no—
130C9⋊C8—62no—
131C9⋊C8—61no—
132#485—271noSylow
133C2×C3×C9#12311yescharacteristic
134C2×C3×C9#13511yescharacteristic
135C2×C3×C9#13711yescharacteristic
136C3×D9—41no—
137C2×C3×Q8—91no—
138C22×C3×C4—91no—
139C2×C4×D3—31no—
140C2×Dic6—31no—
141C2×C3×C8—31no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—31no—
144C3×C4⋊C4—181no—
145C3×C4⋊C4C2×D911yescharacteristic
146C3×C4⋊C4D3211yescharacteristic
147C3×C22⋊C4—181no—
148#217—181no—
149(C2×D3)⋊C4—61no—
150(C2×D3)⋊C4—181no—
151C4⋊Dic3—61no—
152C4⋊Dic3—91no—
153C4⋊Dic3—91no—
154C4⋊Dic3—181no—
155C4⋊Dic3C2×D911yescharacteristic
156C4×Dic3—181no—
157C4×Dic3—181no—
158C2×C3⋊C8—91no—
159C2×C3⋊C8—91no—
160C2×C3⋊C8—181no—
161C2×C3⋊C8C2×D911yescharacteristic
162C22×C32D3×D411yescharacteristic
163C2×C3×D3—61no—
164C2×C3×D3—61no—
165C2×C3×D3—61no—
166C32×C4—41no—
167C32×C4#23511yescharacteristic
168C32×C4#23911yescharacteristic
169C3⋊Dic3—121no—
170C3×Dic3—41no—
171C3×Dic3—61no—
172C3×Dic3—121no—
173C22×C9—21no—
174C22×C9C2×#6711yescharacteristic
175C2×D9—21no—
176C2×D9—21no—
177C2×D9—21no—
178C4×C9—21no—
179C4×C9—21no—
180C4×C9—41no—
181C4×C9—81no—
182C4×C9—121no—
183C4×C9#23711yescharacteristic
184C4×C9#23711yescharacteristic
185Dic9—21no—
186Dic9—41no—
187Dic9—121no—
188Dic9—241no—
189C22⋊Q8—91no—
190#117—271no—
191#107—271no—
192#106—31no—
193#103—271no—
194#103—271no—
195C22⋊C8—271no—
196C3×C9#48511yescharacteristic, complemented, Sylow
197C23×C3—91no—
198C22×D3—31no—
199C3×Q8—181no—
200C3×Q8—181no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—41no—
205C2×C3×C4—61no—
206C2×C3×C4—91no—
207C2×C3×C4—181no—
208C2×C3×C4—181no—
209C2×C3×C4—181no—
210C2×C3×C4C22×D911yescharacteristic
211C2×C3×C4C2×D3211yescharacteristic
212C2×Dic3—21no—
213C2×Dic3—31no—
214C2×Dic3—61no—
215C2×Dic3—181no—
216C2×Dic3—181no—
217C2×Dic3—361no—
218C4×D3—61no—
219C4×D3—61no—
220Dic6—61no—
221Dic6—61no—
222C3×C8—61no—
223C3⋊C8—21no—
224C3⋊C8—182no—
225C3⋊C8—181no—
226C3⋊C8—181no—
227C2×C32#89411yescharacteristic
228C2×C32#91511yescharacteristic
229C2×C32#91611yescharacteristic
230C3×D3—121no—
231C2×C9—21no—
232C2×C9—21no—
233C2×C9—21no—
234C2×C9#93511yescharacteristic
235C2×C9#94211yescharacteristic
236C2×C9#95011yescharacteristic
237D9—41no—
238C2×Q8—91no—
239C22×C4—91no—
240C2×C8—31no—
241C2×C8—271no—
242C4⋊C4—31no—
243C4⋊C4—181no—
244C4⋊C4—271no—
245C4⋊C4D3×D911yescharacteristic
246C22⋊C4—181no—
247C22⋊C4—541no—
248C42—541no—
249C22×C3—21no—
250C22×C3—181no—
251C22×C3—181no—
252C22×C3—181no—
253C22×C3D4×D911yescharacteristic
254C22×C3D3×#6711yescharacteristic
255C2×D3—61no—
256C2×D3—61no—
257C2×D3—61no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—81no—
263C3×C4—121no—
264C3×C4—181no—
265C3×C4—361no—
266C3×C4#369711yescharacteristic
267C3×C4#370111yescharacteristic
268C3×C4#379511yescharacteristic
269C3×C4#379611yescharacteristic
270Dic3—41no—
271Dic3—61no—
272Dic3—121no—
273Dic3—361no—
274Dic3—361no—
275Dic3—721no—
276C32#486811yescharacteristic
277C9—21no—
278C9#504511yescharacteristic, complemented
279C23—91no—
280Q8—181no—
281Q8—181no—
282C2×C4—21no—
283C2×C4—61no—
284C2×C4—91no—
285C2×C4—181no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4—541no—
289C2×C4C2×D3×D911yescharacteristic
290C8—61no—
291C8—541no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—361no—
296C2×C3#6345611yescharacteristic
297C2×C3#6347711yescharacteristic
298C2×C3#6347811yescharacteristic
299C2×C3#6388411yescharacteristic
300C2×C3#6393511yescharacteristic
301C2×C3#6394611yescharacteristic
302D3—121no—
303C22—181no—
304C22—181no—
305C22—181no—
306C22D9×#6711yescharacteristic
307C4—41no—
308C4—121no—
309C4—181no—
310C4—361no—
311C4—1081no—
312C4#8863411yescharacteristic
313C4#8864111yescharacteristic
314C3—21no—
315C3#1058854811yescharacteristic, complemented
316C3#1059044711yescharacteristic
317C2—361no—
318C2#1172098111yescharacteristic
319C2#1172118511yescharacteristic
320C2#1172121411yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC22×C3×D9D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#485
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
3. #11720525C2semidirect
4. C9×#808C2semidirect
6. #11720276C2semidirect
7. #11720268C2semidirect
98. #808D9semidirect
196. C3×C9#485semidirect
278. C9#5045semidirect
315. C3#10588548semidirect

Maximal subgroups

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

subgroupindex
C3×#634592
#117209072
#117205252
C9×#8082
(C2×D9)⋊(C3⋊C8)2
#117202762
#117202682
#105904473
#105885483