Finite Group Databasebeta

← #13138301

Subgroups — #13138301

Order 1728.

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

normal subgroups only

This group has 330 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
1#11720945C211yesmaximal, characteristic
2#11720907C211yesmaximal, characteristic
3#11720539C211yesmaximal, characteristic, complemented
4C3×(C2×D9)⋊C8C211yesmaximal, characteristic
5C9×#818C211yesmaximal, characteristic, complemented
6#11720269C211yesmaximal, characteristic, complemented
7#11720264C211yesmaximal, characteristic, complemented
8#10590708—31nomaximal
9#10588568—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C3×C4×D9C2211yescharacteristic
12C2×C3×C8×C9C2211yescharacteristic
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C9×C4⋊Dic3C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C2×C3×C9⋊C8C2211yescharacteristic
17(C2×D9)⋊Dic3—21no—
18(C2×D9)⋊Dic3—21no—
19Dic3⋊Dic9—21no—
20Dic3×Dic9—21no—
21#63857—31no—
22#63841—31no—
23#63643—31no—
24C3×(C2×D3)⋊C8—31no—
25C3×#818D311yescharacteristic
26#63573—31no—
27#63567—31no—
28#63459—31no—
29#63454—31no—
30C9×#107—31no—
31(C2×D9)⋊C8D311yescharacteristic
32#63376—31no—
33#63376—61no—
34#63368—31no—
35#63367—31no—
36C22×C3×D9D411yescharacteristic
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9—21no—
39C2×C3⋊Dic9—21no—
40#6226—21no—
41#6226—21no—
42C2×C9×Dic3—21no—
43C2×C9×Dic3—21no—
44C2×C3×Dic9D411yescharacteristic
45C3×C4×D9—21no—
46C3×C4×D9—21no—
47C3×C8×C9—21no—
48C3×C9⋊C8—21no—
49#4888—91no—
50#4723—91no—
51C2×#634—31no—
52C2×C3×C4×D3—31no—
53C2×C32×C8C2×D311yescharacteristic
54C4⋊(C3⋊Dic3)—31no—
55C3×C4⋊Dic3C2×D311yescharacteristic
56C3×C4⋊Dic3C2×D311yescharacteristic
57C2×C3×C3⋊C8—31no—
58Dic3⋊Dic3—61no—
59(C2×D3)⋊Dic3—61no—
60(C2×D3)⋊Dic3—61no—
61Dic32—61no—
62C2×C4×D9C2×D311yescharacteristic
63C2×Dic18—31no—
64C2×Dic18—61no—
65C2×C8×C9—21no—
66C2×C8×C9C2×D311yescharacteristic
67C9×C4⋊C4—31no—
68C9×C4⋊C4—31no—
69(C2×D9)⋊C4—61no—
70(C2×D9)⋊C4—61no—
71C4⋊Dic9—31no—
72C4⋊Dic9—61no—
73C4⋊Dic9—61no—
74C4×Dic9—61no—
75C2×C9⋊C8C2×D311yescharacteristic
76C22×C3×C9C2×D411yescharacteristic
77C2×C3×D9—21no—
78C2×C3×D9—21no—
79C2×C3×D9—21no—
80C3×C4×C9#4111yescharacteristic
81C3×C4×C9#4111yescharacteristic
82C3⋊Dic9—41no—
83C3⋊Dic9—41no—
84C9×Dic3—41no—
85C9×Dic3—41no—
86C3×Dic9—21no—
87#926—91no—
88#924—91no—
89#897—91no—
90#892—91no—
91C3×#107—31no—
92C3×C22⋊C8—91no—
93(C2×D3)⋊C8—31no—
94#818D911yescharacteristic, complemented
95#817—91no—
96#816—91no—
97#816—91no—
98#816—91no—
99#816—181no—
100#810—91no—
101#808—91no—
102C22×C3×D3—31no—
103C2×C32×C4C22×D311yescharacteristic
104C2×C3⋊Dic3—61no—
105C2×C3⋊Dic3—61no—
106#634—61no—
107#634—61no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—31no—
111C3×C4×D3—61no—
112C3×C4×D3—61no—
113C22×D9D1211yescharacteristic
114C32×C8—21no—
115C3×C3⋊C8—61no—
116C2×C4×C9—21no—
117C2×C4×C9—61no—
118C2×C4×C9—61no—
119C2×C4×C9C22×D311yescharacteristic
120C2×Dic9—61no—
121C2×Dic9—61no—
122C2×Dic9—121no—
123C2×Dic9—121no—
124C2×Dic9D1211yescharacteristic
125C4×D9—21no—
126C4×D9—21no—
127Dic18—62no—
128Dic18—61no—
129Dic18—61no—
130Dic18—121no—
131C8×C9—21no—
132C8×C9—41no—
133C9⋊C8—21no—
134#485—271noSylow
135C2×C3×C9#12311yescharacteristic
136C2×C3×C9#13511yescharacteristic
137C2×C3×C9#13711yescharacteristic
138C3×D9—41no—
139C22×C3×C4—91no—
140C2×C4×D3—31no—
141C2×Dic6—91no—
142C2×Dic6—91no—
143C2×Dic6—181no—
144C2×C3×C8—21no—
145C2×C3×C8—91no—
146C2×C3×C8C2×D911yescharacteristic
147C2×C3×C8D3211yescharacteristic
148C3×C4⋊C4—31no—
149C3×C4⋊C4—31no—
150#217—181no—
151#217—181no—
152(C2×D3)⋊C4—181no—
153(C2×D3)⋊C4—181no—
154C4⋊Dic3—91no—
155C4⋊Dic3—91no—
156C4⋊Dic3—181no—
157C4⋊Dic3—181no—
158C4⋊Dic3C2×D911yescharacteristic
159C4⋊Dic3C2×D911yescharacteristic
160Dic3⋊C4—181no—
161C4×Dic3—181no—
162C4×Dic3—181no—
163C2×C3⋊C8—31no—
164C22×C32D3×D411yescharacteristic
165C2×C3×D3—61no—
166C2×C3×D3—61no—
167C2×C3×D3—61no—
168C32×C4#23511yescharacteristic
169C32×C4#23911yescharacteristic
170C3⋊Dic3—121no—
171C3⋊Dic3—121no—
172C3×Dic3—41no—
173C3×Dic3—41no—
174C3×Dic3—61no—
175C22×C9—21no—
176C22×C9C2×D1211yescharacteristic
177C2×D9—21no—
178C2×D9—21no—
179C2×D9—21no—
180C4×C9—21no—
181C4×C9—21no—
182C4×C9—121no—
183C4×C9—121no—
184C4×C9#23711yescharacteristic
185C4×C9#23711yescharacteristic
186Dic9—21no—
187Dic9—122no—
188Dic9—121no—
189Dic9—121no—
190Dic9—241no—
191C22⋊Q8—271no—
192#117—271no—
193#107—31no—
194#106—271no—
195#103—271no—
196#103—271no—
197C22⋊C8—91no—
198C3×C9#48511yescharacteristic, complemented, Sylow
199C23×C3—91no—
200C22×D3—31no—
201C2×C3×C4—21no—
202C2×C3×C4—61no—
203C2×C3×C4—61no—
204C2×C3×C4—91no—
205C2×C3×C4—181no—
206C2×C3×C4—181no—
207C2×C3×C4C22×D911yescharacteristic
208C2×C3×C4C2×D3211yescharacteristic
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—31no—
212C2×Dic3—181no—
213C2×Dic3—181no—
214C2×Dic3—181no—
215C2×Dic3—181no—
216C2×Dic3—361no—
217C2×Dic3—361no—
218C4×D3—61no—
219C4×D3—61no—
220Dic6—182no—
221Dic6—181no—
222Dic6—181no—
223Dic6—181no—
224Dic6—181no—
225Dic6—361no—
226C3×C8—21no—
227C3×C8—21no—
228C3×C8—41no—
229C3×C8—181no—
230C3⋊C8—61no—
231C2×C32#89411yescharacteristic
232C2×C32#91211yescharacteristic
233C2×C32#91811yescharacteristic
234C3×D3—121no—
235C2×C9—21no—
236C2×C9—21no—
237C2×C9—21no—
238C2×C9#88611yescharacteristic
239C2×C9#90411yescharacteristic
240C2×C9#90911yescharacteristic
241D9—41no—
242C2×Q8—271no—
243C22×C4—91no—
244C2×C8—91no—
245C2×C8D3×D911yescharacteristic
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—271no—
249C4⋊C4—541no—
250C22⋊C4—541no—
251C22⋊C4—541no—
252C42—541no—
253C22×C3—21no—
254C22×C3—181no—
255C22×C3—181no—
256C22×C3—181no—
257C22×C3D4×D911yescharacteristic
258C22×C3D3×D1211yescharacteristic
259C2×D3—61no—
260C2×D3—61no—
261C2×D3—61no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—121no—
265C3×C4—121no—
266C3×C4—181no—
267C3×C4#369711yescharacteristic
268C3×C4#370111yescharacteristic
269C3×C4#379611yescharacteristic
270C3×C4#380411yescharacteristic
271Dic3—41no—
272Dic3—41no—
273Dic3—61no—
274Dic3—362no—
275Dic3—361no—
276Dic3—361no—
277Dic3—361no—
278Dic3—361no—
279Dic3—721no—
280C32#488811yescharacteristic
281C9—21no—
282C9#472311yescharacteristic, complemented
283C23—91no—
284Q8—541no—
285Q8—541no—
286C2×C4—61no—
287C2×C4—61no—
288C2×C4—91no—
289C2×C4—181no—
290C2×C4—181no—
291C2×C4—541no—
292C2×C4—541no—
293C2×C4C2×D3×D911yescharacteristic
294C8—21no—
295C8—181no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—361no—
300C2×C3#6345611yescharacteristic
301C2×C3#6347411yescharacteristic
302C2×C3#6348011yescharacteristic
303C2×C3#6380111yescharacteristic
304C2×C3#6380711yescharacteristic
305C2×C3#6389011yescharacteristic
306D3—121no—
307C22—181no—
308C22—181no—
309C22—181no—
310C22D9×D1211yescharacteristic
311C4—121no—
312C4—121no—
313C4—181no—
314C4—1081no—
315C4—1081no—
316C4#8863411yescharacteristic
317C4#8865811yescharacteristic
318C3—21no—
319C3#1058856811yescharacteristic, complemented
320C3#1059070811yescharacteristic
321C2—361no—
322C2#1172082211yescharacteristic
323C2#1172082811yescharacteristic
324C2#1172098811yescharacteristic
subgroupG/H
centerC22D9×D12
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC22×C3×D9D4
hypercenterC2×C8D3×D9
hypocenterC3×C9#485
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #11720539C2semidirect
5. C9×#818C2semidirect
6. #11720269C2semidirect
7. #11720264C2semidirect
94. #818D9semidirect
198. C3×C9#485semidirect
282. C9#4723semidirect
319. C3#10588568semidirect

Maximal subgroups

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

subgroupindex
#117209452
#117209072
#117205392
C3×(C2×D9)⋊C82
C9×#8182
#117202692
#117202642
#105907083
#105885683