Finite Group Databasebeta

← #13137732

Subgroups — #13137732

Order 1728.

362 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 85 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#11721746C211yesmaximal, characteristic, complemented
2#11721025C211yesmaximal, characteristic
3C9×#809C211yesmaximal, characteristic
4C3×#63367C211yesmaximal, characteristic, complemented
5#11720295C211yesmaximal, characteristic
6#11720279C211yesmaximal, characteristic, complemented
7#11720273C211yesmaximal, characteristic, complemented
8#10590474—31nomaximal
9#10588446—31nomaximal
10C2×C9×D12C2211yescharacteristic
11C2×D3×Dic9—21no—
12C3×C9×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊Dic9)—21no—
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15#88534—21no—
16C4×C3⋊Dic9C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20(C2×D3)⋊Dic9—21no—
21#64099—31no—
22#63898—31no—
23C3×#809D311yescharacteristic
24C3×#808—31no—
25#63575—31no—
26#63572—31no—
27#63563—31no—
28D4×Dic9—31no—
29#63452—31no—
30#63452—61no—
31C9×#102—31no—
32#63392—31no—
33#63378—31no—
34Dic9⋊C8—31no—
35#63367D311yescharacteristic, complemented
36C22×D3×C9—21no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9D411yescharacteristic
41C2×C3⋊Dic9D411yescharacteristic
42C2×C3×Dic9—21no—
43C9×D12—21no—
44C9×D12Q812yes—
45D3×Dic9—42no—
46C9×C3⋊C8—21no—
47C3×C9⋊C8—21no—
48#4871—91no—
49#4766—91no—
50C2×C3×D12C2×D311yescharacteristic
51C2×D3×Dic3—61no—
52C32×C4⋊C4C2×D311yescharacteristic
53C4⋊(C3⋊Dic3)—31no—
54C4⋊(C3⋊Dic3)—61no—
55#3750—61no—
56C4×C3⋊Dic3—31no—
57C3×C4⋊Dic3—31no—
58C2×C3×C3⋊C8—31no—
59C2×C3×C3⋊C8C2×D311yescharacteristic
60(C2×D3)⋊Dic3—61no—
61C2×D4×C9—31no—
62C22×Dic9—61no—
63C2×C8×C9—31no—
64C9×C4⋊C4—21no—
65C9×C4⋊C4C2×D311yescharacteristic
66#3676—61no—
67C4⋊Dic9—31no—
68C4⋊Dic9—61no—
69C4⋊Dic9—61no—
70C4⋊Dic9—62no—
71C4⋊Dic9C2×D311yescharacteristic
72Dic9⋊C4—61no—
73Dic9⋊C4—62no—
74C4×Dic9—31no—
75C4×Dic9—61no—
76C2×C9⋊C8C2×D311yescharacteristic
77C22×C3×C9C2×D411yescharacteristic
78C2×D3×C9—22no—
79C2×D3×C9—41no—
80C2×D3×C9—41no—
81C3×C4×C9—41no—
82C3×C4×C9C2×Q811yescharacteristic
83C3×C4×C9#4111yescharacteristic
84C3⋊Dic9—21no—
85C3⋊Dic9—21no—
86C3⋊Dic9—41no—
87C3×Dic9—41no—
88D4×Dic3—91no—
89#893—91no—
90#890—91no—
91#890—91no—
92#890—181no—
93C3×#106—91no—
94C3×#102—31no—
95#832—91no—
96#821—91no—
97#818—91no—
98Dic3⋊C8—91no—
99Dic3⋊C8—91no—
100#809D911yescharacteristic
101#808—31no—
102#807—91no—
103C22×C3×D3—21no—
104C2×C32×C4—21no—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—31no—
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—61no—
109C2×C3×Dic3—61no—
110C3×D12—21no—
111C3×D12Dic612yes—
112D3×Dic3—122no—
113C23×C9—61no—
114C3×C3⋊C8—21no—
115C3×C3⋊C8—61no—
116D4×C9—32no—
117D4×C9—61no—
118C2×C4×C9—22no—
119C2×C4×C9—21no—
120C2×C4×C9—21no—
121C2×C4×C9C22×D311yescharacteristic
122C2×Dic9—21no—
123C2×Dic9—31no—
124C2×Dic9—31no—
125C2×Dic9—61no—
126C2×Dic9—61no—
127C2×Dic9—62no—
128C2×Dic9—61no—
129C2×Dic9—122no—
130C8×C9—61no—
131C9⋊C8—21no—
132#478—271noSylow
133C2×C3×C9C22⋊Q811yescharacteristic
134C2×C3×C9#13511yescharacteristic
135C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
136D3×C9—42no—
137C2×C3×D4—31no—
138C22×Dic3—181no—
139C2×D12C2×D911yescharacteristic
140C2×C4×D3—181no—
141C2×C3×C8—31no—
142C2×C3×C8—91no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—91no—
145C3×C4⋊C4C2×D911yescharacteristic
146C3×C4⋊C4D3211yescharacteristic
147#217—181no—
148(C2×D3)⋊C4—181no—
149C4⋊Dic3—31no—
150C4⋊Dic3—91no—
151C4⋊Dic3—91no—
152C4⋊Dic3—181no—
153C4⋊Dic3—181no—
154C4⋊Dic3—181no—
155C4⋊Dic3—182no—
156Dic3⋊C4—181no—
157Dic3⋊C4—181no—
158Dic3⋊C4—182no—
159C4×Dic3—91no—
160C4×Dic3—91no—
161C4×Dic3—181no—
162C2×C3⋊C8—31no—
163C2×C3⋊C8C2×D911yescharacteristic
164C22×C32D3×D411yescharacteristic
165C2×C3×D3—22no—
166C2×C3×D3—41no—
167C2×C3×D3—41no—
168C32×C4—41no—
169C32×C4C2×Dic611yescharacteristic
170C32×C4#23711yescharacteristic
171C3⋊Dic3—61no—
172C3⋊Dic3—61no—
173C3⋊Dic3—121no—
174C3×Dic3—121no—
175C22×C9—21no—
176C22×C9—62no—
177C22×C9—121no—
178C22×C9—121no—
179C22×C9D3×D411yescharacteristic
180C4×C9—21no—
181C4×C9—21no—
182C4×C9—42no—
183C4×C9—41no—
184C4×C9#23511yescharacteristic
185C4×C9D3×Q811yescharacteristic
186Dic9—41no—
187Dic9—61no—
188Dic9—61no—
189Dic9—121no—
190Dic9—121no—
191Dic9—122no—
192Dic9—121no—
193#125—271no—
194C4×D4—271no—
195#107—271no—
196#106—91no—
197C4⋊C8—271no—
198#102—31no—
199#102—271no—
200C3×C9#47811yescharacteristic, complemented, Sylow
201C23×C3—61no—
202C22×D3—21no—
203C3×D4—32no—
204C3×D4—61no—
205C2×C3×C4—22no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—181no—
210C2×C3×C4C22×D911yescharacteristic
211C2×C3×C4C2×D3211yescharacteristic
212C2×Dic3—61no—
213C2×Dic3—91no—
214C2×Dic3—91no—
215C2×Dic3—91no—
216C2×Dic3—91no—
217C2×Dic3—181no—
218C2×Dic3—181no—
219C2×Dic3—182no—
220C2×Dic3—181no—
221C2×Dic3—181no—
222C2×Dic3—362no—
223D12—21no—
224D12Dic1812yes—
225C4×D3—362no—
226C3×C8—61no—
227C3×C8—181no—
228C3⋊C8—21no—
229C3⋊C8—61no—
230C2×C32#87811yescharacteristic
231C2×C32#91211yescharacteristic
232C2×C32#91411yescharacteristic
233C3×D3—42no—
234C2×C9—21no—
235C2×C9—21no—
236C2×C9—21no—
237C2×C9—122no—
238C2×C9#89611yescharacteristic
239C2×C9#91411yescharacteristic
240C2×C9#91611yescharacteristic
241C2×D4—31no—
242C22×C4—541no—
243C2×C8—31no—
244C2×C8—91no—
245C4⋊C4—91no—
246C4⋊C4—271no—
247C4⋊C4—541no—
248C4⋊C4—541no—
249C4⋊C4D3×D911yescharacteristic
250C22⋊C4—541no—
251C42—271no—
252C22×C3—21no—
253C22×C3—62no—
254C22×C3—121no—
255C22×C3—121no—
256C22×C3D4×D911yescharacteristic
257C22×C3#381111yescharacteristic
258C2×D3—22no—
259C2×D3—41no—
260C2×D3—41no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—42no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—361no—
267C3×C4C2×Dic1811yescharacteristic
268C3×C4#369911yescharacteristic
269C3×C4D3×Dic611yescharacteristic
270C3×C4#379511yescharacteristic
271Dic3—121no—
272Dic3—181no—
273Dic3—181no—
274Dic3—181no—
275Dic3—181no—
276Dic3—361no—
277Dic3—361no—
278Dic3—362no—
279Dic3—361no—
280Dic3—361no—
281C32#476611yescharacteristic
282C9—21no—
283C9#487111yescharacteristic, complemented
284C23—61no—
285D4—32no—
286D4—61no—
287C2×C4—21no—
288C2×C4—181no—
289C2×C4—271no—
290C2×C4—271no—
291C2×C4—541no—
292C2×C4—1082no—
293C2×C4C2×D3×D911yescharacteristic
294C8—61no—
295C8—181no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—122no—
300C2×C3#6344011yescharacteristic
301C2×C3#6347411yescharacteristic
302C2×C3#6347611yescharacteristic
303C2×C3#6389611yescharacteristic
304C2×C3#6393611yescharacteristic
305C2×C3#6394111yescharacteristic
306D3—42no—
307C22—62no—
308C22—121no—
309C22—121no—
310C22#8866811yescharacteristic
311C4—41no—
312C4—361no—
313C4—541no—
314C4—541no—
315C4—1081no—
316C4D3×Dic1811yescharacteristic
317C4#8863811yescharacteristic
318C3—21no—
319C3#1058844611yescharacteristic, complemented
320C3#1059047411yescharacteristic
321C2—122no—
322C2#1172102411yescharacteristic
323C2#1172117811yescharacteristic
324C2#1172121311yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×C3×D12C2×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#478
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
1. #11721746C2semidirect
4. C3×#63367C2semidirect
6. #11720279C2semidirect
7. #11720273C2semidirect
35. #63367D3semidirect
200. C3×C9#478semidirect
283. C9#4871semidirect
319. C3#10588446semidirect

Maximal subgroups

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

subgroupindex
#117217462
#117210252
C9×#8092
C3×#633672
#117202952
#117202792
#117202732
#105904743
#105884463