Finite Group Databasebeta

← #13138530

Subgroups — #13138530

Order 1728.

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

normal subgroups only

This group has 344 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#11721025C211yesmaximal, characteristic
2#11720937C211yesmaximal, characteristic, complemented
3#11720537C211yesmaximal, characteristic, complemented
4C9×#821C211yesmaximal, characteristic
5C3×#63378C211yesmaximal, characteristic, complemented
6#11720294C211yesmaximal, characteristic
7#11720265C211yesmaximal, characteristic, complemented
8#10590780—31nomaximal
9#10588446—31nomaximal
10C2×C9×D12C2211yescharacteristic
11C2×D3×Dic9—21no—
12C2×C3×C8×C9C2211yescharacteristic
13C4×C3⋊Dic9C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18(C2×D3)⋊Dic9—21no—
19Dic3⋊Dic9—21no—
20Dic9⋊Dic3—21no—
21#63898—31no—
22#63872—31no—
23#63641—31no—
24C3×#821D311yescharacteristic
25C3×#818—31no—
26#63574—31no—
27#63562—31no—
28D4×Dic9—31no—
29#63452—31no—
30C9×#102—31no—
31#63392—31no—
32#63378D311yescharacteristic, complemented
33Dic9⋊C8—31no—
34Dic9⋊C8—61no—
35#63367—31no—
36C22×D3×C9—21no—
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9D411yescharacteristic
39C2×C3⋊Dic9D411yescharacteristic
40C2×C9×Dic3—21no—
41C2×C3×Dic9—21no—
42C2×C3×Dic9—21no—
43C9×D12—21no—
44C9×D12Q812yes—
45D3×Dic9—42no—
46C3×C8×C9—21no—
47C3⋊(C9⋊C8)—21no—
48#4891—91no—
49#4766—91no—
50C2×C3×D12C2×D311yescharacteristic
51C2×D3×Dic3—61no—
52C2×C32×C8C2×D311yescharacteristic
53C4×C3⋊Dic3—31no—
54C2×C3⋊(C3⋊C8)—31no—
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3—31no—
57C3×C4⋊Dic3C2×D311yescharacteristic
58Dic3⋊Dic3—61no—
59Dic3⋊Dic3—61no—
60(C2×D3)⋊Dic3—61no—
61C2×D4×C9—31no—
62C22×Dic9—61no—
63C2×C8×C9—21no—
64C2×C8×C9C2×D311yescharacteristic
65C9×C4⋊C4—31no—
66#3676—61no—
67C4⋊Dic9—61no—
68C4⋊Dic9C2×D311yescharacteristic
69C4⋊Dic9C2×D311yescharacteristic
70Dic9⋊C4—61no—
71C4×Dic9—31no—
72C4×Dic9—61no—
73C2×C9⋊C8—31no—
74C2×C9⋊C8—61no—
75C22×C3×C9C2×D411yescharacteristic
76C2×D3×C9—22no—
77C2×D3×C9—41no—
78C2×D3×C9—41no—
79C3×C4×C9C2×Q811yescharacteristic
80C3×C4×C9#4111yescharacteristic
81C3⋊Dic9—21no—
82C3⋊Dic9—21no—
83C9×Dic3—41no—
84C3×Dic9—41no—
85C3×Dic9—41no—
86D4×Dic3—91no—
87#893—91no—
88#890—91no—
89#890—91no—
90C3×#107—91no—
91C3×#102—31no—
92#832—91no—
93#821D911yescharacteristic
94#818—31no—
95Dic3⋊C8—91no—
96Dic3⋊C8—91no—
97Dic3⋊C8—181no—
98#809—91no—
99#808—91no—
100#808—91no—
101C22×C3×D3—21no—
102C2×C32×C4C22×D311yescharacteristic
103C2×C3⋊Dic3—31no—
104C2×C3⋊Dic3—31no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C3×D12—21no—
109C3×D12Dic612yes—
110D3×Dic3—122no—
111C23×C9—61no—
112C32×C8—21no—
113C3⋊(C3⋊C8)—61no—
114D4×C9—32no—
115D4×C9—61no—
116C2×C4×C9—21no—
117C2×C4×C9—61no—
118C2×C4×C9C22×D311yescharacteristic
119C2×Dic9—21no—
120C2×Dic9—21no—
121C2×Dic9—31no—
122C2×Dic9—31no—
123C2×Dic9—61no—
124C2×Dic9—61no—
125C2×Dic9—122no—
126C8×C9—21no—
127C8×C9—41no—
128C9⋊C8—61no—
129C9⋊C8—121no—
130#478—271noSylow
131C2×C3×C9C22⋊Q811yescharacteristic
132C2×C3×C9#13511yescharacteristic
133C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
134D3×C9—42no—
135C2×C3×D4—31no—
136C22×Dic3—181no—
137C2×D12C2×D911yescharacteristic
138C2×C4×D3—181no—
139C2×C3×C8—21no—
140C2×C3×C8C2×D911yescharacteristic
141C2×C3×C8D3211yescharacteristic
142C3×C4⋊C4—31no—
143C3×C4⋊C4—91no—
144C3×C4⋊C4—91no—
145#217—181no—
146(C2×D3)⋊C4—181no—
147C4⋊Dic3—31no—
148C4⋊Dic3—31no—
149C4⋊Dic3—181no—
150C4⋊Dic3—181no—
151C4⋊Dic3C2×D911yescharacteristic
152Dic3⋊C4—181no—
153Dic3⋊C4—181no—
154C4×Dic3—91no—
155C4×Dic3—91no—
156C4×Dic3—181no—
157C2×C3⋊C8—91no—
158C2×C3⋊C8—91no—
159C2×C3⋊C8—181no—
160C22×C32D3×D411yescharacteristic
161C2×C3×D3—22no—
162C2×C3×D3—41no—
163C2×C3×D3—41no—
164C32×C4C2×Dic611yescharacteristic
165C32×C4#23711yescharacteristic
166C3⋊Dic3—61no—
167C3⋊Dic3—61no—
168C3×Dic3—41no—
169C3×Dic3—121no—
170C3×Dic3—121no—
171C22×C9—21no—
172C22×C9—62no—
173C22×C9—121no—
174C22×C9—121no—
175C22×C9D3×D411yescharacteristic
176C4×C9—21no—
177C4×C9—21no—
178C4×C9—121no—
179C4×C9#23511yescharacteristic
180C4×C9D3×Q811yescharacteristic
181Dic9—41no—
182Dic9—41no—
183Dic9—62no—
184Dic9—61no—
185Dic9—61no—
186Dic9—121no—
187#125—271no—
188C4×D4—271no—
189#107—91no—
190#106—271no—
191C4⋊C8—271no—
192#102—31no—
193#102—271no—
194C3×C9#47811yescharacteristic, complemented, Sylow
195C23×C3—61no—
196C22×D3—21no—
197C3×D4—32no—
198C3×D4—61no—
199C2×C3×C4—21no—
200C2×C3×C4—61no—
201C2×C3×C4—181no—
202C2×C3×C4—181no—
203C2×C3×C4C22×D911yescharacteristic
204C2×C3×C4C2×D3211yescharacteristic
205C2×Dic3—21no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—91no—
209C2×Dic3—91no—
210C2×Dic3—91no—
211C2×Dic3—91no—
212C2×Dic3—181no—
213C2×Dic3—181no—
214C2×Dic3—362no—
215D12—21no—
216D12Dic1812yes—
217C4×D3—362no—
218C3×C8—21no—
219C3×C8—21no—
220C3×C8—41no—
221C3⋊C8—181no—
222C3⋊C8—181no—
223C3⋊C8—361no—
224C2×C32#87811yescharacteristic
225C2×C32#91211yescharacteristic
226C2×C32#91411yescharacteristic
227C3×D3—42no—
228C2×C9—21no—
229C2×C9—21no—
230C2×C9—21no—
231C2×C9—122no—
232C2×C9#89611yescharacteristic
233C2×C9#91111yescharacteristic
234C2×C9#91911yescharacteristic
235C2×D4—31no—
236C22×C4—541no—
237C2×C8—271no—
238C2×C8D3×D911yescharacteristic
239C4⋊C4—31no—
240C4⋊C4—91no—
241C4⋊C4—91no—
242C4⋊C4—541no—
243C4⋊C4—541no—
244C22⋊C4—541no—
245C42—271no—
246C22×C3—21no—
247C22×C3—62no—
248C22×C3—121no—
249C22×C3—121no—
250C22×C3D4×D911yescharacteristic
251C22×C3#380211yescharacteristic
252C2×D3—22no—
253C2×D3—41no—
254C2×D3—41no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—121no—
258C3×C4—361no—
259C3×C4—361no—
260C3×C4C2×Dic1811yescharacteristic
261C3×C4#369911yescharacteristic
262C3×C4D3×Dic611yescharacteristic
263C3×C4#380411yescharacteristic
264Dic3—41no—
265Dic3—121no—
266Dic3—121no—
267Dic3—182no—
268Dic3—181no—
269Dic3—181no—
270Dic3—181no—
271Dic3—181no—
272Dic3—361no—
273C32#476611yescharacteristic
274C9—21no—
275C9#489111yescharacteristic, complemented
276C23—61no—
277D4—32no—
278D4—61no—
279C2×C4—61no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4—271no—
283C2×C4—271no—
284C2×C4—1082no—
285C2×C4C2×D3×D911yescharacteristic
286C8—21no—
287C8—541no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—122no—
292C2×C3#6344011yescharacteristic
293C2×C3#6347411yescharacteristic
294C2×C3#6347611yescharacteristic
295C2×C3#6379811yescharacteristic
296C2×C3#6381011yescharacteristic
297C2×C3#6389911yescharacteristic
298D3—42no—
299C22—62no—
300C22—121no—
301C22—121no—
302C22#8864611yescharacteristic
303C4—121no—
304C4—361no—
305C4—361no—
306C4—541no—
307C4—541no—
308C4D3×Dic1811yescharacteristic
309C4#8866211yescharacteristic
310C3—21no—
311C3#1058844611yescharacteristic, complemented
312C3#1059078011yescharacteristic
313C2—122no—
314C2#1172082011yescharacteristic
315C2#1172082411yescharacteristic
316C2#1172102611yescharacteristic
subgroupG/H
centerC22#88646
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×D12C2×D3
hypercenterC2×C8D3×D9
hypocenterC3×C9#478
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
2. #11720937C2semidirect
3. #11720537C2semidirect
5. C3×#63378C2semidirect
7. #11720265C2semidirect
32. #63378D3semidirect
194. C3×C9#478semidirect
275. C9#4891semidirect
311. C3#10588446semidirect

Maximal subgroups

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

subgroupindex
#117210252
#117209372
#117205372
C9×#8212
C3×#633782
#117202942
#117202652
#105907803
#105884463