Finite Group Databasebeta

← #10594022⋊C3 #13166122

Subgroups — #10594022⋊C3 #13166122

Order 1728.

318 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 34 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⋊(C3⋊(#126⋊C3))C211yesmaximal, characteristic, complemented
2C3×C3⋊(#126⋊C3)C211yesmaximal, characteristic, complemented
3C3×C3⋊(#127⋊C3)C211yesmaximal, characteristic, complemented
4#10594022C311yesmaximal, characteristic, complemented
5#5720⋊C3—31nomaximal
6(#126⋊C3)⋊D3—31nomaximal
7D32×A4—41nomaximal
8C32×C42⋊C3C2211yescharacteristic, complemented
9#64085C2×C311yescharacteristic, complemented
10C3×#876C2×C311yescharacteristic, complemented
11C3×#874C2×C311yescharacteristic, complemented
12C3×#127⋊C3—31no—
13C3×#126⋊C3—31no—
14#63922—31no—
15#63886—31no—
16#63885—31no—
17#63884—31no—
18C3⋊(#126⋊C3)—21no—
19C3⋊(#126⋊C3)—31no—
20C3⋊(#126⋊C3)—31no—
21C3⋊(#126⋊C3)—61no—
22C3⋊(#126⋊C3)D311yescharacteristic, complemented
23C3⋊(#127⋊C3)—21no—
24C3⋊(#127⋊C3)D311yescharacteristic, complemented
25A4×C3⋊D3—41no—
26C3×D3×A4—41no—
27C3×D3×A4—41no—
28#5720—31no—
29#5711—31no—
30#560⋊C3—91no—
31C22×D32A411yescharacteristic
32C2×D3×A4—121no—
33C2×D3×A4—121no—
34C2×C3×D12—31no—
35C2×C3×D12—31no—
36C2×#626—31no—
37C2×#625—31no—
38C2×D3×Dic3—31no—
39C32×C42C22×C311yescharacteristic, complemented
40#3752—31no—
41#3750—31no—
42C3×(C2×D3)⋊C4—31no—
43C3×Dic3⋊C4—31no—
44C3×C42⋊C3—21no—
45C3×C42⋊C3—21no—
46C3×C42⋊C3—21no—
47C3×C42⋊C3—41no—
48C3×C42⋊C3C2×D311yescharacteristic, complemented
49C3×C42⋊C3C2×D311yescharacteristic, complemented
50(C2×D3)⋊Dic3—31no—
51(C2×D3)⋊Dic3—31no—
52C32×A4—41no—
53C3×D32—161no—
54C3×#127—31no—
55C3×#126—31no—
56#939—91no—
57#939—91no—
58#937—91no—
59#935—91no—
60#895—91no—
61#894—91no—
62#884—91no—
63#882—91no—
64#876—31no—
65#876—31no—
66#876—61no—
67#876C3×D311yescharacteristic, complemented
68#874C3×D311yescharacteristic, complemented
69#127⋊C3—31no—
70#127⋊C3—61no—
71#126⋊C3—31no—
72#126⋊C3—61no—
73#126⋊C3—91no—
74C22×C3⋊D3C2×A411yescharacteristic
75C22×C3×D3—31no—
76C22×C3×D3C2×A411yescharacteristic
77C22×C3×D3C2×A411yescharacteristic
78C2×C3×A4—121no—
79C2×C3×A4—121no—
80C2×D32—121no—
81D3×A4—41no—
82D3×A4—41no—
83D3×A4—81no—
84D3×A4—81no—
85D3×A4—121no—
86D3×A4—121no—
87D3×A4—241no—
88C2×C32×C4—31no—
89C2×C3⋊Dic3—31no—
90C2×C3×Dic3—31no—
91C3×D12—121no—
92C3×D12—121no—
93#626—121no—
94#625—121no—
95D3×Dic3—121no—
96#560—91noSylow
97C3×C3⋊D3—161no—
98C32×D3—161no—
99C32×D3—161no—
100C23×D3—31no—
101C23×D3—31no—
102C22×A4—361no—
103C2×C3×D4—91no—
104C2×C3×D4—91no—
105C2×#67—91no—
106C2×#67—91no—
107C2×#67—91no—
108C22×Dic3—91no—
109C2×D12—31no—
110C2×D12—31no—
111C2×D12—91no—
112C2×C4×D3—91no—
113C3×C4⋊C4—91no—
114C3×C22⋊C4—91no—
115C3×C42—21no—
116C3×C42C2×C3×D311yescharacteristic, complemented
117C3×C42C2×C3×D311yescharacteristic, complemented
118#217—91no—
119#217—91no—
120(C2×D3)⋊C4—31no—
121(C2×D3)⋊C4—91no—
122(C2×D3)⋊C4—91no—
123(C2×D3)⋊C4—91no—
124(C2×D3)⋊C4—91no—
125(C2×D3)⋊C4—181no—
126Dic3⋊C4—31no—
127Dic3⋊C4—91no—
128Dic3⋊C4—91no—
129Dic3⋊C4—181no—
130C42⋊C3—21no—
131C42⋊C3—21no—
132C42⋊C3—41no—
133C42⋊C3D3211yescharacteristic, complemented
134C22×C32C22×A411yescharacteristic
135C2×C3⋊D3—61no—
136C2×C3×D3—61no—
137C2×C3×D3—61no—
138C2×C3×D3—61no—
139C2×C3×D3—61no—
140C2×C3×D3—61no—
141C2×C3×D3—481no—
142C2×C3×D3—481no—
143C3×A4—41no—
144C3×A4—41no—
145C3×A4—81no—
146C3×A4—81no—
147C3×A4—81no—
148C3×A4—161no—
149D32—41no—
150D32—121no—
151C32×C4—61no—
152C3⋊Dic3—61no—
153C3×Dic3—61no—
154#127—31no—
155#126—31no—
156#126—91no—
157#123—271no—
158#121—271no—
159#121—271no—
160#120—271no—
161C33—161noSylow
162C23×C3—31no—
163C23×C3—31no—
164C23×C3—91no—
165C22×D3—31no—
166C22×D3—31no—
167C22×D3—31no—
168C22×D3—61no—
169C22×D3—361no—
170C22×D3—361no—
171C22×D3D3×A411yescharacteristic
172C22×D3D3×A411yescharacteristic
173C2×A4—121no—
174C2×A4—121no—
175C2×A4—241no—
176C2×A4—241no—
177C2×A4—361no—
178C3×D4—361no—
179C3×D4—361no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—61no—
183C2×C3×C4—91no—
184#67—361no—
185#67—361no—
186#67—361no—
187C2×Dic3—31no—
188C2×Dic3—91no—
189C2×Dic3—91no—
190C2×Dic3—181no—
191C2×Dic3—361no—
192D12—121no—
193D12—121no—
194D12—361no—
195C4×D3—361no—
196C2×C32—31no—
197C2×C32—481no—
198C2×C32—481no—
199C3⋊D3—41no—
200C3×D3—41no—
201C3×D3—41no—
202C3×D3—121no—
203C3×D3—161no—
204C3×D3—161no—
205C3×D3—321no—
206C3×D3—321no—
207C3×D3—481no—
208C3×D3—481no—
209C3×D3—961no—
210C24—91no—
211C2×D4—91no—
212C2×D4—91no—
213C2×D4—271no—
214C2×D4—271no—
215C22×C4—271no—
216C4⋊C4—91no—
217C4⋊C4—271no—
218C22⋊C4—91no—
219C22⋊C4—271no—
220C22⋊C4—271no—
221C22⋊C4—271no—
222C42C3×D3211yescharacteristic, complemented
223C22×C3—21no—
224C22×C3—181no—
225C22×C3—181no—
226C22×C3—181no—
227C22×C3—181no—
228C22×C3—181no—
229C22×C3—1441no—
230C22×C3C2×D3×A411yescharacteristic
231C22×C3C2×D3×A411yescharacteristic
232C2×D3—61no—
233C2×D3—61no—
234C2×D3—61no—
235C2×D3—61no—
236C2×D3—61no—
237C2×D3—121no—
238C2×D3—121no—
239C2×D3—181no—
240C2×D3—181no—
241C2×D3—361no—
242C2×D3—361no—
243C2×D3—361no—
244A4—41no—
245A4—81no—
246A4—81no—
247A4—161no—
248C3×C4—61no—
249C3×C4—61no—
250C3×C4—121no—
251C3×C4—181no—
252Dic3—61no—
253Dic3—181no—
254Dic3—181no—
255Dic3—361no—
256C32—161no—
257C32—161no—
258C32—321no—
259C32—321no—
260C32—321no—
261C32—641no—
262C32#560⋊C311yescharacteristic, complemented
263C23—31no—
264C23—31no—
265C23—91no—
266C23—91no—
267C23—1081no—
268D4—361no—
269D4—361no—
270D4—1081no—
271D4—1081no—
272C2×C4—31no—
273C2×C4—91no—
274C2×C4—271no—
275C2×C4—1081no—
276C2×C3—31no—
277C2×C3—31no—
278C2×C3—61no—
279C2×C3—121no—
280C2×C3—121no—
281C2×C3—361no—
282C2×C3—481no—
283C2×C3—481no—
284C2×C3—961no—
285C2×C3—961no—
286C2×C3—1441no—
287D3—41no—
288D3—41no—
289D3—121no—
290D3—121no—
291D3—121no—
292D3—241no—
293C22—181no—
294C22—181no—
295C22—181no—
296C22—181no—
297C22—181no—
298C22—361no—
299C22—541no—
300C22—1081no—
301C22D32×A411yescharacteristic
302C4—61no—
303C4—181no—
304C4—541no—
305C3—21no—
306C3—161no—
307C3—321no—
308C3—321no—
309C3—641no—
310C3(#126⋊C3)⋊D311yescharacteristic, complemented
311C3#5720⋊C311yescharacteristic, complemented
312C2—31no—
313C2—121no—
314C2—121no—
315C2—361no—
316C2—361no—
subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC32×C42C22×C3
FrattiniC22D32×A4
socleC22×C32C22×A4
LCSG ▷ C32×C42
UCSC1

Complements

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

NH ≅ G/Ntype
1. C3⋊(C3⋊(#126⋊C3))C2semidirect
2. C3×C3⋊(#126⋊C3)C2semidirect
3. C3×C3⋊(#127⋊C3)C2semidirect
4. #10594022C3semidirect
8. C32×C42⋊C3C22semidirect
9. #64085C2×C3semidirect
10. C3×#876C2×C3semidirect
11. C3×#874C2×C3semidirect
22. C3⋊(#126⋊C3)D3semidirect
24. C3⋊(#127⋊C3)D3semidirect
39. C32×C42C22×C3semidirect
48. C3×C42⋊C3C2×D3semidirect
49. C3×C42⋊C3C2×D3semidirect
67. #876C3×D3semidirect
68. #874C3×D3semidirect
116. C3×C42C2×C3×D3semidirect
117. C3×C42C2×C3×D3semidirect
133. C42⋊C3D32semidirect
222. C42C3×D32semidirect
262. C32#560⋊C3semidirect
310. C3(#126⋊C3)⋊D3semidirect
311. C3#5720⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3⋊(C3⋊(#126⋊C3))2
C3×C3⋊(#126⋊C3)2
C3×C3⋊(#127⋊C3)2
#105940223
#5720⋊C33
(#126⋊C3)⋊D33
D32×A44