Finite Group Databasebeta

← #13137555

Subgroups — #13137555

Order 1728.

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

normal subgroups only

This group has 342 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
1C3×#63455C211yesmaximal, characteristic
2#11720935C211yesmaximal, characteristic, complemented
3#11720525C211yesmaximal, characteristic, complemented
4C9×#808C211yesmaximal, characteristic, complemented
5#11720283C211yesmaximal, characteristic
6Dic9⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic
8#10590545—31nomaximal
9#10588551—31nomaximal
10C2×C3×D36C2211yescharacteristic
11C2×C3×C4×D9—21no—
12C3×C9×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C3×(C2×D9)⋊C4—21no—
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4×Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20#88442—21no—
21C3×#893—31no—
22#63869—31no—
23#63634—31no—
24C3×#808D311yescharacteristic
25Dic3⋊(C3⋊C8)—31no—
26#63563—31no—
27#63562—31no—
28#63455D311yescharacteristic
29#63452—31no—
30C9×#106—31no—
31#63380—31no—
32Dic9⋊C8—31no—
33#63369—31no—
34#63366—31no—
35#63366—61no—
36C22×C3×D9—21no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C9×Dic3—21no—
41C2×C3×Dic9D411yescharacteristic
42C2×C3×Dic9D411yescharacteristic
43C3×D36—21no—
44C3×D36Q812yes—
45C3×C4×D9—42no—
46C3⋊(C9⋊C8)—21no—
47C9×C3⋊C8—21no—
48#5009—91no—
49#4871—91no—
50C2×C3×D12—31no—
51C2×C3×C4×D3—61no—
52C32×C4⋊C4C2×D311yescharacteristic
53C4⋊(C3⋊Dic3)—31no—
54C2×C3⋊(C3⋊C8)—31no—
55C3×(C2×D3)⋊C4—61no—
56C3×C4⋊Dic3C2×D311yescharacteristic
57C3×C4×Dic3—31no—
58C2×C3×C3⋊C8C2×D311yescharacteristic
59#3724—61no—
60Dic3⋊Dic3—61no—
61C2×D36C2×D311yescharacteristic
62C2×C4×D9—21no—
63C2×C8×C9—31no—
64C9×C4⋊C4—21no—
65C9×C4⋊C4—31no—
66C9×C4⋊C4C2×D311yescharacteristic
67(C2×D9)⋊C4—21no—
68C4⋊Dic9—31no—
69C4⋊Dic9—61no—
70C4⋊Dic9—61no—
71Dic9⋊C4—61no—
72C4×Dic9C2×D311yescharacteristic
73C2×C9⋊C8—31no—
74C2×C9⋊C8—61no—
75C22×C3×C9C2×D411yescharacteristic
76C2×C3×D9—22no—
77C2×C3×D9—41no—
78C2×C3×D9—41no—
79C3×C4×C9—41no—
80C3×C4×C9C2×Q811yescharacteristic
81C3×C4×C9#4111yescharacteristic
82C3⋊Dic9—41no—
83C9×Dic3—41no—
84C3×Dic9—21no—
85C3×Dic9—21no—
86C3×C4×D4—91no—
87#893—31no—
88#890—91no—
89#870—91no—
90C3×#106—31no—
91#832—91no—
92#832—91no—
93#821—91no—
94Dic3⋊C8—91no—
95#809—91no—
96#808D911yescharacteristic, complemented
97#807—91no—
98#807—91no—
99#807—181no—
100C4⋊(C3⋊C8)—91no—
101C22×C3×D3—61no—
102C2×C32×C4—21no—
103C2×C32×C4C22×D311yescharacteristic
104C2×C3⋊Dic3—61no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—31no—
107C2×C3×Dic3—31no—
108C3×D12—32no—
109C3×D12—61no—
110C3×C4×D3—122no—
111C22×D9—21no—
112C3⋊(C3⋊C8)—61no—
113C3×C3⋊C8—21no—
114C2×C4×C9—21no—
115C2×C4×C9—21no—
116C2×C4×C9—41no—
117C2×C4×C9—61no—
118C2×C4×C9C22×D311yescharacteristic
119C2×Dic9—61no—
120C2×Dic9—121no—
121C2×Dic9#6711yescharacteristic
122C2×Dic9#6711yescharacteristic
123D36—21no—
124D36Dic612yes—
125C4×D9—42no—
126C8×C9—61no—
127C9⋊C8—62no—
128C9⋊C8—61no—
129#478—271noSylow
130C2×C3×C9C22⋊Q811yescharacteristic
131C2×C3×C9#13511yescharacteristic
132C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
133C3×D9—42no—
134C2×C3×D4—91no—
135C22×C3×C4—181no—
136C2×D12—31no—
137C2×C4×D3—61no—
138C2×C3×C8—31no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—31no—
141C3×C4⋊C4C2×D911yescharacteristic
142C3×C4⋊C4D3211yescharacteristic
143C3×C22⋊C4—181no—
144C3×C42—91no—
145(C2×D3)⋊C4—61no—
146C4⋊Dic3—91no—
147C4⋊Dic3—91no—
148C4⋊Dic3—181no—
149C4⋊Dic3—181no—
150C4⋊Dic3C2×D911yescharacteristic
151Dic3⋊C4—181no—
152Dic3⋊C4—181no—
153Dic3⋊C4—181no—
154C4×Dic3—31no—
155C2×C3⋊C8—91no—
156C2×C3⋊C8—91no—
157C2×C3⋊C8—181no—
158C2×C3⋊C8C2×D911yescharacteristic
159C22×C32D3×D411yescharacteristic
160C2×C3×D3—62no—
161C2×C3×D3—121no—
162C2×C3×D3—121no—
163C32×C4—41no—
164C32×C4#23511yescharacteristic
165C32×C4D3×Q811yescharacteristic
166C3⋊Dic3—121no—
167C3×Dic3—41no—
168C3×Dic3—61no—
169C3×Dic3—61no—
170C22×C9—21no—
171C22×C9C2×#6711yescharacteristic
172C2×D9—22no—
173C2×D9—41no—
174C2×D9—41no—
175C4×C9—21no—
176C4×C9—21no—
177C4×C9—41no—
178C4×C9—81no—
179C4×C9—121no—
180C4×C9C2×Dic611yescharacteristic
181C4×C9#23511yescharacteristic
182Dic9—21no—
183Dic9—21no—
184Dic9—121no—
185Dic9—241no—
186#125—271no—
187C4×D4—91no—
188#107—271no—
189#106—31no—
190C4⋊C8—271no—
191#102—271no—
192#102—271no—
193C3×C9#47811yescharacteristic, complemented, Sylow
194C23×C3—181no—
195C22×D3—61no—
196C3×D4—92no—
197C3×D4—181no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—41no—
202C2×C3×C4—61no—
203C2×C3×C4—91no—
204C2×C3×C4—91no—
205C2×C3×C4—362no—
206C2×C3×C4C22×D911yescharacteristic
207C2×C3×C4C2×D3211yescharacteristic
208C2×Dic3—21no—
209C2×Dic3—31no—
210C2×Dic3—31no—
211C2×Dic3—181no—
212C2×Dic3—181no—
213C2×Dic3—361no—
214D12—32no—
215D12—61no—
216C4×D3—122no—
217C3×C8—61no—
218C3⋊C8—21no—
219C3⋊C8—182no—
220C3⋊C8—181no—
221C3⋊C8—181no—
222C2×C32#89611yescharacteristic
223C2×C32#91411yescharacteristic
224C2×C32#91611yescharacteristic
225C3×D3—122no—
226C2×C9—21no—
227C2×C9—21no—
228C2×C9—21no—
229C2×C9#92411yescharacteristic
230C2×C9#93211yescharacteristic
231C2×C9#95011yescharacteristic
232D9—42no—
233C2×D4—91no—
234C22×C4—181no—
235C2×C8—31no—
236C2×C8—271no—
237C4⋊C4—31no—
238C4⋊C4—271no—
239C4⋊C4—541no—
240C4⋊C4—541no—
241C4⋊C4D3×D911yescharacteristic
242C22⋊C4—181no—
243C42—91no—
244C22×C3—21no—
245C22×C3—182no—
246C22×C3—361no—
247C22×C3—361no—
248C22×C3D4×D911yescharacteristic
249C22×C3D3×#6711yescharacteristic
250C2×D3—62no—
251C2×D3—121no—
252C2×D3—121no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—81no—
258C3×C4—121no—
259C3×C4—181no—
260C3×C4—181no—
261C3×C4#369711yescharacteristic
262C3×C4Q8×D911yescharacteristic
263C3×C4D3×Dic611yescharacteristic
264C3×C4#379811yescharacteristic
265Dic3—41no—
266Dic3—61no—
267Dic3—61no—
268Dic3—361no—
269Dic3—361no—
270Dic3—721no—
271C32#487111yescharacteristic
272C9—21no—
273C9#500911yescharacteristic, complemented
274C23—181no—
275D4—92no—
276D4—181no—
277C2×C4—21no—
278C2×C4—61no—
279C2×C4—91no—
280C2×C4—91no—
281C2×C4—362no—
282C2×C4—541no—
283C2×C4C2×D3×D911yescharacteristic
284C8—61no—
285C8—541no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—362no—
290C2×C3#6345811yescharacteristic
291C2×C3#6347611yescharacteristic
292C2×C3#6347811yescharacteristic
293C2×C3#6385111yescharacteristic
294C2×C3#6393411yescharacteristic
295C2×C3#6394711yescharacteristic
296D3—122no—
297C22—182no—
298C22—361no—
299C22—361no—
300C22D9×#6711yescharacteristic
301C4—41no—
302C4—121no—
303C4—181no—
304C4—181no—
305C4—1081no—
306C4D9×Dic611yescharacteristic
307C4#8864011yescharacteristic
308C3—21no—
309C3#1058855111yescharacteristic, complemented
310C3#1059054511yescharacteristic
311C2—362no—
312C2#1172093911yescharacteristic
313C2#1172118411yescharacteristic
314C2#1172121511yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×C3×D36C22
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
2. #11720935C2semidirect
3. #11720525C2semidirect
4. C9×#808C2semidirect
6. Dic9⋊(C3⋊C8)C2semidirect
96. #808D9semidirect
193. C3×C9#478semidirect
273. C9#5009semidirect
309. C3#10588551semidirect

Maximal subgroups

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

subgroupindex
C3×#634552
#117209352
#117205252
C9×#8082
#117202832
Dic9⋊(C3⋊C8)2
#117202662
#105905453
#105885513