Finite Group Databasebeta

← #10588692⋊C3 #13142756

Subgroups — #10588692⋊C3 #13142756

Order 1728.

353 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C4×D4×3−1+2C211yesmaximal, characteristic
2#63432⋊C3C211yesmaximal, characteristic, complemented
3#63392⋊C3C211yesmaximal, characteristic
4#63392⋊C3C211yesmaximal, characteristic
5#63367⋊C3C211yesmaximal, characteristic, complemented
6#63366⋊C3C211yesmaximal, characteristic, complemented
7C4⋊(3−1+2⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5009—31nomaximal
9#10588692C311yesmaximal, characteristic, complemented
10C2×D4×3−1+2C2211yescharacteristic
11C22×C4×3−1+2—21no—
12C4⋊C4×3−1+2C2211yescharacteristic
13C22⋊C4×3−1+2—21no—
14C42×3−1+2C2211yescharacteristic
15C4⋊(3−1+2⋊C4)C2211yescharacteristic
16C4⋊(3−1+2⋊C4)C2211yescharacteristic
17(3−1+2⋊C4)⋊C4—21no—
18(3−1+2⋊C4)⋊C4—21no—
19C2×3−1+2⋊C8C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21C32×C4×D4D311yescharacteristic
22C3×#870—31no—
23C3×#832—31no—
24C3×#832—31no—
25C3×#808—31no—
26C3×#807—31no—
27C3×C4⋊(C3⋊C8)—31no—
28C4×D4×C9—21no—
29C4×D4×C9C2×C311yescharacteristic
30#63432C2×C311yescharacteristic, complemented
31#63392C2×C311yescharacteristic
32#63392C2×C311yescharacteristic
33#63367C2×C311yescharacteristic, complemented
34#63366C2×C311yescharacteristic, complemented
35C4⋊(C9⋊C8)C2×C311yescharacteristic, complemented
36C23×3−1+2—21no—
37D4×3−1+2—21no—
38D4×3−1+2Q812yes—
39C2×C4×3−1+2—21no—
40C2×C4×3−1+2—42no—
41C2×C4×3−1+2D411yescharacteristic
42C2×C4×3−1+2D411yescharacteristic
43C2×C4×3−1+2C2311yescharacteristic
44C2×3−1+2⋊C4—21no—
45C2×3−1+2⋊C4—21no—
463−1+2⋊C8—21no—
473−1+2⋊C8—21no—
48C3×#478—91no—
49#5009—31no—
50C2×C32×D4C2×D311yescharacteristic
51C22×C32×C4—21no—
52C32×C4⋊C4C2×D311yescharacteristic
53C32×C22⋊C4—21no—
54C32×C42C2×D311yescharacteristic
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3—31no—
57C3×Dic3⋊C4—61no—
58C3×Dic3⋊C4—61no—
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊C8—31no—
61C2×D4×C9—21no—
62C2×D4×C9C22×C311yescharacteristic
63C22×C4×C9—22no—
64C22×C4×C9—21no—
65C9×C4⋊C4—21no—
66C9×C4⋊C4C22×C311yescharacteristic
67C9×C22⋊C4—22no—
68C9×C22⋊C4—21no—
69C42×C9—21no—
70C42×C9C22×C311yescharacteristic
71C4⋊Dic9C22×C311yescharacteristic
72C4⋊Dic9C22×C311yescharacteristic
73Dic9⋊C4—21no—
74Dic9⋊C4—21no—
75C2×C9⋊C8C22×C311yescharacteristic
76C2×C9⋊C8C22×C311yescharacteristic
77C22×3−1+2—22no—
78C22×3−1+2—41no—
79C22×3−1+2—41no—
80C22×3−1+2C2×D411yescharacteristic
81C4×3−1+2—21no—
82C4×3−1+2—21no—
83C4×3−1+2—41no—
84C4×3−1+2C2×Q811yescharacteristic
85C4×3−1+2#4111yescharacteristic
863−1+2⋊C4—41no—
873−1+2⋊C4—41no—
88C3×#125—91no—
89C3×C4×D4—31no—
90C3×C4×D4C3×D311yescharacteristic
91#870—31no—
92C3×#107—91no—
93C3×#106—91no—
94C3×C4⋊C8—91no—
95C3×#102—91no—
96C3×#102—91no—
97#832—31no—
98#832—31no—
99#808—31no—
100#807—31no—
101C4⋊(C3⋊C8)—31no—
102C23×C32—21no—
103C32×D4—21no—
104C32×D4Dic612yes—
105C2×C32×C4—21no—
106C2×C32×C4—42no—
107C2×C32×C4#6711yescharacteristic
108C2×C32×C4#6711yescharacteristic
109C2×C32×C4C22×D311yescharacteristic
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C23×C9—22no—
113C23×C9—21no—
114C3×C3⋊C8—61no—
115C3×C3⋊C8—61no—
116D4×C9—22no—
117D4×C9—21no—
118D4×C9—22no—
119D4×C9C3×Q812yes—
120C2×C4×C9—22no—
121C2×C4×C9—21no—
122C2×C4×C9—21no—
123C2×C4×C9—21no—
124C2×C4×C9—21no—
125C2×C4×C9—44no—
126C2×C4×C9—42no—
127C2×C4×C9C3×D411yescharacteristic
128C2×C4×C9C3×D411yescharacteristic
129C2×C4×C9C23×C311yescharacteristic
130C2×Dic9—21no—
131C2×Dic9—21no—
132C9⋊C8—21no—
133C9⋊C8—21no—
134#478—91noSylow
135C2×3−1+2—42no—
136C2×3−1+2C22⋊Q811yescharacteristic
137C2×3−1+2#13511yescharacteristic
138C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
139C2×C3×D4—31no—
140C2×C3×D4C2×C3×D311yescharacteristic
141C22×C3×C4—21no—
142C22×C3×C4—61no—
143C2×C3×C8—91no—
144C2×C3×C8—91no—
145C3×C4⋊C4—31no—
146C3×C4⋊C4—91no—
147C3×C4⋊C4—91no—
148C3×C4⋊C4—181no—
149C3×C4⋊C4—181no—
150C3×C4⋊C4C2×C3×D311yescharacteristic
151C3×C22⋊C4—21no—
152C3×C22⋊C4—61no—
153C3×C42—31no—
154C3×C42C2×C3×D311yescharacteristic
155C4⋊Dic3—31no—
156C4⋊Dic3—31no—
157Dic3⋊C4—61no—
158Dic3⋊C4—61no—
159C2×C3⋊C8—31no—
160C2×C3⋊C8—31no—
161C22×C32—22no—
162C22×C32—41no—
163C22×C32—41no—
164C22×C32C2×#6711yescharacteristic
165C32×C4—21no—
166C32×C4—21no—
167C32×C4—41no—
168C32×C4C2×Dic611yescharacteristic
169C32×C4#23511yescharacteristic
170C3×Dic3—121no—
171C3×Dic3—121no—
172C22×C9—24no—
173C22×C9—22no—
174C22×C9—21no—
175C22×C9—42no—
176C22×C9—42no—
177C22×C9—41no—
178C22×C9—41no—
179C22×C9C2×C3×D411yescharacteristic
180C4×C9—22no—
181C4×C9—21no—
182C4×C9—21no—
183C4×C9—21no—
184C4×C9—21no—
185C4×C9—41no—
186C4×C9—42no—
187C4×C9—41no—
188C4×C9C2×C3×Q811yescharacteristic
189C4×C9C3×#4111yescharacteristic
190Dic9—41no—
191Dic9—41no—
192#125—91no—
193C4×D4AGL(1,ℤ/9)11yescharacteristic
194#107—91no—
195#106—91no—
196C4⋊C8—91no—
197#102—91no—
198#102—91no—
1993−1+2#47811yescharacteristic, complemented, Sylow
200C23×C3—21no—
201C23×C3—61no—
202C3×D4—21no—
203C3×D4—32no—
204C3×D4—61no—
205C3×D4C3×Dic612yes—
206C2×C3×C4—21no—
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—31no—
210C2×C3×C4—42no—
211C2×C3×C4—61no—
212C2×C3×C4—122no—
213C2×C3×C4—181no—
214C2×C3×C4—181no—
215C2×C3×C4C3×#6711yescharacteristic
216C2×C3×C4C3×#6711yescharacteristic
217C2×C3×C4C22×C3×D311yescharacteristic
218C2×Dic3—61no—
219C2×Dic3—61no—
220C3×C8—181no—
221C3×C8—181no—
222C3⋊C8—61no—
223C3⋊C8—61no—
224C2×C32—42no—
225C2×C32#92411yescharacteristic
226C2×C32#93211yescharacteristic
227C2×C32#95011yescharacteristic
228C2×C9—21no—
229C2×C9—21no—
230C2×C9—21no—
231C2×C9—44no—
232C2×C9—42no—
233C2×C9C3×C22⋊Q811yescharacteristic
234C2×C9C3×#13511yescharacteristic
235C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
236C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
237C22×C4—21no—
238C2×C8—91no—
239C2×C8—91no—
240C4⋊C4—91no—
241C4⋊C4—91no—
242C4⋊C4—181no—
243C4⋊C4—181no—
244C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
245C22⋊C4—21no—
246C42C2×AGL(1,ℤ/9)11yescharacteristic
247C22×C3—22no—
248C22×C3—31no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—62no—
252C22×C3—121no—
253C22×C3—121no—
254C22×C3C2×C3×#6711yescharacteristic
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—31no—
258C3×C4—31no—
259C3×C4—41no—
260C3×C4—61no—
261C3×C4—61no—
262C3×C4—121no—
263C3×C4—361no—
264C3×C4—361no—
265C3×C4C2×C3×Dic611yescharacteristic
266C3×C4C3×#23511yescharacteristic
267Dic3—121no—
268Dic3—121no—
269C32#500911yescharacteristic
270C9—21no—
271C9C3×#47811yescharacteristic, complemented
272C23—21no—
273D4—21no—
274D43−1+2⋊Q812yes—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—181no—
278C2×C4—181no—
279C2×C4#611⋊C311yescharacteristic
280C2×C4#611⋊C311yescharacteristic
281C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
282C8—181no—
283C8—181no—
284C2×C3—31no—
285C2×C3—31no—
286C2×C3—31no—
287C2×C3—42no—
288C2×C3—122no—
289C2×C3C3×#92411yescharacteristic
290C2×C3C3×#93211yescharacteristic
291C2×C3C3×#95011yescharacteristic
292C22—22no—
293C22—41no—
294C22—41no—
295C22C2×#611⋊C311yescharacteristic
296C4—21no—
297C4—21no—
298C4—41no—
299C4—361no—
300C4—361no—
301C4C2×3−1+2⋊Q811yescharacteristic
302C4#3697⋊C311yescharacteristic
303C3—31no—
304C3C3×#500911yescharacteristic
305C2—42no—
306C2#63486⋊C311yescharacteristic
307C2#63495⋊C311yescharacteristic
308C2#63513⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×D4×3−1+2C2
socleC22×C3C2×C3×#67
prime radicalC2×C32×D4C2×D3
hypercenterC4×D4AGL(1,ℤ/9)
hypocenterC9C3×#478
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. #63432⋊C3C2semidirect
5. #63367⋊C3C2semidirect
6. #63366⋊C3C2semidirect
7. C4⋊(3−1+2⋊C8)C2semidirect
9. #10588692C3semidirect
30. #63432C2×C3semidirect
33. #63367C2×C3semidirect
34. #63366C2×C3semidirect
35. C4⋊(C9⋊C8)C2×C3semidirect
199. 3−1+2#478semidirect
271. C9C3×#478semidirect

Maximal subgroups

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

subgroupindex
C4×D4×3−1+22
#63432⋊C32
#63392⋊C32
#63367⋊C32
#63366⋊C32
C4⋊(3−1+2⋊C8)2
C3×#50093
#105886923