Finite Group Databasebeta

← #10588725⋊C3 #13142892

Subgroups — #10588725⋊C3 #13142892

Order 1728.

325 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 92 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
13−1+2×C22⋊Q8C211yesmaximal, characteristic
2#63488⋊C3C211yesmaximal, characteristic
3#63395⋊C3C211yesmaximal, characteristic, complemented
4#63395⋊C3C211yesmaximal, characteristic, complemented
5#63389⋊C3C211yesmaximal, characteristic
6#63367⋊C3C211yesmaximal, characteristic, complemented
7#63366⋊C3C211yesmaximal, characteristic, complemented
8C3×#5045—31nomaximal
9#10588725C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C22×C4×3−1+2C2211yescharacteristic
12C4⋊C4×3−1+2—21no—
13C4⋊C4×3−1+2C2211yescharacteristic
14C22⋊C4×3−1+2—21no—
15#3676⋊C3—21no—
16C4⋊(3−1+2⋊C4)C2211yescharacteristic
17C4⋊(3−1+2⋊C4)C2211yescharacteristic
18C4×3−1+2⋊C4—21no—
19C2×3−1+2⋊C8C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21C32×C22⋊Q8D311yescharacteristic
22C3×#926—31no—
23C3×#835—31no—
24C3×#835—31no—
25C3×#830—31no—
26C3×#808—31no—
27C3×#807—31no—
28C9×C22⋊Q8—21no—
29C9×C22⋊Q8C2×C311yescharacteristic
30#63488C2×C311yescharacteristic
31#63395C2×C311yescharacteristic, complemented
32#63395C2×C311yescharacteristic, complemented
33#63389C2×C311yescharacteristic
34#63367C2×C311yescharacteristic, complemented
35#63366C2×C311yescharacteristic, complemented
36C23×3−1+2D411yescharacteristic
37Q8×3−1+2—21no—
38Q8×3−1+2—21no—
39C2×C4×3−1+2—21no—
40C2×C4×3−1+2—21no—
41C2×C4×3−1+2—21no—
42C2×C4×3−1+2—21no—
43C2×C4×3−1+2D411yescharacteristic
44C2×C4×3−1+2C2311yescharacteristic
45C2×3−1+2⋊C4—21no—
46C2×3−1+2⋊C4—21no—
473−1+2⋊C8—21no—
483−1+2⋊C8—21no—
49C3×#485—91no—
50#5045—31no—
51C2×C32×Q8C2×D311yescharacteristic
52C22×C32×C4C2×D311yescharacteristic
53C32×C4⋊C4—21no—
54C32×C4⋊C4C2×D311yescharacteristic
55C32×C22⋊C4—21no—
56C3×#217—61no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—31no—
59C3×C4×Dic3—61no—
60C2×C3×C3⋊C8—31no—
61C2×C3×C3⋊C8—31no—
62C2×Q8×C9—21no—
63C2×Q8×C9C22×C311yescharacteristic
64C22×C4×C9—21no—
65C22×C4×C9C22×C311yescharacteristic
66C9×C4⋊C4—22no—
67C9×C4⋊C4—21no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4C22×C311yescharacteristic
70C9×C22⋊C4—22no—
71C9×C22⋊C4—21no—
72#3676—21no—
73C4⋊Dic9C22×C311yescharacteristic
74C4⋊Dic9C22×C311yescharacteristic
75C4×Dic9—21no—
76C2×C9⋊C8C22×C311yescharacteristic
77C2×C9⋊C8C22×C311yescharacteristic
78C22×3−1+2—21no—
79C22×3−1+2—21no—
80C22×3−1+2—21no—
81C22×3−1+2C2×D411yescharacteristic
82C4×3−1+2—21no—
83C4×3−1+2—41no—
84C4×3−1+2—41no—
85C4×3−1+2#4111yescharacteristic
86C4×3−1+2#4111yescharacteristic
873−1+2⋊C4—41no—
883−1+2⋊C4—41no—
89C3×C22⋊Q8—31no—
90C3×C22⋊Q8C3×D311yescharacteristic
91C3×#117—91no—
92#926—31no—
93C3×#107—91no—
94C3×#106—91no—
95C3×#103—91no—
96C3×#103—91no—
97C3×C22⋊C8—91no—
98#835—31no—
99#835—31no—
100#830—31no—
101#808—31no—
102#807—31no—
103C23×C32#6711yescharacteristic
104C32×Q8—21no—
105C32×Q8—21no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4#6711yescharacteristic
111C2×C32×C4C22×D311yescharacteristic
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C23×C9—21no—
115C23×C9C3×D411yescharacteristic
116C3×C3⋊C8—61no—
117C3×C3⋊C8—61no—
118Q8×C9—21no—
119Q8×C9—21no—
120Q8×C9—41no—
121Q8×C9—41no—
122C2×C4×C9—21no—
123C2×C4×C9—21no—
124C2×C4×C9—22no—
125C2×C4×C9—22no—
126C2×C4×C9—21no—
127C2×C4×C9—21no—
128C2×C4×C9—21no—
129C2×C4×C9—21no—
130C2×C4×C9—41no—
131C2×C4×C9—41no—
132C2×C4×C9C3×D411yescharacteristic
133C2×C4×C9C23×C311yescharacteristic
134C2×Dic9—21no—
135C2×Dic9—21no—
136C9⋊C8—21no—
137C9⋊C8—21no—
138#485—91noSylow
139C2×3−1+2—41no—
140C2×3−1+2#12311yescharacteristic
141C2×3−1+2#13511yescharacteristic
142C2×3−1+2#13711yescharacteristic
143C2×C3×Q8—31no—
144C2×C3×Q8C2×C3×D311yescharacteristic
145C22×C3×C4—31no—
146C22×C3×C4C2×C3×D311yescharacteristic
147C2×C3×C8—91no—
148C2×C3×C8—91no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—31no—
151C3×C4⋊C4—61no—
152C3×C4⋊C4—91no—
153C3×C4⋊C4—91no—
154C3×C4⋊C4C2×C3×D311yescharacteristic
155C3×C22⋊C4—21no—
156C3×C22⋊C4—61no—
157C3×C22⋊C4—181no—
158C3×C42—181no—
159#217—61no—
160C4⋊Dic3—31no—
161C4⋊Dic3—31no—
162C4×Dic3—61no—
163C2×C3⋊C8—31no—
164C2×C3⋊C8—31no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32C2×#6711yescharacteristic
169C32×C4—21no—
170C32×C4—41no—
171C32×C4—41no—
172C32×C4#23711yescharacteristic
173C32×C4#23711yescharacteristic
174C3×Dic3—121no—
175C3×Dic3—121no—
176C22×C9—22no—
177C22×C9—21no—
178C22×C9—21no—
179C22×C9—21no—
180C22×C9—21no—
181C22×C9—41no—
182C22×C9—41no—
183C22×C9C2×C3×D411yescharacteristic
184C4×C9—21no—
185C4×C9—21no—
186C4×C9—21no—
187C4×C9—41no—
188C4×C9—42no—
189C4×C9—42no—
190C4×C9—41no—
191C4×C9—41no—
192C4×C9C3×#4111yescharacteristic
193C4×C9C3×#4111yescharacteristic
194Dic9—41no—
195Dic9—41no—
196C22⋊Q8AGL(1,ℤ/9)11yescharacteristic
197#117—91no—
198#107—91no—
199#106—91no—
200#103—91no—
201#103—91no—
202C22⋊C8—91no—
2033−1+2#48511yescharacteristic, complemented, Sylow
204C23×C3—31no—
205C23×C3C3×#6711yescharacteristic
206C3×Q8—21no—
207C3×Q8—21no—
208C3×Q8—61no—
209C3×Q8—61no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—31no—
215C2×C3×C4—31no—
216C2×C3×C4—61no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—181no—
221C2×C3×C4—181no—
222C2×C3×C4C3×#6711yescharacteristic
223C2×C3×C4C22×C3×D311yescharacteristic
224C2×Dic3—61no—
225C2×Dic3—61no—
226C3×C8—181no—
227C3×C8—181no—
228C3⋊C8—61no—
229C3⋊C8—61no—
230C2×C32—41no—
231C2×C32#93511yescharacteristic
232C2×C32#94211yescharacteristic
233C2×C32#95011yescharacteristic
234C2×C9—21no—
235C2×C9—21no—
236C2×C9—21no—
237C2×C9—42no—
238C2×C9—41no—
239C2×C9C3×#12311yescharacteristic
240C2×C9C3×#13511yescharacteristic
241C2×C9C3×#13711yescharacteristic
242C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
243C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
244C2×C8—91no—
245C2×C8—91no—
246C4⋊C4—21no—
247C4⋊C4—91no—
248C4⋊C4—91no—
249C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
250C22⋊C4—21no—
251C22⋊C4—181no—
252C42—181no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—31no—
257C22×C3—61no—
258C22×C3—61no—
259C22×C3—61no—
260C22×C3C2×C3×#6711yescharacteristic
261C3×C4—21no—
262C3×C4—31no—
263C3×C4—31no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—61no—
267C3×C4—121no—
268C3×C4—121no—
269C3×C4—361no—
270C3×C4—361no—
271C3×C4C3×#23711yescharacteristic
272C3×C4C3×#23711yescharacteristic
273Dic3—121no—
274Dic3—121no—
275C32#504511yescharacteristic
276C9—21no—
277C9C3×#48511yescharacteristic, complemented
278C23#611⋊C311yescharacteristic
279Q8—21no—
280Q8—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—181no—
286C2×C4—181no—
287C2×C4#611⋊C311yescharacteristic
288C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
289C8—181no—
290C8—181no—
291C2×C3—31no—
292C2×C3—31no—
293C2×C3—31no—
294C2×C3—41no—
295C2×C3—121no—
296C2×C3C3×#93511yescharacteristic
297C2×C3C3×#94211yescharacteristic
298C2×C3C3×#95011yescharacteristic
299C22—21no—
300C22—21no—
301C22—21no—
302C22C2×#611⋊C311yescharacteristic
303C4—21no—
304C4—41no—
305C4—41no—
306C4—361no—
307C4—361no—
308C4#3699⋊C311yescharacteristic
309C4#3699⋊C311yescharacteristic
310C3—31no—
311C3C3×#504511yescharacteristic
312C2—41no—
313C2#63497⋊C311yescharacteristic
314C2#63505⋊C311yescharacteristic
315C2#63513⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C22⋊Q8C2
socleC22×C3C2×C3×#67
prime radicalC23×C32#67
hypercenterC22⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#485
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #63395⋊C3C2semidirect
4. #63395⋊C3C2semidirect
6. #63367⋊C3C2semidirect
7. #63366⋊C3C2semidirect
9. #10588725C3semidirect
31. #63395C2×C3semidirect
32. #63395C2×C3semidirect
34. #63367C2×C3semidirect
35. #63366C2×C3semidirect
203. 3−1+2#485semidirect
277. C9C3×#485semidirect

Maximal subgroups

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

subgroupindex
3−1+2×C22⋊Q82
#63488⋊C32
#63395⋊C32
#63389⋊C32
#63367⋊C32
#63366⋊C32
C3×#50453
#105887253