Finite Group Databasebeta

← #10588403⋊C3 #13141628

Subgroups — #10588403⋊C3 #13141628

Order 1728.

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

normal subgroups only

This group has 313 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
1#63488⋊C3C211yesmaximal, characteristic
2#63486⋊C3C211yesmaximal, characteristic
33−1+2×C22⋊C8C211yesmaximal, characteristic
4#63378⋊C3C211yesmaximal, characteristic, complemented
5#63377⋊C3C211yesmaximal, characteristic, complemented
6#63376⋊C3C211yesmaximal, characteristic, complemented
7#63376⋊C3C211yesmaximal, characteristic, complemented
8C3×#4723—31nomaximal
9#10588403C311yesmaximal, characteristic, complemented
10C22×C4×3−1+2C2211yescharacteristic
11C2×3−1+2⋊Q8C2211yescharacteristic
12C2×C8×3−1+2C2211yescharacteristic
13C2×C8×3−1+2C2211yescharacteristic
14#3676⋊C3—21no—
15#3676⋊C3—21no—
16C4⋊(3−1+2⋊C4)C2211yescharacteristic
17C4⋊(3−1+2⋊C4)C2211yescharacteristic
18C4⋊(3−1+2⋊C4)C2211yescharacteristic
19(3−1+2⋊C4)⋊C4—21no—
20C4×3−1+2⋊C4—21no—
21C3×#926—31no—
22C3×#924—31no—
23C32×C22⋊C8D311yescharacteristic
24C3×#818—31no—
25C3×#817—31no—
26C3×#816—31no—
27C3×#816—31no—
28#63488C2×C311yescharacteristic
29#63486C2×C311yescharacteristic
30C9×C22⋊C8—21no—
31C9×C22⋊C8C2×C311yescharacteristic
32#63378C2×C311yescharacteristic, complemented
33#63377C2×C311yescharacteristic, complemented
34#63376C2×C311yescharacteristic, complemented
35#63376C2×C311yescharacteristic, complemented
36C23×3−1+2D411yescharacteristic
37C2×C4×3−1+2—21no—
38C2×C4×3−1+2—21no—
39C2×C4×3−1+2D411yescharacteristic
40C2×C4×3−1+2C2311yescharacteristic
41C2×3−1+2⋊C4—21no—
42C2×3−1+2⋊C4—21no—
43C2×3−1+2⋊C4—21no—
44C2×3−1+2⋊C4—21no—
453−1+2⋊Q8—21no—
463−1+2⋊Q8—21no—
47C8×3−1+2—21no—
48C8×3−1+2—21no—
49C3×#485—91no—
50#4723—31no—
51C22×C32×C4C2×D311yescharacteristic
52C2×C3×Dic6—31no—
53C2×C32×C8C2×D311yescharacteristic
54C2×C32×C8C2×D311yescharacteristic
55C3×#217—61no—
56C3×#217—61no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3—31no—
60C3×Dic3⋊C4—61no—
61C3×C4×Dic3—61no—
62C22×C4×C9—21no—
63C22×C4×C9C22×C311yescharacteristic
64C2×Dic18C22×C311yescharacteristic
65C2×C8×C9—21no—
66C2×C8×C9—21no—
67C2×C8×C9C22×C311yescharacteristic
68C2×C8×C9C22×C311yescharacteristic
69#3676—21no—
70#3676—21no—
71C4⋊Dic9C22×C311yescharacteristic
72C4⋊Dic9C22×C311yescharacteristic
73C4⋊Dic9C22×C311yescharacteristic
74Dic9⋊C4—21no—
75C4×Dic9—21no—
76C22×3−1+2—21no—
77C22×3−1+2—21no—
78C22×3−1+2—21no—
79C22×3−1+2C2×D411yescharacteristic
80C4×3−1+2—21no—
81C4×3−1+2#4111yescharacteristic
82C4×3−1+2#4111yescharacteristic
833−1+2⋊C4—41no—
843−1+2⋊C4—41no—
853−1+2⋊C4—41no—
863−1+2⋊C4—41no—
87C3×C22⋊Q8—91no—
88C3×#117—91no—
89#926—31no—
90#924—31no—
91C3×#107—91no—
92C3×#106—91no—
93C3×#103—91no—
94C3×#103—91no—
95C3×C22⋊C8—31no—
96C3×C22⋊C8C3×D311yescharacteristic
97#818—31no—
98#817—31no—
99#816—31no—
100#816—31no—
101C23×C32D1211yescharacteristic
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4D1211yescharacteristic
105C2×C32×C4C22×D311yescharacteristic
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—61no—
110C3×Dic6—61no—
111C3×Dic6—61no—
112C23×C9—21no—
113C23×C9C3×D411yescharacteristic
114C32×C8—21no—
115C32×C8—21no—
116C2×C4×C9—21no—
117C2×C4×C9—21no—
118C2×C4×C9—21no—
119C2×C4×C9—21no—
120C2×C4×C9—41no—
121C2×C4×C9—41no—
122C2×C4×C9C3×D411yescharacteristic
123C2×C4×C9C23×C311yescharacteristic
124C2×Dic9—21no—
125C2×Dic9—21no—
126C2×Dic9—21no—
127C2×Dic9—21no—
128Dic18—21no—
129Dic18—21no—
130C8×C9—21no—
131C8×C9—21no—
132C8×C9—41no—
133C8×C9—41no—
134#485—91noSylow
135C2×3−1+2—41no—
136C2×3−1+2#12311yescharacteristic
137C2×3−1+2#13511yescharacteristic
138C2×3−1+2#13711yescharacteristic
139C2×C3×Q8—91no—
140C22×C3×C4—31no—
141C22×C3×C4C2×C3×D311yescharacteristic
142C2×Dic6—31no—
143C2×C3×C8—31no—
144C2×C3×C8—31no—
145C2×C3×C8C2×C3×D311yescharacteristic
146C2×C3×C8C2×C3×D311yescharacteristic
147C3×C4⋊C4—91no—
148C3×C4⋊C4—91no—
149C3×C4⋊C4—91no—
150C3×C4⋊C4—181no—
151C3×C22⋊C4—181no—
152C3×C22⋊C4—181no—
153C3×C42—181no—
154#217—61no—
155#217—61no—
156C4⋊Dic3—31no—
157C4⋊Dic3—31no—
158C4⋊Dic3—31no—
159Dic3⋊C4—61no—
160C4×Dic3—61no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32—21no—
164C22×C32C2×D1211yescharacteristic
165C32×C4—21no—
166C32×C4#23711yescharacteristic
167C32×C4#23711yescharacteristic
168C3×Dic3—121no—
169C3×Dic3—121no—
170C3×Dic3—121no—
171C3×Dic3—121no—
172C22×C9—22no—
173C22×C9—21no—
174C22×C9—21no—
175C22×C9—21no—
176C22×C9—21no—
177C22×C9—41no—
178C22×C9—41no—
179C22×C9C2×C3×D411yescharacteristic
180C4×C9—21no—
181C4×C9—21no—
182C4×C9—21no—
183C4×C9—41no—
184C4×C9C3×#4111yescharacteristic
185C4×C9C3×#4111yescharacteristic
186Dic9—41no—
187Dic9—41no—
188Dic9—41no—
189Dic9—41no—
190C22⋊Q8—91no—
191#117—91no—
192#107—91no—
193#106—91no—
194#103—91no—
195#103—91no—
196C22⋊C8AGL(1,ℤ/9)11yescharacteristic
1973−1+2#48511yescharacteristic, complemented, Sylow
198C23×C3—31no—
199C23×C3C3×D1211yescharacteristic
200C3×Q8—181no—
201C3×Q8—181no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—61no—
207C2×C3×C4—61no—
208C2×C3×C4—181no—
209C2×C3×C4—181no—
210C2×C3×C4—181no—
211C2×C3×C4—181no—
212C2×C3×C4C3×D1211yescharacteristic
213C2×C3×C4C22×C3×D311yescharacteristic
214C2×Dic3—61no—
215C2×Dic3—61no—
216C2×Dic3—61no—
217C2×Dic3—61no—
218Dic6—61no—
219Dic6—61no—
220C3×C8—21no—
221C3×C8—21no—
222C3×C8—61no—
223C3×C8—61no—
224C2×C32—41no—
225C2×C32#88611yescharacteristic
226C2×C32#90411yescharacteristic
227C2×C32#90911yescharacteristic
228C2×C9—21no—
229C2×C9—21no—
230C2×C9—21no—
231C2×C9—42no—
232C2×C9—41no—
233C2×C9C3×#12311yescharacteristic
234C2×C9C3×#13511yescharacteristic
235C2×C9C3×#13711yescharacteristic
236C2×Q8—91no—
237C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
238C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
239C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
240C4⋊C4—91no—
241C4⋊C4—91no—
242C4⋊C4—91no—
243C4⋊C4—181no—
244C22⋊C4—181no—
245C22⋊C4—181no—
246C42—181no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—31no—
251C22×C3—61no—
252C22×C3—61no—
253C22×C3—61no—
254C22×C3C2×C3×D1211yescharacteristic
255C3×C4—21no—
256C3×C4—31no—
257C3×C4—31no—
258C3×C4—61no—
259C3×C4—361no—
260C3×C4—361no—
261C3×C4—361no—
262C3×C4—361no—
263C3×C4C3×#23711yescharacteristic
264C3×C4C3×#23711yescharacteristic
265Dic3—121no—
266Dic3—121no—
267Dic3—121no—
268Dic3—121no—
269C32#472311yescharacteristic
270C9—21no—
271C9C3×#48511yescharacteristic, complemented
272C23C4⋊AGL(1,ℤ/9)11yescharacteristic
273Q8—181no—
274Q8—181no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—181no—
278C2×C4—181no—
279C2×C4—181no—
280C2×C4—181no—
281C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
282C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
283C8—21no—
284C8—21no—
285C2×C3—31no—
286C2×C3—31no—
287C2×C3—31no—
288C2×C3—41no—
289C2×C3—121no—
290C2×C3C3×#88611yescharacteristic
291C2×C3C3×#90411yescharacteristic
292C2×C3C3×#90911yescharacteristic
293C22—21no—
294C22—21no—
295C22—21no—
296C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
297C4—21no—
298C4—361no—
299C4—361no—
300C4—361no—
301C4—361no—
302C4#3699⋊C311yescharacteristic
303C4#3699⋊C311yescharacteristic
304C3—31no—
305C3C3×#472311yescharacteristic
306C2—41no—
307C2#63448⋊C311yescharacteristic
308C2#63467⋊C311yescharacteristic
309C2#63471⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C22⋊C8C2
socleC22×C3C2×C3×D12
prime radicalC23×C32D12
hypercenterC22⋊C8AGL(1,ℤ/9)
hypocenterC9C3×#485
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
4. #63378⋊C3C2semidirect
5. #63377⋊C3C2semidirect
6. #63376⋊C3C2semidirect
7. #63376⋊C3C2semidirect
9. #10588403C3semidirect
32. #63378C2×C3semidirect
33. #63377C2×C3semidirect
34. #63376C2×C3semidirect
35. #63376C2×C3semidirect
197. 3−1+2#485semidirect
271. C9C3×#485semidirect

Maximal subgroups

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

subgroupindex
#63488⋊C32
#63486⋊C32
3−1+2×C22⋊C82
#63378⋊C32
#63377⋊C32
#63376⋊C32
C3×#47233
#105884033