Finite Group Databasebeta

← #10588550⋊C3 #13142152

Subgroups — #10588550⋊C3 #13142152

Order 1728.

342 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 94 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#63455⋊C3C211yesmaximal, characteristic
2#63450⋊C3C211yesmaximal, characteristic, complemented
33−1+2×#106C211yesmaximal, characteristic, complemented
4#63380⋊C3C211yesmaximal, characteristic
5(3−1+2⋊C4)⋊C8C211yesmaximal, characteristic, complemented
6#63369⋊C3C211yesmaximal, characteristic
7#63367⋊C3C211yesmaximal, characteristic, complemented
8C3×#4870—31nomaximal
9#10588550C311yesmaximal, characteristic, complemented
10C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
11C2×C4×AGL(1,ℤ/9)—21no—
12C2×3−1+2⋊Q8—21no—
13C2×C8×3−1+2C2211yescharacteristic
14C4⋊C4×3−1+2C2211yescharacteristic
15C4⋊C4×3−1+2C2211yescharacteristic
16(C2×AGL(1,ℤ/9))⋊C4—21no—
17C4⋊(3−1+2⋊C4)C2211yescharacteristic
18(3−1+2⋊C4)⋊C4—21no—
19C4×3−1+2⋊C4C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21C3×#893—31no—
22C3×#888—31no—
23C32×#106D311yescharacteristic
24C3×#821—31no—
25C3×Dic3⋊C8—31no—
26C3×#809—31no—
27C3×#808—31no—
28#63455C2×C311yescharacteristic
29#63450C2×C311yescharacteristic, complemented
30C9×#106—21no—
31C9×#106C2×C311yescharacteristic, complemented
32#63380C2×C311yescharacteristic
33Dic9⋊C8C2×C311yescharacteristic, complemented
34#63369C2×C311yescharacteristic
35#63367C2×C311yescharacteristic, complemented
36C22×AGL(1,ℤ/9)—21no—
37C2×C4×3−1+2—21no—
38C2×C4×3−1+2—21no—
39C2×C4×3−1+2C2311yescharacteristic
40C2×3−1+2⋊C4—21no—
41C2×3−1+2⋊C4D411yescharacteristic
42C2×3−1+2⋊C4D411yescharacteristic
43C4⋊AGL(1,ℤ/9)—21no—
44C4⋊AGL(1,ℤ/9)Q812yescomplemented
45C4×AGL(1,ℤ/9)—42no—
463−1+2⋊Q8—42no—
47C8×3−1+2—21no—
483−1+2⋊C8—21no—
49C3×#476—91no—
50#4870—31no—
51C2×C3×D12—31no—
52C2×C3×C4×D3—61no—
53C2×C3×Dic6—61no—
54C2×C32×C8C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56C32×C4⋊C4C2×D311yescharacteristic
57C3×(C2×D3)⋊C4—61no—
58C3×C4⋊Dic3—31no—
59C3×Dic3⋊C4—61no—
60C3×C4×Dic3—31no—
61C2×C3×C3⋊C8—31no—
62C2×D36C22×C311yescharacteristic
63C2×C4×D9—21no—
64C2×Dic18—21no—
65C2×C8×C9—21no—
66C2×C8×C9C22×C311yescharacteristic
67C9×C4⋊C4—21no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4C22×C311yescharacteristic
70C9×C4⋊C4C22×C311yescharacteristic
71(C2×D9)⋊C4—21no—
72C4⋊Dic9C22×C311yescharacteristic
73Dic9⋊C4—21no—
74C4×Dic9C22×C311yescharacteristic
75C2×C9⋊C8C22×C311yescharacteristic
76C22×3−1+2C2×D411yescharacteristic
77C2×AGL(1,ℤ/9)—22no—
78C2×AGL(1,ℤ/9)—41no—
79C2×AGL(1,ℤ/9)—41no—
80C4×3−1+2—41no—
81C4×3−1+2—41no—
82C4×3−1+2C2×Q811yescharacteristic
83C4×3−1+2#4111yescharacteristic
843−1+2⋊C4—21no—
853−1+2⋊C4—41no—
863−1+2⋊C4SD1612yes—
87C3×C4⋊Q8—91no—
88C3×C4×D4—91no—
89#893—31no—
90#888—31no—
91C3×#106—31no—
92C3×#106—91no—
93C3×#106C3×D311yescharacteristic
94C3×C4⋊C8—91no—
95C3×#102—91no—
96C3×#102—91no—
97#821—31no—
98Dic3⋊C8—31no—
99#809—31no—
100#808—31no—
101C22×C3×D3—61no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3×Dic3—31no—
106C2×C3×Dic3—31no—
107C2×C3×Dic3—61no—
108C3×D12—32no—
109C3×D12—61no—
110C3×C4×D3—122no—
111C3×Dic6—122no—
112C22×D9—21no—
113C32×C8—21no—
114C3×C3⋊C8—61no—
115C2×C4×C9—21no—
116C2×C4×C9—21no—
117C2×C4×C9—21no—
118C2×C4×C9—41no—
119C2×C4×C9—41no—
120C2×C4×C9C23×C311yescharacteristic
121C2×Dic9—21no—
122C2×Dic9C3×D411yescharacteristic
123C2×Dic9C3×D411yescharacteristic
124D36—21no—
125D36C3×Q812yescomplemented
126C4×D9—42no—
127Dic18—42no—
128C8×C9—22no—
129C8×C9—21no—
130C9⋊C8—21no—
131#476—91noSylow
132C2×3−1+2C22⋊Q811yescharacteristic
133C2×3−1+2C2×SD1611yescharacteristic
134C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
135AGL(1,ℤ/9)—42no—
136C2×C3×Q8—181no—
137C2×C3×D4—91no—
138C22×C3×C4—181no—
139C2×D12—31no—
140C2×C4×D3—61no—
141C2×Dic6—61no—
142C2×C3×C8—31no—
143C2×C3×C8—91no—
144C2×C3×C8C2×C3×D311yescharacteristic
145C3×C4⋊C4—31no—
146C3×C4⋊C4—31no—
147C3×C4⋊C4—91no—
148C3×C4⋊C4—181no—
149C3×C4⋊C4C2×C3×D311yescharacteristic
150C3×C4⋊C4C2×C3×D311yescharacteristic
151C3×C22⋊C4—181no—
152C3×C42—91no—
153(C2×D3)⋊C4—61no—
154C4⋊Dic3—31no—
155Dic3⋊C4—61no—
156C4×Dic3—31no—
157C2×C3⋊C8—31no—
158C22×C32D3×D411yescharacteristic
159C2×C3×D3—62no—
160C2×C3×D3—121no—
161C2×C3×D3—121no—
162C32×C4—41no—
163C32×C4—41no—
164C32×C4#23511yescharacteristic
165C32×C4D3×Q811yescharacteristic
166C3×Dic3—32no—
167C3×Dic3—61no—
168C3×Dic3—121no—
169C22×C9—21no—
170C22×C9C2×C3×D411yescharacteristic
171C2×D9—22no—
172C2×D9—41no—
173C2×D9—41no—
174C4×C9—21no—
175C4×C9—21no—
176C4×C9—41no—
177C4×C9—41no—
178C4×C9—81no—
179C4×C9—81no—
180C4×C9C2×C3×Q811yescharacteristic
181C4×C9C3×#4111yescharacteristic
182Dic9—21no—
183Dic9—41no—
184Dic9C3×SD1612yes—
185C4⋊Q8—91no—
186C4×D4—91no—
187#106—91no—
188#106AGL(1,ℤ/9)11yescharacteristic, complemented
189C4⋊C8—91no—
190#102—91no—
191#102—91no—
1923−1+2#47611yescharacteristic, complemented, Sylow
193C23×C3—181no—
194C22×D3—61no—
195C3×Q8—362no—
196C3×D4—92no—
197C3×D4—181no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—31no—
201C2×C3×C4—61no—
202C2×C3×C4—61no—
203C2×C3×C4—91no—
204C2×C3×C4—91no—
205C2×C3×C4—181no—
206C2×C3×C4—362no—
207C2×C3×C4C22×C3×D311yescharacteristic
208C2×Dic3—31no—
209C2×Dic3—31no—
210C2×Dic3—61no—
211D12—32no—
212D12—61no—
213C4×D3—122no—
214Dic6—122no—
215C3×C8—21no—
216C3×C8—61no—
217C3×C8—181no—
218C3⋊C8—61no—
219C2×C32#89611yescharacteristic
220C2×C32D3×SD1611yescharacteristic
221C2×C32#91411yescharacteristic
222C3×D3—122no—
223C2×C9—21no—
224C2×C9—21no—
225C2×C9—21no—
226C2×C9C3×C22⋊Q811yescharacteristic
227C2×C9C2×C3×SD1611yescharacteristic
228C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
229D9—42no—
230C2×Q8—181no—
231C2×D4—91no—
232C22×C4—181no—
233C2×C8—91no—
234C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
235C4⋊C4—91no—
236C4⋊C4—181no—
237C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
238C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
239C22⋊C4—181no—
240C42—91no—
241C22×C3—31no—
242C22×C3—182no—
243C22×C3—361no—
244C22×C3—361no—
245C22×C3C3×D3×D411yescharacteristic
246C2×D3—62no—
247C2×D3—121no—
248C2×D3—121no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4—92no—
254C3×C4—121no—
255C3×C4—121no—
256C3×C4—181no—
257C3×C4—361no—
258C3×C4C3×#23511yescharacteristic
259C3×C4C3×D3×Q811yescharacteristic
260Dic3—32no—
261Dic3—61no—
262Dic3—121no—
263C32#487011yescharacteristic
264C9—21no—
265C9C3×#47611yescharacteristic, complemented
266C23—181no—
267Q8—362no—
268D4—92no—
269D4—181no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—91no—
273C2×C4—91no—
274C2×C4—181no—
275C2×C4—362no—
276C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
277C8—21no—
278C8—181no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—362no—
283C2×C3C3×#89611yescharacteristic
284C2×C3C3×D3×SD1611yescharacteristic
285C2×C3C3×#91411yescharacteristic
286D3—122no—
287C22—182no—
288C22—361no—
289C22—361no—
290C22D4×AGL(1,ℤ/9)11yescharacteristic
291C4—41no—
292C4—41no—
293C4—92no—
294C4—181no—
295C4—361no—
296C4#3697⋊C311yescharacteristic
297C4Q8×AGL(1,ℤ/9)11yescharacteristic
298C3—31no—
299C3C3×#487011yescharacteristic
300C2—362no—
301C2#63458⋊C311yescharacteristic
302C2SD16×AGL(1,ℤ/9)11yescharacteristic
303C2#63476⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#106C2
socleC22×C3C3×D3×D4
prime radicalC2×C4⋊AGL(1,ℤ/9)C22
hypercenter#106AGL(1,ℤ/9)
hypocenterC9C3×#476
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
2. #63450⋊C3C2semidirect
3. 3−1+2×#106C2semidirect
5. (3−1+2⋊C4)⋊C8C2semidirect
7. #63367⋊C3C2semidirect
9. #10588550C3semidirect
29. #63450C2×C3semidirect
31. C9×#106C2×C3semidirect
33. Dic9⋊C8C2×C3semidirect
35. #63367C2×C3semidirect
44. C4⋊AGL(1,ℤ/9)Q8semidirect
125. D36C3×Q8semidirect
188. #106AGL(1,ℤ/9)semidirect
192. 3−1+2#476semidirect
265. C9C3×#476semidirect

Maximal subgroups

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

subgroupindex
#63455⋊C32
#63450⋊C32
3−1+2×#1062
#63380⋊C32
(3−1+2⋊C4)⋊C82
#63369⋊C32
#63367⋊C32
C3×#48703
#105885503