Finite Group Databasebeta

← #10588551⋊C3 #13142155

Subgroups — #10588551⋊C3 #13142155

Order 1728.

324 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 90 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#63452⋊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#63366⋊C3C211yesmaximal, characteristic, complemented
8C3×#4871—31nomaximal
9#10588551C311yesmaximal, characteristic, complemented
10C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
11C2×C4×AGL(1,ℤ/9)—21no—
12C2×C8×3−1+2C2211yescharacteristic
13C4⋊C4×3−1+2C2211yescharacteristic
14C4⋊C4×3−1+2C2211yescharacteristic
15(C2×AGL(1,ℤ/9))⋊C4—21no—
16C4⋊(3−1+2⋊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×#890—31no—
23C32×#106D311yescharacteristic
24C3×#821—31no—
25C3×Dic3⋊C8—31no—
26C3×#809—31no—
27C3×#807—31no—
28#63455C2×C311yescharacteristic
29#63452C2×C311yescharacteristic, complemented
30C9×#106—21no—
31C9×#106C2×C311yescharacteristic, complemented
32#63380C2×C311yescharacteristic
33Dic9⋊C8C2×C311yescharacteristic, complemented
34#63369C2×C311yescharacteristic
35#63366C2×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)Q812yes—
45C4×AGL(1,ℤ/9)—42no—
46C8×3−1+2—21no—
473−1+2⋊C8—21no—
48C3×#478—91no—
49#4871—31no—
50C2×C3×D12—31no—
51C2×C3×C4×D3—61no—
52C2×C32×C8C2×D311yescharacteristic
53C32×C4⋊C4C2×D311yescharacteristic
54C32×C4⋊C4C2×D311yescharacteristic
55C3×(C2×D3)⋊C4—61no—
56C3×C4⋊Dic3—31no—
57C3×C4⋊Dic3—61no—
58C3×Dic3⋊C4—61no—
59C3×C4×Dic3—31no—
60C2×C3×C3⋊C8—31no—
61C2×D36C22×C311yescharacteristic
62C2×C4×D9—21no—
63C2×C8×C9—21no—
64C2×C8×C9C22×C311yescharacteristic
65C9×C4⋊C4—21no—
66C9×C4⋊C4—21no—
67C9×C4⋊C4C22×C311yescharacteristic
68C9×C4⋊C4C22×C311yescharacteristic
69(C2×D9)⋊C4—21no—
70C4⋊Dic9—21no—
71C4⋊Dic9C22×C311yescharacteristic
72Dic9⋊C4—21no—
73C4×Dic9C22×C311yescharacteristic
74C2×C9⋊C8C22×C311yescharacteristic
75C22×3−1+2C2×D411yescharacteristic
76C2×AGL(1,ℤ/9)—22no—
77C2×AGL(1,ℤ/9)—41no—
78C2×AGL(1,ℤ/9)—41no—
79C4×3−1+2—41no—
80C4×3−1+2—41no—
81C4×3−1+2C2×Q811yescharacteristic
82C4×3−1+2#4111yescharacteristic
833−1+2⋊C4—21no—
843−1+2⋊C4—21no—
853−1+2⋊C4—41no—
86C3×#125—91no—
87C3×C4×D4—91no—
88#893—31no—
89#890—31no—
90C3×#107—91no—
91C3×#106—31no—
92C3×#106C3×D311yescharacteristic
93C3×C4⋊C8—91no—
94C3×#102—91no—
95C3×#102—91no—
96#821—31no—
97Dic3⋊C8—31no—
98#809—31no—
99#807—31no—
100C22×C3×D3—61no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4C22×D311yescharacteristic
104C2×C3×Dic3—31no—
105C2×C3×Dic3—31no—
106C2×C3×Dic3—61no—
107C3×D12—32no—
108C3×D12—61no—
109C3×C4×D3—122no—
110C22×D9—21no—
111C32×C8—21no—
112C3×C3⋊C8—61no—
113C2×C4×C9—21no—
114C2×C4×C9—21no—
115C2×C4×C9—21no—
116C2×C4×C9—41no—
117C2×C4×C9—41no—
118C2×C4×C9C23×C311yescharacteristic
119C2×Dic9—21no—
120C2×Dic9C3×D411yescharacteristic
121C2×Dic9C3×D411yescharacteristic
122D36—21no—
123D36C3×Q812yes—
124C4×D9—42no—
125C8×C9—22no—
126C8×C9—21no—
127C9⋊C8—21no—
128#478—91noSylow
129C2×3−1+2C22⋊Q811yescharacteristic
130C2×3−1+2#13511yescharacteristic
131C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
132AGL(1,ℤ/9)—42no—
133C2×C3×D4—91no—
134C22×C3×C4—181no—
135C2×D12—31no—
136C2×C4×D3—61no—
137C2×C3×C8—31no—
138C2×C3×C8—91no—
139C2×C3×C8C2×C3×D311yescharacteristic
140C3×C4⋊C4—31no—
141C3×C4⋊C4—31no—
142C3×C4⋊C4—91no—
143C3×C4⋊C4—181no—
144C3×C4⋊C4—181no—
145C3×C4⋊C4C2×C3×D311yescharacteristic
146C3×C4⋊C4C2×C3×D311yescharacteristic
147C3×C22⋊C4—181no—
148C3×C42—91no—
149(C2×D3)⋊C4—61no—
150C4⋊Dic3—31no—
151C4⋊Dic3—61no—
152Dic3⋊C4—61no—
153C4×Dic3—31no—
154C2×C3⋊C8—31no—
155C22×C32D3×D411yescharacteristic
156C2×C3×D3—62no—
157C2×C3×D3—121no—
158C2×C3×D3—121no—
159C32×C4—41no—
160C32×C4—41no—
161C32×C4#23511yescharacteristic
162C32×C4D3×Q811yescharacteristic
163C3×Dic3—61no—
164C3×Dic3—61no—
165C3×Dic3—121no—
166C22×C9—21no—
167C22×C9C2×C3×D411yescharacteristic
168C2×D9—22no—
169C2×D9—41no—
170C2×D9—41no—
171C4×C9—21no—
172C4×C9—21no—
173C4×C9—41no—
174C4×C9—41no—
175C4×C9—81no—
176C4×C9—81no—
177C4×C9C2×C3×Q811yescharacteristic
178C4×C9C3×#4111yescharacteristic
179Dic9—21no—
180Dic9—21no—
181Dic9—41no—
182#125—91no—
183C4×D4—91no—
184#107—91no—
185#106AGL(1,ℤ/9)11yescharacteristic, complemented
186C4⋊C8—91no—
187#102—91no—
188#102—91no—
1893−1+2#47811yescharacteristic, complemented, Sylow
190C23×C3—181no—
191C22×D3—61no—
192C3×D4—92no—
193C3×D4—181no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—31no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—91no—
200C2×C3×C4—91no—
201C2×C3×C4—181no—
202C2×C3×C4—362no—
203C2×C3×C4C22×C3×D311yescharacteristic
204C2×Dic3—31no—
205C2×Dic3—31no—
206C2×Dic3—61no—
207D12—32no—
208D12—61no—
209C4×D3—122no—
210C3×C8—21no—
211C3×C8—61no—
212C3×C8—181no—
213C3⋊C8—61no—
214C2×C32#89611yescharacteristic
215C2×C32#91411yescharacteristic
216C2×C32#91611yescharacteristic
217C3×D3—122no—
218C2×C9—21no—
219C2×C9—21no—
220C2×C9—21no—
221C2×C9C3×C22⋊Q811yescharacteristic
222C2×C9C3×#13511yescharacteristic
223C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
224D9—42no—
225C2×D4—91no—
226C22×C4—181no—
227C2×C8—91no—
228C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
229C4⋊C4—91no—
230C4⋊C4—181no—
231C4⋊C4—181no—
232C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
233C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
234C22⋊C4—181no—
235C42—91no—
236C22×C3—31no—
237C22×C3—182no—
238C22×C3—361no—
239C22×C3—361no—
240C22×C3C3×D3×D411yescharacteristic
241C2×D3—62no—
242C2×D3—121no—
243C2×D3—121no—
244C3×C4—31no—
245C3×C4—31no—
246C3×C4—41no—
247C3×C4—41no—
248C3×C4—121no—
249C3×C4—121no—
250C3×C4—181no—
251C3×C4—181no—
252C3×C4—361no—
253C3×C4C3×#23511yescharacteristic
254C3×C4C3×D3×Q811yescharacteristic
255Dic3—61no—
256Dic3—61no—
257Dic3—121no—
258C32#487111yescharacteristic
259C9—21no—
260C9C3×#47811yescharacteristic, complemented
261C23—181no—
262D4—92no—
263D4—181no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—91no—
267C2×C4—91no—
268C2×C4—181no—
269C2×C4—362no—
270C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
271C8—21no—
272C8—181no—
273C2×C3—31no—
274C2×C3—31no—
275C2×C3—31no—
276C2×C3—362no—
277C2×C3C3×#89611yescharacteristic
278C2×C3C3×#91411yescharacteristic
279C2×C3C3×#91611yescharacteristic
280D3—122no—
281C22—182no—
282C22—361no—
283C22—361no—
284C22D4×AGL(1,ℤ/9)11yescharacteristic
285C4—41no—
286C4—41no—
287C4—181no—
288C4—181no—
289C4—361no—
290C4#3697⋊C311yescharacteristic
291C4Q8×AGL(1,ℤ/9)11yescharacteristic
292C3—31no—
293C3C3×#487111yescharacteristic
294C2—362no—
295C2#63458⋊C311yescharacteristic
296C2#63476⋊C311yescharacteristic
297C2#63478⋊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×#478
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. #63452⋊C3C2semidirect
3. 3−1+2×#106C2semidirect
5. (3−1+2⋊C4)⋊C8C2semidirect
7. #63366⋊C3C2semidirect
9. #10588551C3semidirect
29. #63452C2×C3semidirect
31. C9×#106C2×C3semidirect
33. Dic9⋊C8C2×C3semidirect
35. #63366C2×C3semidirect
185. #106AGL(1,ℤ/9)semidirect
189. 3−1+2#478semidirect
260. C9C3×#478semidirect

Maximal subgroups

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

subgroupindex
#63455⋊C32
#63452⋊C32
3−1+2×#1062
#63380⋊C32
(3−1+2⋊C4)⋊C82
#63369⋊C32
#63366⋊C32
C3×#48713
#105885513