Finite Group Databasebeta

← #10588419⋊C3 #13141642

Subgroups — #10588419⋊C3 #13141642

Order 1728.

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

normal subgroups only

This group has 343 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#63492⋊C3C211yesmaximal, characteristic
2#63488⋊C3C211yesmaximal, characteristic, complemented
33−1+2×C22⋊C8C211yesmaximal, characteristic, complemented
4#63380⋊C3C211yesmaximal, characteristic, complemented
5#63380⋊C3C211yesmaximal, characteristic, complemented
6#63378⋊C3C211yesmaximal, characteristic, complemented
7#63377⋊C3C211yesmaximal, characteristic, complemented
8C3×#4737—31nomaximal
9#10588419C311yesmaximal, characteristic, complemented
10C22×C4×3−1+2C2211yescharacteristic
11C2×#611⋊C3—21no—
12C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
13C2×C8×3−1+2C2211yescharacteristic
14C2×C8×3−1+2C2211yescharacteristic
15#3676⋊C3—21no—
16(C2×AGL(1,ℤ/9))⋊C4—21no—
17C4⋊(3−1+2⋊C4)C2211yescharacteristic
18C4⋊(3−1+2⋊C4)C2211yescharacteristic
19C4⋊(3−1+2⋊C4)C2211yescharacteristic
20C4×3−1+2⋊C4—21no—
21C3×#930—31no—
22C3×#926—31no—
23C32×C22⋊C8D311yescharacteristic
24C3×#821—31no—
25C3×#821—31no—
26C3×#818—31no—
27C3×#817—31no—
28#63492C2×C311yescharacteristic
29#63488C2×C311yescharacteristic, complemented
30C9×C22⋊C8—21no—
31C9×C22⋊C8C2×C311yescharacteristic, complemented
32#63380C2×C311yescharacteristic, complemented
33#63380C2×C311yescharacteristic, complemented
34#63378C2×C311yescharacteristic, complemented
35#63377C2×C311yescharacteristic, complemented
36C23×3−1+2D411yescharacteristic
37C22×AGL(1,ℤ/9)—21no—
38C2×C4×3−1+2—21no—
39C2×C4×3−1+2—21no—
40C2×C4×3−1+2D411yescharacteristic
41C2×C4×3−1+2C2311yescharacteristic
42#611⋊C3—42no—
43C2×3−1+2⋊C4—21no—
44C2×3−1+2⋊C4—21no—
45C2×3−1+2⋊C4—21no—
46C4⋊AGL(1,ℤ/9)—21no—
47C4⋊AGL(1,ℤ/9)—21no—
48C8×3−1+2—21no—
49C8×3−1+2—21no—
50C3×#482—91no—
51#4737—31no—
52C22×C32×C4C2×D311yescharacteristic
53C2×C3×#67—61no—
54C2×C3×D12—31no—
55C2×C32×C8C2×D311yescharacteristic
56C2×C32×C8C2×D311yescharacteristic
57C3×#217—61no—
58C3×(C2×D3)⋊C4—61no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3—31no—
61C3×C4⋊Dic3—31no—
62C3×C4×Dic3—61no—
63C22×C4×C9—21no—
64C22×C4×C9C22×C311yescharacteristic
65C2×#611—21no—
66C2×D36C22×C311yescharacteristic
67C2×C8×C9—21no—
68C2×C8×C9—21no—
69C2×C8×C9C22×C311yescharacteristic
70C2×C8×C9C22×C311yescharacteristic
71#3676—21no—
72(C2×D9)⋊C4—21no—
73C4⋊Dic9C22×C311yescharacteristic
74C4⋊Dic9C22×C311yescharacteristic
75C4⋊Dic9C22×C311yescharacteristic
76C4×Dic9—21no—
77C22×3−1+2—21no—
78C22×3−1+2—21no—
79C22×3−1+2—21no—
80C22×3−1+2C2×D411yescharacteristic
81C2×AGL(1,ℤ/9)—41no—
82C2×AGL(1,ℤ/9)—41no—
83C2×AGL(1,ℤ/9)—41no—
84C4×3−1+2—21no—
85C4×3−1+2#4111yescharacteristic
86C4×3−1+2#4111yescharacteristic
873−1+2⋊C4—41no—
883−1+2⋊C4—41no—
893−1+2⋊C4—41no—
90C3×#121—91no—
91C3×#117—91no—
92#930—31no—
93#926—31no—
94C3×#107—91no—
95C3×#106—91no—
96C3×#102—91no—
97C3×#102—91no—
98C3×C22⋊C8—31no—
99C3×C22⋊C8C3×D311yescharacteristic
100#821—31no—
101#821—31no—
102#818—31no—
103#817—31no—
104C23×C32D1211yescharacteristic
105C22×C3×D3—61no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4D1211yescharacteristic
109C2×C32×C4C22×D311yescharacteristic
110C3×#67—122no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C3×D12—61no—
115C3×D12—61no—
116C23×C9—21no—
117C23×C9C3×D411yescharacteristic
118C22×D9—21no—
119C32×C8—21no—
120C32×C8—21no—
121C2×C4×C9—21no—
122C2×C4×C9—21no—
123C2×C4×C9—21no—
124C2×C4×C9—21no—
125C2×C4×C9—41no—
126C2×C4×C9—41no—
127C2×C4×C9C3×D411yescharacteristic
128C2×C4×C9C23×C311yescharacteristic
129#611—42no—
130C2×Dic9—21no—
131C2×Dic9—21no—
132C2×Dic9—21no—
133D36—21no—
134D36—21no—
135C8×C9—21no—
136C8×C9—21no—
137C8×C9—41no—
138C8×C9—41no—
139#482—91noSylow
140C2×3−1+2—41no—
141C2×3−1+2#12311yescharacteristic
142C2×3−1+2#13511yescharacteristic
143C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
144AGL(1,ℤ/9)—81no—
145C2×C3×D4—91no—
146C2×C3×D4—181no—
147C22×C3×C4—31no—
148C22×C3×C4C2×C3×D311yescharacteristic
149C2×#67—61no—
150C2×D12—31no—
151C2×C3×C8—31no—
152C2×C3×C8—31no—
153C2×C3×C8C2×C3×D311yescharacteristic
154C2×C3×C8C2×C3×D311yescharacteristic
155C3×C4⋊C4—91no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—91no—
158C3×C22⋊C4—181no—
159C3×C22⋊C4—181no—
160C3×C42—181no—
161#217—61no—
162(C2×D3)⋊C4—61no—
163C4⋊Dic3—31no—
164C4⋊Dic3—31no—
165C4⋊Dic3—31no—
166C4×Dic3—61no—
167C22×C32—21no—
168C22×C32—21no—
169C22×C32—21no—
170C22×C32C2×D1211yescharacteristic
171C2×C3×D3—121no—
172C2×C3×D3—121no—
173C2×C3×D3—121no—
174C32×C4—21no—
175C32×C4#23711yescharacteristic
176C32×C4#23711yescharacteristic
177C3×Dic3—121no—
178C3×Dic3—121no—
179C3×Dic3—121no—
180C22×C9—22no—
181C22×C9—21no—
182C22×C9—21no—
183C22×C9—21no—
184C22×C9—21no—
185C22×C9—41no—
186C22×C9—41no—
187C22×C9C2×C3×D411yescharacteristic
188C2×D9—41no—
189C2×D9—41no—
190C2×D9—41no—
191C4×C9—21no—
192C4×C9—21no—
193C4×C9—21no—
194C4×C9—41no—
195C4×C9C3×#4111yescharacteristic
196C4×C9C3×#4111yescharacteristic
197Dic9—41no—
198Dic9—41no—
199Dic9—41no—
200#121—91no—
201#117—91no—
202#107—91no—
203#106—91no—
204#102—91no—
205#102—91no—
206C22⋊C8AGL(1,ℤ/9)11yescharacteristic, complemented
2073−1+2#48211yescharacteristic, complemented, Sylow
208C23×C3—31no—
209C23×C3—181no—
210C23×C3C3×D1211yescharacteristic
211C22×D3—61no—
212C3×D4—181no—
213C3×D4—181no—
214C3×D4—362no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—181no—
222C2×C3×C4—181no—
223C2×C3×C4—181no—
224C2×C3×C4C3×D1211yescharacteristic
225C2×C3×C4C22×C3×D311yescharacteristic
226#67—122no—
227C2×Dic3—61no—
228C2×Dic3—61no—
229C2×Dic3—61no—
230D12—61no—
231D12—61no—
232C3×C8—21no—
233C3×C8—21no—
234C3×C8—61no—
235C3×C8—61no—
236C2×C32—41no—
237C2×C32#88611yescharacteristic
238C2×C32#90411yescharacteristic
239C2×C32#90811yescharacteristic
240C3×D3—241no—
241C2×C9—21no—
242C2×C9—21no—
243C2×C9—21no—
244C2×C9—42no—
245C2×C9—41no—
246C2×C9C3×#12311yescharacteristic
247C2×C9C3×#13511yescharacteristic
248C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
249D9—81no—
250C2×D4—91no—
251C2×D4—181no—
252C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
253C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
254C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
255C4⋊C4—91no—
256C4⋊C4—91no—
257C4⋊C4—91no—
258C22⋊C4—181no—
259C22⋊C4—181no—
260C42—181no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—31no—
265C22×C3—61no—
266C22×C3—61no—
267C22×C3—61no—
268C22×C3—361no—
269C22×C3—361no—
270C22×C3—361no—
271C22×C3C2×C3×D1211yescharacteristic
272C2×D3—121no—
273C2×D3—121no—
274C2×D3—121no—
275C3×C4—21no—
276C3×C4—31no—
277C3×C4—31no—
278C3×C4—61no—
279C3×C4—361no—
280C3×C4—361no—
281C3×C4—361no—
282C3×C4C3×#23711yescharacteristic
283C3×C4C3×#23711yescharacteristic
284Dic3—121no—
285Dic3—121no—
286Dic3—121no—
287C32#473711yescharacteristic
288C9—21no—
289C9C3×#48211yescharacteristic, complemented
290C23—181no—
291C23C4⋊AGL(1,ℤ/9)11yescharacteristic
292D4—181no—
293D4—181no—
294D4—362no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—181no—
298C2×C4—181no—
299C2×C4—181no—
300C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
301C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
302C8—21no—
303C8—21no—
304C2×C3—31no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—41no—
308C2×C3—121no—
309C2×C3—721no—
310C2×C3C3×#88611yescharacteristic
311C2×C3C3×#90411yescharacteristic
312C2×C3C3×#90811yescharacteristic
313D3—241no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—361no—
318C22—361no—
319C22—361no—
320C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
321C4—21no—
322C4—361no—
323C4—361no—
324C4—361no—
325C4#3699⋊C311yescharacteristic
326C4#3699⋊C311yescharacteristic
327C3—31no—
328C3C3×#473711yescharacteristic
329C2—41no—
330C2—721no—
331C2#63448⋊C311yescharacteristic
332C2#63467⋊C311yescharacteristic
333C2#63470⋊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 radical#63492⋊C3C2
hypercenterC22⋊C8AGL(1,ℤ/9)
hypocenterC9C3×#482
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
2. #63488⋊C3C2semidirect
3. 3−1+2×C22⋊C8C2semidirect
4. #63380⋊C3C2semidirect
5. #63380⋊C3C2semidirect
6. #63378⋊C3C2semidirect
7. #63377⋊C3C2semidirect
9. #10588419C3semidirect
29. #63488C2×C3semidirect
31. C9×C22⋊C8C2×C3semidirect
32. #63380C2×C3semidirect
33. #63380C2×C3semidirect
34. #63378C2×C3semidirect
35. #63377C2×C3semidirect
206. C22⋊C8AGL(1,ℤ/9)semidirect
207. 3−1+2#482semidirect
289. C9C3×#482semidirect

Maximal subgroups

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

subgroupindex
#63492⋊C32
#63488⋊C32
3−1+2×C22⋊C82
#63380⋊C32
#63378⋊C32
#63377⋊C32
C3×#47373
#105884193