Finite Group Databasebeta

← C3×#10588747 #13142943

Subgroups — C3×#10588747 #13142943

Order 1728.

362 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#125C211yesmaximal, characteristic, complemented
2C3×C4×D36C211yesmaximal, characteristic
3C3×#63369C211yesmaximal, characteristic
4C3×#63369C211yesmaximal, characteristic
5C3×#63367C211yesmaximal, characteristic, complemented
6C3×#63366C211yesmaximal, characteristic, complemented
7C3×C4⋊(C9⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5066—31nomaximal
9#10588747C311yesmaximal, characteristic, direct product factor
10C2×C3×D36C2211yescharacteristic
11C2×C3×C4×D9—21no—
12C3×C9×C4⋊C4—21no—
13C3×C9×C4⋊C4—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16C3×C42×C9C2211yescharacteristic
17C3×(C2×D9)⋊C4—21no—
18C3×C4⋊Dic9C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21C32×#125D311yescharacteristic
22C3×C4×D12—31no—
23C3×#809—31no—
24C3×#809—31no—
25C3×#808—31no—
26C3×#807—31no—
27C3×C4⋊(C3⋊C8)—31no—
28C9×#125—21no—
29C9×#125C2×C311yescharacteristic, complemented
30C4×D36C2×C311yescharacteristic
31#63369C2×C311yescharacteristic
32#63369C2×C311yescharacteristic
33#63367C2×C311yescharacteristic, complemented
34#63366C2×C311yescharacteristic, complemented
35C4⋊(C9⋊C8)C2×C311yescharacteristic, complemented
36C22×C3×D9—21no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9D411yescharacteristic
40C2×C3×C4×C9D411yescharacteristic
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3×Dic9—21no—
43C3×D36—21no—
44C3×D36Q812yes—
45C3×C4×D9—42no—
46C3×C9⋊C8—21no—
47C3×C9⋊C8—21no—
48C3×#478—91no—
49#5066—31no—
50C2×C3×D12—31no—
51C2×C3×C4×D3—61no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56C32×C42C2×D311yescharacteristic
57C3×(C2×D3)⋊C4—61no—
58C3×C4⋊Dic3—31no—
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊C8—31no—
61C2×D36C22×C311yescharacteristic
62C2×C4×D9—21no—
63C9×C4⋊C4—22no—
64C9×C4⋊C4—22no—
65C9×C4⋊C4—21no—
66C9×C4⋊C4—21no—
67C9×C4⋊C4—21no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4C22×C311yescharacteristic
70C9×C4⋊C4C22×C311yescharacteristic
71C42×C9—21no—
72C42×C9C22×C311yescharacteristic
73(C2×D9)⋊C4—21no—
74C4⋊Dic9C22×C311yescharacteristic
75C2×C9⋊C8C22×C311yescharacteristic
76C2×C9⋊C8C22×C311yescharacteristic
77C22×C3×C9C2×D411yescharacteristic
78C2×C3×D9—22no—
79C2×C3×D9—41no—
80C2×C3×D9—41no—
81C3×C4×C9—21no—
82C3×C4×C9—21no—
83C3×C4×C9—41no—
84C3×C4×C9—41no—
85C3×C4×C9C2×Q811yescharacteristic
86C3×C4×C9#4111yescharacteristic
87C3×Dic9—41no—
88C3×#125—21no—
89C3×#125D911yescharacteristic, complemented
90C3×#125C3×D311yescharacteristic
91C3×C4×D4—91no—
92C4×D12—31no—
93C3×#107—91no—
94C3×#106—91no—
95C3×C4⋊C8—91no—
96C3×#102—91no—
97C3×#102—91no—
98#809—31no—
99#809—31no—
100#808—31no—
101#807—31no—
102C4⋊(C3⋊C8)—31no—
103C22×C3×D3—61no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4#6711yescharacteristic
107C2×C32×C4#6711yescharacteristic
108C2×C32×C4C22×D311yescharacteristic
109C2×C3×Dic3—61no—
110C3×D12—32no—
111C3×D12—61no—
112C3×C4×D3—122no—
113C22×D9—21no—
114C3×C3⋊C8—61no—
115C3×C3⋊C8—61no—
116C2×C4×C9—22no—
117C2×C4×C9—22no—
118C2×C4×C9—21no—
119C2×C4×C9—21no—
120C2×C4×C9—21no—
121C2×C4×C9—21no—
122C2×C4×C9—21no—
123C2×C4×C9C3×D411yescharacteristic
124C2×C4×C9C3×D411yescharacteristic
125C2×C4×C9C23×C311yescharacteristic
126C2×Dic9—21no—
127D36—21no—
128D36C3×Q812yes—
129C4×D9—42no—
130C9⋊C8—21no—
131C9⋊C8—21no—
132#478—91noSylow
133C2×C3×C9C22⋊Q811yescharacteristic
134C2×C3×C9#13511yescharacteristic
135C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
136C3×D9—42no—
137C2×C3×D4—91no—
138C22×C3×C4—181no—
139C2×D12—31no—
140C2×C4×D3—61no—
141C2×C3×C8—91no—
142C2×C3×C8—91no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4C2×D911yescharacteristic
153C3×C4⋊C4C2×D911yescharacteristic
154C3×C4⋊C4C2×C3×D311yescharacteristic
155C3×C4⋊C4C2×C3×D311yescharacteristic
156C3×C22⋊C4—181no—
157C3×C42—21no—
158C3×C42C2×D911yescharacteristic
159C3×C42C2×C3×D311yescharacteristic
160(C2×D3)⋊C4—61no—
161C4⋊Dic3—31no—
162C2×C3⋊C8—31no—
163C2×C3⋊C8—31no—
164C22×C32C2×#6711yescharacteristic
165C2×C3×D3—62no—
166C2×C3×D3—121no—
167C2×C3×D3—121no—
168C32×C4—21no—
169C32×C4—21no—
170C32×C4—41no—
171C32×C4—41no—
172C32×C4D3×Q811yescharacteristic
173C32×C4#23911yescharacteristic
174C3×Dic3—121no—
175C22×C9—21no—
176C22×C9C2×C3×D411yescharacteristic
177C2×D9—22no—
178C2×D9—41no—
179C2×D9—41no—
180C4×C9—21no—
181C4×C9—21no—
182C4×C9—21no—
183C4×C9—21no—
184C4×C9—41no—
185C4×C9—41no—
186C4×C9—42no—
187C4×C9—42no—
188C4×C9—41no—
189C4×C9—41no—
190C4×C9C2×C3×Q811yescharacteristic
191C4×C9C3×#4111yescharacteristic
192Dic9—41no—
193#125C3×D911yescharacteristic, complemented
194C4×D4—91no—
195#107—91no—
196#106—91no—
197C4⋊C8—91no—
198#102—91no—
199#102—91no—
200C3×C9#47811yescharacteristic, complemented, Sylow
201C23×C3—181no—
202C22×D3—61no—
203C3×D4—92no—
204C3×D4—181no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—181no—
215C2×C3×C4—362no—
216C2×C3×C4#61111yescharacteristic
217C2×C3×C4#61111yescharacteristic
218C2×C3×C4C22×D911yescharacteristic
219C2×C3×C4C3×#6711yescharacteristic
220C2×C3×C4C3×#6711yescharacteristic
221C2×C3×C4C22×C3×D311yescharacteristic
222C2×Dic3—61no—
223D12—32no—
224D12—61no—
225C4×D3—122no—
226C3×C8—181no—
227C3×C8—181no—
228C3⋊C8—61no—
229C3⋊C8—61no—
230C2×C32#94611yescharacteristic
231C2×C32#94911yescharacteristic
232C2×C32#95011yescharacteristic
233C3×D3—122no—
234C2×C9—21no—
235C2×C9—21no—
236C2×C9—21no—
237C2×C9C3×C22⋊Q811yescharacteristic
238C2×C9C3×#13511yescharacteristic
239C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
240D9—42no—
241C2×D4—91no—
242C22×C4—181no—
243C2×C8—91no—
244C2×C8—91no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—91no—
248C4⋊C4C2×C3×D911yescharacteristic
249C4⋊C4C2×C3×D911yescharacteristic
250C22⋊C4—181no—
251C42C2×C3×D911yescharacteristic
252C22×C3—21no—
253C22×C3—182no—
254C22×C3—361no—
255C22×C3—361no—
256C22×C3C2×#61111yescharacteristic
257C22×C3C2×C3×#6711yescharacteristic
258C2×D3—62no—
259C2×D3—121no—
260C2×D3—121no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4—42no—
270C3×C4—42no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—361no—
276C3×C4Q8×D911yescharacteristic
277C3×C4#370111yescharacteristic
278C3×C4C3×D3×Q811yescharacteristic
279C3×C4C3×#23911yescharacteristic
280Dic3—121no—
281C32#506611yescharacteristic
282C9—21no—
283C9C3×#47811yescharacteristic, complemented
284C23—181no—
285D4—92no—
286D4—181no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—181no—
290C2×C4—362no—
291C2×C4C3×#61111yescharacteristic
292C2×C4C3×#61111yescharacteristic
293C2×C4C22×C3×D911yescharacteristic
294C8—181no—
295C8—181no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—362no—
300C2×C3#6350811yescharacteristic
301C2×C3#6351211yescharacteristic
302C2×C3#6351311yescharacteristic
303C2×C3C3×#94611yescharacteristic
304C2×C3C3×#94911yescharacteristic
305C2×C3C3×#95011yescharacteristic
306D3—122no—
307C22—182no—
308C22—361no—
309C22—361no—
310C22C2×C3×#61111yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—361no—
316C4C3×Q8×D911yescharacteristic
317C4C3×#370111yescharacteristic
318C3—21no—
319C3#1058874711yescharacteristic, direct product factor
320C3C3×#506611yescharacteristic
321C2—362no—
322C2C3×#6350811yescharacteristic
323C2C3×#6351211yescharacteristic
324C2C3×#6351311yescharacteristic
subgroupG/H
centerC22×C3C2×#611
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#125C2
socleC22×C32C2×#67
prime radicalC2×C3×D36C22
hypercenterC3×#125D9
hypocenterC9C3×#478
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
1. C3×C9×#125C2semidirect
5. C3×#63367C2semidirect
6. C3×#63366C2semidirect
7. C3×C4⋊(C9⋊C8)C2semidirect
9. #10588747C3direct (N × H)
29. C9×#125C2×C3semidirect
33. #63367C2×C3semidirect
34. #63366C2×C3semidirect
35. C4⋊(C9⋊C8)C2×C3semidirect
89. C3×#125D9semidirect
193. #125C3×D9semidirect
200. C3×C9#478semidirect
283. C9C3×#478semidirect
319. C3#10588747direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1252
C3×C4×D362
C3×#633692
C3×#633672
C3×#633662
C3×C4⋊(C9⋊C8)2
C3×#50663
#105887473