Finite Group Databasebeta

← #13139326

Subgroups — #13139326

Order 1728.

550 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3697C211yesmaximal, characteristic
2C22×C3⋊(C9⋊C8)C211yesmaximal, characteristic, complemented
3C9×#926C211yesmaximal, characteristic, complemented
4#11720268C212yesmaximal, complemented
5#11720266C212yesmaximal, complemented
6#10591083—31nomaximal
7#10588683—31nomaximal
8C22×C3×C4×C9C2211yescharacteristic
9C2×C3×#611—21no—
10C3×#3697—22no—
11C3×#3697C414yescomplemented
12C2×C3×D36C2211yescharacteristic
13C2×C3×C4×D9—21no—
14C2×C3×Dic18C2211yescharacteristic
15C2×C3⋊(C9⋊C8)—22no—
16C2×C3⋊(C9⋊C8)C2212yes—
17C9×#217—21no—
18C9×C4⋊Dic3C2212yes—
19C4×C9×Dic3—21no—
20C2×C3×#235—31no—
21C22×C3⋊(C3⋊C8)—31no—
22C3×#926D311yescharacteristic
23C2×#3697D311yescharacteristic
24#63565—32no—
25#63562—32no—
26C9×#117—31no—
27C22×C9⋊C8—31no—
28C22×C9⋊C8—61no—
29#63369—32no—
30#63368—32no—
31C23×C3×C9D411yescharacteristic
32C22×C3×D9—21no—
33C2×C3×C4×C9C2×C412yes—
34C2×C3×C4×C9D411yescharacteristic
35C2×C3×C4×C9D412yes—
36C2×C3×C4×C9C2311yescharacteristic
37C3×#611—24no—
38C2×C9×Dic3—22no—
39C2×C3×Dic9—21no—
40C3×D36—21no—
41C3×D36C2×C412yes—
42C3×C4×D9—24no—
43C3×Dic18—21no—
44C3×Dic18C2×C412yes—
45C3⋊(C9⋊C8)—22no—
46#5233—91no—
47#5002—91no—
48C22×C32×C4C2×D311yescharacteristic
49C2×C3×#67—61no—
50C3×#235—34no—
51C3×#235—62no—
52C2×C3×D12—31no—
53C2×C3×C4×D3—61no—
54C2×C3×Dic6—31no—
55C2×C3⋊(C3⋊C8)—32no—
56C2×C3⋊(C3⋊C8)—62no—
57C3×#217—21no—
58C3×C4⋊Dic3C2×D312yes—
59C3×C4×Dic3—21no—
60C22×C4×C9—21no—
61C22×C4×C9C2×D311yescharacteristic
62C2×#611—21no—
63#3697—22no—
64#3697Dic314yescomplemented
65C2×D36C2×D311yescharacteristic
66C2×C4×D9—21no—
67C2×Dic18C2×D311yescharacteristic
68C9×C4⋊C4—32no—
69C9×C22⋊C4—61no—
70C42×C9—61no—
71C2×C9⋊C8—32no—
72C2×C9⋊C8—62no—
73C2×C9⋊C8—64no—
74C2×C9⋊C8—62no—
75C22×C3×C9—22no—
76C22×C3×C9C22⋊C412yes—
77C22×C3×C9C2×D411yescharacteristic
78C2×C3×D9—22no—
79C2×C3×D9—42no—
80C3×C4×C9C22⋊C412yes—
81C3×C4×C9C22×C411yescharacteristic
82C3×C4×C9C2×D411yescharacteristic
83C9×Dic3—42no—
84C3×Dic9—22no—
85C2×C3×#41—91no—
86C2×#235—31no—
87C3×#117—31no—
88#926D911yescharacteristic, complemented
89C22×C3⋊C8—91no—
90C22×C3⋊C8—91no—
91C22×C3⋊C8—181no—
92#835—92no—
93#832—92no—
94#810—92no—
95#809—92no—
96C23×C32#6711yescharacteristic
97C22×C3×D3—61no—
98C2×C32×C4C4×D312yes—
99C2×C32×C4D1212yes—
100C2×C32×C4#6711yescharacteristic
101C2×C32×C4C22×D311yescharacteristic
102C3×#67—64no—
103C2×C3×Dic3—22no—
104C2×C3×Dic3—61no—
105C3×D12—32no—
106C3×D12—61no—
107C3×C4×D3—64no—
108C3×Dic6—32no—
109C3×Dic6—61no—
110C23×C9—21no—
111C23×C9#6711yescharacteristic
112C22×D9—21no—
113C3⋊(C3⋊C8)—62no—
114C2×C4×C9—22no—
115C2×C4×C9—21no—
116C2×C4×C9—22no—
117C2×C4×C9—21no—
118C2×C4×C9—62no—
119C2×C4×C9C2×Dic312yes—
120C2×C4×C9#6711yescharacteristic
121C2×C4×C9#6712yes—
122C2×C4×C9C22×D311yescharacteristic
123#611—24no—
124C2×Dic9—21no—
125D36—21no—
126D36C2×Dic312yes—
127C4×D9—24no—
128Dic18—21no—
129Dic18C2×Dic312yes—
130C9⋊C8—64no—
131C9⋊C8—62no—
132#416—271noSylow
133C2×C3×C9—22no—
134C2×C3×C9C2×C22⋊C411yescharacteristic
135C2×C3×C9#13512yes—
136C3×D9—42no—
137C3×#41—94no—
138C3×#41—182no—
139C2×C3×Q8—91no—
140C2×C3×D4—91no—
141C2×C3×D4—181no—
142C22×C3×C4—21no—
143C22×C3×C4—181no—
144C22×C3×C4C2×D911yescharacteristic
145C22×C3×C4D3211yescharacteristic
146C2×#67—61no—
147#235—34no—
148#235—62no—
149C2×D12—31no—
150C2×C4×D3—61no—
151C2×Dic6—31no—
152C3×C4⋊C4—32no—
153C3×C22⋊C4—61no—
154C3×C42—61no—
155#217—21no—
156C4⋊Dic3C2×D912yes—
157C4×Dic3—21no—
158C2×C3⋊C8—92no—
159C2×C3⋊C8—92no—
160C2×C3⋊C8—182no—
161C2×C3⋊C8—184no—
162C2×C3⋊C8—182no—
163C2×C3⋊C8—182no—
164C22×C32—22no—
165C22×C32(C2×D3)⋊C412yes—
166C22×C32C2×#6711yescharacteristic
167C2×C3×D3—62no—
168C2×C3×D3—122no—
169C32×C4(C2×D3)⋊C412yes—
170C32×C4C2×C4×D311yescharacteristic
171C32×C4C2×D1211yescharacteristic
172C3×Dic3—42no—
173C3×Dic3—62no—
174C22×C9—24no—
175C22×C9—22no—
176C22×C9—22no—
177C22×C9—21no—
178C22×C9#21712yes—
179C22×C9C2×#6711yescharacteristic
180C2×D9—22no—
181C2×D9—42no—
182C4×C9—22no—
183C4×C9—21no—
184C4×C9—21no—
185C4×C9—122no—
186C4×C9#21712yes—
187C4×C9C22×Dic311yescharacteristic
188C4×C9C2×#6711yescharacteristic
189Dic9—22no—
190C2×#41—91no—
191C22×C8—271no—
192#117—31no—
193#103—272no—
194#102—272no—
195C3×C9#41611yescharacteristic, complemented, Sylow
196C23×C3—21no—
197C23×C3—181no—
198C23×C3#61111yescharacteristic
199C23×C3#62511yescharacteristic
200C22×D3—61no—
201C3×Q8—92no—
202C3×Q8—181no—
203C3×D4—92no—
204C3×D4—184no—
205C3×D4—181no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—62no—
211C2×C3×C4—181no—
212C2×C3×C4—184no—
213C2×C3×C4C4×D912yes—
214C2×C3×C4D3612yes—
215C2×C3×C4#61111yescharacteristic
216C2×C3×C4C22×D911yescharacteristic
217C2×C3×C4D3×Dic312yes—
218C2×C3×C4#62511yescharacteristic
219C2×C3×C4#62612yes—
220C2×C3×C4C2×D3211yescharacteristic
221#67—64no—
222C2×Dic3—22no—
223C2×Dic3—61no—
224D12—32no—
225D12—61no—
226C4×D3—64no—
227Dic6—32no—
228Dic6—61no—
229C3⋊C8—184no—
230C3⋊C8—182no—
231C3⋊C8—182no—
232C2×C32—22no—
233C2×C32C2×(C2×D3)⋊C411yescharacteristic
234C2×C32#95012yes—
235C3×D3—122no—
236C2×C9—24no—
237C2×C9—22no—
238C2×C9—22no—
239C2×C9—21no—
240C2×C9#95012yes—
241C2×C9C2×#21711yescharacteristic
242D9—42no—
243#41—94no—
244#41—182no—
245C2×Q8—91no—
246C2×D4—91no—
247C2×D4—181no—
248C22×C4—181no—
249C22×C4D3×D911yescharacteristic
250C2×C8—272no—
251C2×C8—542no—
252C4⋊C4—32no—
253C22⋊C4—61no—
254C42—61no—
255C22×C3—24no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—21no—
259C22×C3—22no—
260C22×C3—182no—
261C22×C3—362no—
262C22×C3(C2×D9)⋊C412yes—
263C22×C3C2×#61111yescharacteristic
264C22×C3(C2×D3)⋊Dic312yes—
265C22×C3C2×#62511yescharacteristic
266C2×D3—62no—
267C2×D3—122no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—22no—
271C3×C4—122no—
272C3×C4—182no—
273C3×C4(C2×D9)⋊C412yes—
274C3×C4C2×C4×D911yescharacteristic
275C3×C4C2×D3611yescharacteristic
276C3×C4(C2×D3)⋊Dic312yes—
277C3×C4C2×D3×Dic311yescharacteristic
278C3×C4C2×#62611yescharacteristic
279Dic3—42no—
280Dic3—62no—
281C32#500211yescharacteristic
282C9—21no—
283C9#523311yescharacteristic, complemented
284C23—181no—
285C23#619411yescharacteristic
286Q8—92no—
287Q8—181no—
288D4—92no—
289D4—184no—
290D4—181no—
291C2×C4—62no—
292C2×C4—181no—
293C2×C4—184no—
294C2×C4Dic3×D912yes—
295C2×C4#619212yes—
296C2×C4#619411yescharacteristic
297C2×C4C2×D3×D911yescharacteristic
298C8—542no—
299C2×C3—24no—
300C2×C3—22no—
301C2×C3—22no—
302C2×C3—21no—
303C2×C3—22no—
304C2×C3—362no—
305C2×C3C2×(C2×D9)⋊C411yescharacteristic
306C2×C3#6351312yes—
307C2×C3#6382212yes—
308C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
309D3—122no—
310C22—22no—
311C22—182no—
312C22—362no—
313C22(C2×D9)⋊Dic312yes—
314C22C2×#619411yescharacteristic
315C4—122no—
316C4—182no—
317C4(C2×D9)⋊Dic312yes—
318C4C2×Dic3×D911yescharacteristic
319C4C2×#619211yescharacteristic
320C3—21no—
321C3#1058868311yescharacteristic, complemented
322C3#1059108311yescharacteristic
323C2—22no—
324C2—362no—
325C2#1172088112yes—
326C2C2×(C2×D9)⋊Dic311yescharacteristic
subgroupG/H
centerC2×C4#6194
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×#67
prime radicalC2×C3×#3697C2
cosocleC2×C3×C4×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#416
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊(C9⋊C8)C2semidirect
3. C9×#926C2semidirect
4. #11720268C2semidirect
5. #11720266C2semidirect
11. C3×#3697C4semidirect
64. #3697Dic3semidirect
88. #926D9semidirect
195. C3×C9#416semidirect
283. C9#5233semidirect
321. C3#10588683semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36972
C22×C3⋊(C9⋊C8)2
C9×#9262
#117202682
#117202662
#105910833
#105886833