Finite Group Databasebeta

← #67×C3⋊D9 #13048705

Subgroups — #67×C3⋊D9 #13048705

Order 1296.

660 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C22×C3×C3⋊D9C211yesmaximal, characteristic, complemented
2C2×D3×C3⋊D9C211yesmaximal, characteristic, complemented
3#10619992C211yesmaximal, characteristic, complemented
4C3×C9×#67C211yesmaximal, characteristic, complemented
5#10619879C211yesmaximal, characteristic, complemented
6#10619874C211yesmaximal, characteristic, complemented
7Dic3×C3⋊D9C211yesmaximal, characteristic, complemented
8C3⋊D3×#67—31nomaximal
9D4×C3⋊D9—31nomaximal
10D9×#67—33nomaximal
11C22×C32×C9C2211yescharacteristic, complemented
12C2×C3⋊(C3⋊D9)C2211yescharacteristic, complemented
13C2×C3×C3⋊D9—22no—
14C2×C3×C3⋊D9C2211yescharacteristic
15C2×C3×C3⋊D9C2211yescharacteristic, complemented
16C2×C3×D3×C9C2211yescharacteristic, complemented
17D3×C3⋊D9—22no—
18C3⋊(C3⋊Dic9)C2211yescharacteristic, complemented
19C3×C9×Dic3C2211yescharacteristic, complemented
20C22×C3×C3⋊D3—31no—
21C2×D3×C3⋊D3—31no—
22#6312—31no—
23C32×#67D311yescharacteristic
24#6292—31no—
25#6290—31no—
26Dic3×C3⋊D3—31no—
27C22×C3⋊D9—31no—
28C22×C3⋊D9D311yescharacteristic, complemented
29C22×C3×D9—33no—
30C2×D3×D9—33no—
31C3×D4×C9—31no—
32#6233—31no—
33#6233—33no—
34#6233—31no—
35#6233—63no—
36#6233—61no—
37C4⋊(C3⋊D9)—31no—
38C4×C3⋊D9—31no—
39C9×#67D313yescomplemented
40#6194—33no—
41#6192—33no—
42Dic3×D9—33no—
43C2×C32×C9—21no—
44C2×C32×C9C2311yescharacteristic
45C3⋊(C3⋊D9)—21no—
46C3×C3⋊D9—21no—
47C3×C3⋊D9D412yescomplemented
48C3×D3×C9—21no—
49D4×C3⋊D3—91no—
50D3×#67—93no—
51D3×#67—91no—
52D4×D9—93no—
53C22×C33C2×D311yescharacteristic
54C2×C3⋊(C3⋊D3)—31no—
55C2×C3×C3⋊D3—31no—
56C2×C3×C3⋊D3—31no—
57C2×C3×C3⋊D3—62no—
58C2×C32×D3C2×D311yescharacteristic
59D3×C3⋊D3—62no—
60C3⋊(C3⋊Dic3)—31no—
61C32×Dic3C2×D311yescharacteristic
62C22×C3×C9—21no—
63C22×C3×C9—23no—
64C22×C3×C9—31no—
65C22×C3×C9C2×D313yescomplemented
66C22×C3×C9C2×D311yescharacteristic, complemented
67C2×C3⋊D9—22no—
68C2×C3⋊D9—31no—
69C2×C3⋊D9—33no—
70C2×C3⋊D9—62no—
71C2×C3⋊D9—63no—
72C2×C3⋊D9—61no—
73C2×C3⋊D9C2×D311yescharacteristic
74C2×C3⋊D9C2×D311yescharacteristic, complemented
75C2×D3×C9C2×D313yescomplemented
76C2×C3×D9—33no—
77C2×C3×D9—33no—
78C2×C3×D9—66no—
79D3×D9—66no—
80C3×C4×C9—31no—
81C3⋊Dic9—31no—
82C3⋊Dic9—33no—
83C3⋊Dic9—63no—
84C3⋊Dic9—61no—
85C9×Dic3C2×D313yescomplemented
86C32×C9C2×D411yescharacteristic, complemented, Sylow
87C22×C3⋊D3—31no—
88C22×C3⋊D3—91no—
89C22×C3×D3—93no—
90C22×C3×D3—91no—
91C2×D32—93no—
92C2×D32—91no—
93C32×D4—31no—
94#638—91no—
95#638—91no—
96#638—93no—
97#638—91no—
98#638—181no—
99#638—183no—
100C4⋊(C3⋊D3)—91no—
101C4×C3⋊D3—91no—
102C3×#67D913yescomplemented
103C3×#67C3⋊D311yescharacteristic
104#626—93no—
105#626—91no—
106#625—93no—
107#625—91no—
108D3×Dic3—93no—
109D3×Dic3—91no—
110C22×D9—33no—
111C22×D9—93no—
112D4×C9—33no—
113#611—93no—
114#611—93no—
115#611—183no—
116D36—93no—
117C4×D9—93no—
118C2×C33—21no—
119C2×C33C22×D311yescharacteristic
120C3⋊(C3⋊D3)—61no—
121C3×C3⋊D3—32no—
122C3×C3⋊D3—61no—
123C32×D3—21no—
124C2×C3×C9—21no—
125C2×C3×C9—21no—
126C2×C3×C9—23no—
127C2×C3×C9—23no—
128C2×C3×C9—43no—
129C2×C3×C9—41no—
130C2×C3×C9—61no—
131C2×C3×C9C22×D313yes—
132C2×C3×C9C22×D311yescharacteristic
133C3⋊D9—21no—
134C3⋊D9—61no—
135C3⋊D9—63no—
136C3⋊D9—123no—
137C3⋊D9—121no—
138C3⋊D9#6712yesdirect product factor
139D3×C9—23no—
140C3×D9—36no—
141C3×D9—63no—
142C2×#67—271no—
143D3×D4—271no—
144D3×D4—273no—
145C22×C32—23no—
146C22×C32—21no—
147C22×C32—31no—
148C22×C32C2×D913yescomplemented
149C22×C32D3211yescharacteristic
150C22×C32C2×C3⋊D311yescharacteristic
151C2×C3⋊D3—31no—
152C2×C3⋊D3—31no—
153C2×C3⋊D3—62no—
154C2×C3⋊D3—91no—
155C2×C3⋊D3—93no—
156C2×C3⋊D3—91no—
157C2×C3⋊D3—181no—
158C2×C3⋊D3—183no—
159C2×C3⋊D3—182no—
160C2×C3×D3—93no—
161C2×C3×D3—91no—
162C2×C3×D3—93no—
163C2×C3×D3—91no—
164C2×C3×D3—182no—
165C2×C3×D3—186no—
166C2×C3×D3C2×D913yescomplemented
167C2×C3×D3C2×C3⋊D311yescharacteristic
168D32—182no—
169D32—186no—
170C32×C4—31no—
171C3⋊Dic3—91no—
172C3⋊Dic3—93no—
173C3⋊Dic3—91no—
174C3⋊Dic3—181no—
175C3⋊Dic3—183no—
176C3×Dic3C2×D913yescomplemented
177C3×Dic3C2×C3⋊D311yescharacteristic
178C22×C9—23no—
179C22×C9—33no—
180C22×C9D3213yescomplemented
181C2×D9—33no—
182C2×D9—33no—
183C2×D9—66no—
184C2×D9—93no—
185C2×D9—183no—
186C2×D9—186no—
187C4×C9—33no—
188Dic9—93no—
189Dic9—183no—
190C33D3×D411yescharacteristic
191C3×C9—21no—
192C3×C9—23no—
193C3×C9D3×D413yescomplemented
194C3×C9C2×#6711yescharacteristic, complemented
195C23×C3—271no—
196C22×D3—93no—
197C22×D3—91no—
198C22×D3—271no—
199C22×D3—273no—
200C22×D3—271no—
201C3×D4—33no—
202C3×D4—31no—
203#67—271no—
204#67—273no—
205#67—273no—
206#67—271no—
207#67—271no—
208#67—271no—
209#67—271no—
210#67—543no—
211#67—541no—
212#67C3⋊D911yescharacteristic, direct product factor
213C2×Dic3—271no—
214D12—271no—
215D12—273no—
216C4×D3—271no—
217C4×D3—273no—
218C2×C32—21no—
219C2×C32—23no—
220C2×C32—21no—
221C2×C32—23no—
222C2×C32—21no—
223C2×C32—41no—
224C2×C32—43no—
225C2×C32—61no—
226C2×C32C22×D913yes—
227C2×C32C2×D3211yescharacteristic
228C2×C32C22×C3⋊D311yescharacteristic
229C3⋊D3—32no—
230C3⋊D3—61no—
231C3⋊D3—181no—
232C3⋊D3—183no—
233C3⋊D3—181no—
234C3⋊D3—361no—
235C3⋊D3—363no—
236C3×D3—23no—
237C3×D3—21no—
238C3×D3—96no—
239C3×D3—92no—
240C3×D3—183no—
241C3×D3—181no—
242C2×C9—23no—
243C2×C9—23no—
244C2×C9—43no—
245C2×C9—63no—
246C2×C9C2×D3213yes—
247D9—36no—
248D9—63no—
249D9—183no—
250D9—363no—
251C2×D4—811noSylow
252C22×C3—23no—
253C22×C3—21no—
254C22×C3—33no—
255C22×C3—31no—
256C22×C3—271no—
257C22×C3—271no—
258C22×C3—542no—
259C22×C3D3×D913yescomplemented
260C22×C3C2×C3⋊D911yescharacteristic, complemented
261C22×C3D3×C3⋊D311yescharacteristic
262C2×D3—93no—
263C2×D3—91no—
264C2×D3—93no—
265C2×D3—91no—
266C2×D3—182no—
267C2×D3—186no—
268C2×D3—273no—
269C2×D3—271no—
270C2×D3—271no—
271C2×D3—543no—
272C2×D3—541no—
273C2×D3—542no—
274C2×D3—542no—
275C2×D3—546no—
276C2×D3C2×C3⋊D911yescharacteristic, complemented
277C3×C4—33no—
278C3×C4—31no—
279Dic3—273no—
280Dic3—271no—
281Dic3—271no—
282Dic3—543no—
283Dic3—541no—
284Dic3C2×C3⋊D911yescharacteristic, complemented
285C32—23no—
286C32—21no—
287C32D4×D913yescomplemented
288C32D3×#6711yescharacteristic
289C32D4×C3⋊D311yescharacteristic
290C9—23no—
291C9D3×#6713yescomplemented
292C23—271no—
293C23—811no—
294D4—31no—
295D4—811no—
296D4—811no—
297D4—811no—
298C2×C4—811no—
299C2×C3—23no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—23no—
303C2×C3—21no—
304C2×C3—43no—
305C2×C3—41no—
306C2×C3—63no—
307C2×C3—61no—
308C2×C3—272no—
309C2×C3—541no—
310C2×C3C2×D3×D913yes—
311C2×C3C22×C3⋊D911yescharacteristic
312C2×C3C2×D3×C3⋊D311yescharacteristic
313D3—21no—
314D3—96no—
315D3—92no—
316D3—183no—
317D3—181no—
318D3—543no—
319D3—541no—
320D3—541no—
321D3—1083no—
322D3—1081no—
323C22—31no—
324C22—271no—
325C22—271no—
326C22—542no—
327C22—811no—
328C22—1622no—
329C22D3×C3⋊D911yescharacteristic, complemented
330C4—31no—
331C4—811no—
332C3—23no—
333C3—21no—
334C3D9×#6713yescomplemented
335C3D4×C3⋊D911yescharacteristic, complemented
336C3C3⋊D3×#6711yescharacteristic
337C2—21no—
338C2—61no—
339C2—272no—
340C2—541no—
341C2—1621no—
342C2C2×D3×C3⋊D911yescharacteristic
subgroupG/H
centerC2C2×D3×C3⋊D9
commutator, cosocleC2×C32×C9C23
FrattiniC2×C3C2×D3×C3⋊D3
FittingC22×C32×C9C22
socleC2×C33C22×D3
hypercenterC22D3×C3⋊D9
hypocenterC32×C9C2×D4
LCSG ▷ C2×C32×C9 ▷ C32×C9
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×C3⋊D9C2semidirect
2. C2×D3×C3⋊D9C2semidirect
3. #10619992C2semidirect
4. C3×C9×#67C2semidirect
5. #10619879C2semidirect
6. #10619874C2semidirect
7. Dic3×C3⋊D9C2semidirect
11. C22×C32×C9C22semidirect
12. C2×C3⋊(C3⋊D9)C22semidirect
15. C2×C3×C3⋊D9C22semidirect
16. C2×C3×D3×C9C22semidirect
18. C3⋊(C3⋊Dic9)C22semidirect
19. C3×C9×Dic3C22semidirect
28. C22×C3⋊D9D3semidirect
39. C9×#67D3semidirect
47. C3×C3⋊D9D4semidirect
65. C22×C3×C9C2×D3semidirect
66. C22×C3×C9C2×D3semidirect
74. C2×C3⋊D9C2×D3semidirect
75. C2×D3×C9C2×D3semidirect
85. C9×Dic3C2×D3semidirect
86. C32×C9C2×D4semidirect
102. C3×#67D9semidirect
138. C3⋊D9#67direct (N × H)
148. C22×C32C2×D9semidirect
166. C2×C3×D3C2×D9semidirect
176. C3×Dic3C2×D9semidirect
180. C22×C9D32semidirect
193. C3×C9D3×D4semidirect
194. C3×C9C2×#67semidirect
212. #67C3⋊D9direct (N × H)
259. C22×C3D3×D9semidirect
260. C22×C3C2×C3⋊D9semidirect
276. C2×D3C2×C3⋊D9semidirect
284. Dic3C2×C3⋊D9semidirect
287. C32D4×D9semidirect
291. C9D3×#67semidirect
329. C22D3×C3⋊D9semidirect
334. C3D9×#67semidirect
335. C3D4×C3⋊D9semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊D92
C2×D3×C3⋊D92
#106199922
C3×C9×#672
#106198792
#106198742
Dic3×C3⋊D92
C3⋊D3×#673
D4×C3⋊D93
D9×#673