Finite Group Databasebeta

← #13048709

Subgroups — #13048709

Order 1296.

780 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 106 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⋊D9)C211yesmaximal, characteristic, complemented
2C2×D3×C3⋊D9C211yesmaximal, characteristic, complemented
3C3×#6233C211yesmaximal, characteristic, complemented
4C3×C9×#67C211yesmaximal, characteristic, complemented
5#10619880C211yesmaximal, characteristic, complemented
6#10619874C211yesmaximal, characteristic, complemented
7#10619872C211yesmaximal, characteristic, complemented
8#89039—31nomaximal
9D4×C3⋊D9—31nomaximal
10#88668—33nomaximal
11C22×C32×C9C2211yescharacteristic, complemented
12C2×C3⋊(C3⋊D9)—22no—
13C2×C3⋊(C3⋊D9)C2211yescharacteristic
14C2×C3⋊(C3⋊D9)C2211yescharacteristic, complemented
15C2×C3×C3⋊D9C2211yescharacteristic, complemented
16C2×C3×D3×C9C2211yescharacteristic, complemented
17D3×C3⋊D9—22no—
18C3×C3⋊Dic9C2211yescharacteristic, complemented
19C3×C9×Dic3C2211yescharacteristic, complemented
20C22×C3⋊(C3⋊D3)—31no—
21C2×D3×C3⋊D3—31no—
22C3×#638—31no—
23C32×#67D311yescharacteristic
24#6292—31no—
25#6291—31no—
26#6289—31no—
27C22×C3⋊D9—31no—
28C22×C3⋊D9—33no—
29C22×C3⋊D9—31no—
30C22×C3⋊D9—63no—
31C22×C3⋊D9—61no—
32C2×D3×D9—33no—
33C3×D4×C9—31no—
34#6233—31no—
35#6233D311yescharacteristic, complemented
36C4⋊(C3⋊D9)—31no—
37C4×C3⋊D9—31no—
38C9×#67D313yescomplemented
39C3×#611—33no—
40#6195—33no—
41#6192—33no—
42(C3⋊D9)⋊C4—33no—
43C2×C32×C9—21no—
44C2×C32×C9C2311yescharacteristic
45C3⋊(C3⋊D9)—21no—
46C3⋊(C3⋊D9)D412yescomplemented
47C3×C3⋊D9—21no—
48C3×D3×C9—21no—
49D4×C3⋊D3—91no—
50#3811—93no—
51#3811—91no—
52D4×D9—93no—
53C22×C33C2×D311yescharacteristic
54C2×C3⋊(C3⋊D3)—31no—
55C2×C3⋊(C3⋊D3)—31no—
56C2×C3⋊(C3⋊D3)—62no—
57C2×C3×C3⋊D3—31no—
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—31no—
68C2×C3⋊D9—33no—
69C2×C3⋊D9—31no—
70C2×C3⋊D9—33no—
71C2×C3⋊D9—62no—
72C2×C3⋊D9—62no—
73C2×C3⋊D9—66no—
74C2×C3⋊D9—63no—
75C2×C3⋊D9—63no—
76C2×C3⋊D9—61no—
77C2×C3⋊D9—61no—
78C2×C3⋊D9—612no—
79C2×C3⋊D9—64no—
80C2×C3⋊D9C2×D311yescharacteristic, complemented
81C2×D3×C9C2×D313yescomplemented
82C2×C3×D9—33no—
83D3×D9—66no—
84C3×C4×C9—31no—
85C3⋊Dic9C2×D311yescharacteristic, complemented
86C9×Dic3C2×D313yescomplemented
87C3×Dic9—33no—
88C32×C9C2×D411yescharacteristic, complemented, Sylow
89C22×C3⋊D3—91no—
90C22×C3⋊D3—91no—
91C22×C3⋊D3—93no—
92C22×C3⋊D3—91no—
93C22×C3⋊D3—181no—
94C22×C3⋊D3—183no—
95C2×D32—93no—
96C2×D32—91no—
97C32×D4—31no—
98#638—31no—
99#638—91no—
100C4⋊(C3⋊D3)—91no—
101C4×C3⋊D3—91no—
102C3×#67—93no—
103C3×#67—91no—
104C3×#67D913yescomplemented
105C3×#67C3⋊D311yescharacteristic
106#626—93no—
107#626—91no—
108#626—93no—
109#626—91no—
110#624—93no—
111#624—91no—
112C22×D9—93no—
113C22×D9—93no—
114C22×D9—183no—
115D4×C9—33no—
116#611—33no—
117#611—93no—
118D36—93no—
119C4×D9—93no—
120C2×C33—21no—
121C2×C33C22×D311yescharacteristic
122C3⋊(C3⋊D3)—32no—
123C3⋊(C3⋊D3)—61no—
124C3×C3⋊D3—61no—
125C32×D3—21no—
126C2×C3×C9—22no—
127C2×C3×C9—26no—
128C2×C3×C9—23no—
129C2×C3×C9—21no—
130C2×C3×C9—21no—
131C2×C3×C9—23no—
132C2×C3×C9—61no—
133C2×C3×C9C22×D313yes—
134C2×C3×C9C22×D311yescharacteristic
135C3⋊D9—21no—
136C3⋊D9—32no—
137C3⋊D9—36no—
138C3⋊D9—66no—
139C3⋊D9—62no—
140C3⋊D9—62no—
141C3⋊D9—66no—
142C3⋊D9—63no—
143C3⋊D9—61no—
144D3×C9—23no—
145C3×D9—63no—
146D3×D4—271no—
147D3×D4—273no—
148D3×D4—271no—
149C22×C32—23no—
150C22×C32—21no—
151C22×C32—31no—
152C22×C32C2×D913yescomplemented
153C22×C32D3211yescharacteristic
154C22×C32C2×C3⋊D311yescharacteristic
155C2×C3⋊D3—31no—
156C2×C3⋊D3—91no—
157C2×C3⋊D3—91no—
158C2×C3⋊D3—93no—
159C2×C3⋊D3—91no—
160C2×C3⋊D3—93no—
161C2×C3⋊D3—91no—
162C2×C3⋊D3—181no—
163C2×C3⋊D3—181no—
164C2×C3⋊D3—183no—
165C2×C3⋊D3—183no—
166C2×C3⋊D3—182no—
167C2×C3⋊D3—182no—
168C2×C3⋊D3—182no—
169C2×C3⋊D3—186no—
170C2×C3⋊D3—184no—
171C2×C3⋊D3—1812no—
172C2×C3×D3—93no—
173C2×C3×D3—91no—
174C2×C3×D3C2×D913yescomplemented
175C2×C3×D3C2×C3⋊D311yescharacteristic
176D32—182no—
177D32—186no—
178C32×C4—31no—
179C3⋊Dic3—31no—
180C3×Dic3—93no—
181C3×Dic3—91no—
182C3×Dic3C2×D913yescomplemented
183C3×Dic3C2×C3⋊D311yescharacteristic
184C22×C9—23no—
185C22×C9—33no—
186C22×C9D3213yescomplemented
187C2×D9—33no—
188C2×D9—93no—
189C2×D9—93no—
190C2×D9—183no—
191C2×D9—183no—
192C2×D9—186no—
193C2×D9—186no—
194C2×D9—1812no—
195C4×C9—33no—
196Dic9—33no—
197C33D3×D411yescharacteristic
198C3×C9—21no—
199C3×C9—23no—
200C3×C9D3×D413yescomplemented
201C3×C9D3×D411yescharacteristic, complemented
202C22×D3—271no—
203C22×D3—273no—
204C22×D3—273no—
205C22×D3—271no—
206C22×D3—271no—
207C22×D3—271no—
208C22×D3—543no—
209C22×D3—541no—
210C3×D4—33no—
211C3×D4—31no—
212C3×D4—271no—
213#67—93no—
214#67—91no—
215#67—271no—
216#67—273no—
217#67—271no—
218#67C3⋊D911yescharacteristic, complemented
219D12—271no—
220D12—273no—
221D12—271no—
222C4×D3—271no—
223C4×D3—273no—
224C4×D3—271no—
225C2×C32—22no—
226C2×C32—26no—
227C2×C32—23no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—23no—
231C2×C32—21no—
232C2×C32—61no—
233C2×C32C22×D913yes—
234C2×C32C2×D3211yescharacteristic
235C2×C32C22×C3⋊D311yescharacteristic
236C3⋊D3—61no—
237C3⋊D3—92no—
238C3⋊D3—96no—
239C3⋊D3—92no—
240C3⋊D3—182no—
241C3⋊D3—186no—
242C3⋊D3—182no—
243C3⋊D3—186no—
244C3⋊D3—183no—
245C3⋊D3—181no—
246C3⋊D3—181no—
247C3×D3—23no—
248C3×D3—21no—
249C3×D3—183no—
250C3×D3—181no—
251C2×C9—26no—
252C2×C9—23no—
253C2×C9—23no—
254C2×C9—63no—
255C2×C9C2×D3213yes—
256D9—63no—
257D9—96no—
258D9—186no—
259D9—186no—
260D9—183no—
261C2×D4—811noSylow
262C22×C3—23no—
263C22×C3—21no—
264C22×C3—33no—
265C22×C3—31no—
266C22×C3—271no—
267C22×C3D3×D913yescomplemented
268C22×C3C2×C3⋊D911yescharacteristic, complemented
269C22×C3D3×C3⋊D311yescharacteristic
270C2×D3—93no—
271C2×D3—91no—
272C2×D3—273no—
273C2×D3—271no—
274C2×D3—271no—
275C2×D3—273no—
276C2×D3—271no—
277C2×D3—271no—
278C2×D3—543no—
279C2×D3—541no—
280C2×D3—543no—
281C2×D3—541no—
282C2×D3—542no—
283C2×D3—542no—
284C2×D3—546no—
285C2×D3—542no—
286C2×D3—546no—
287C2×D3—542no—
288C2×D3—544no—
289C2×D3—5412no—
290C2×D3C2×C3⋊D911yescharacteristic, complemented
291C3×C4—33no—
292C3×C4—31no—
293C3×C4—271no—
294Dic3—93no—
295Dic3—91no—
296Dic3C2×C3⋊D911yescharacteristic, complemented
297C32—23no—
298C32—21no—
299C32D4×D913yescomplemented
300C32#381111yescharacteristic
301C32D4×C3⋊D311yescharacteristic
302C9—23no—
303C9#381113yescomplemented
304C23—811no—
305C23—811no—
306D4—31no—
307D4—271no—
308D4—811no—
309D4—811no—
310C2×C4—811no—
311C2×C3—26no—
312C2×C3—22no—
313C2×C3—23no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—23no—
317C2×C3—21no—
318C2×C3—63no—
319C2×C3—61no—
320C2×C3—541no—
321C2×C3C2×D3×D913yes—
322C2×C3C22×C3⋊D911yescharacteristic
323C2×C3C2×D3×C3⋊D311yescharacteristic
324D3—21no—
325D3—183no—
326D3—181no—
327D3—276no—
328D3—272no—
329D3—272no—
330D3—546no—
331D3—542no—
332D3—546no—
333D3—542no—
334D3—543no—
335D3—541no—
336D3—541no—
337C22—31no—
338C22—271no—
339C22—811no—
340C22—811no—
341C22—1622no—
342C22—1622no—
343C22D3×C3⋊D911yescharacteristic, complemented
344C4—31no—
345C4—271no—
346C3—23no—
347C3—21no—
348C3#8866813yescomplemented
349C3D4×C3⋊D911yescharacteristic, complemented
350C3#8903911yescharacteristic
351C2—21no—
352C2—61no—
353C2—541no—
354C2—812no—
355C2—1621no—
356C2C2×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⋊D9)C2semidirect
2. C2×D3×C3⋊D9C2semidirect
3. C3×#6233C2semidirect
4. C3×C9×#67C2semidirect
5. #10619880C2semidirect
6. #10619874C2semidirect
7. #10619872C2semidirect
11. C22×C32×C9C22semidirect
14. C2×C3⋊(C3⋊D9)C22semidirect
15. C2×C3×C3⋊D9C22semidirect
16. C2×C3×D3×C9C22semidirect
18. C3×C3⋊Dic9C22semidirect
19. C3×C9×Dic3C22semidirect
35. #6233D3semidirect
38. C9×#67D3semidirect
46. C3⋊(C3⋊D9)D4semidirect
65. C22×C3×C9C2×D3semidirect
66. C22×C3×C9C2×D3semidirect
80. C2×C3⋊D9C2×D3semidirect
81. C2×D3×C9C2×D3semidirect
85. C3⋊Dic9C2×D3semidirect
86. C9×Dic3C2×D3semidirect
88. C32×C9C2×D4semidirect
104. C3×#67D9semidirect
152. C22×C32C2×D9semidirect
174. C2×C3×D3C2×D9semidirect
182. C3×Dic3C2×D9semidirect
186. C22×C9D32semidirect
200. C3×C9D3×D4semidirect
201. C3×C9D3×D4semidirect
218. #67C3⋊D9semidirect
267. C22×C3D3×D9semidirect
268. C22×C3C2×C3⋊D9semidirect
290. C2×D3C2×C3⋊D9semidirect
296. Dic3C2×C3⋊D9semidirect
299. C32D4×D9semidirect
303. C9#3811semidirect
343. C22D3×C3⋊D9semidirect
348. C3#88668semidirect
349. 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⋊D9)2
C2×D3×C3⋊D92
C3×#62332
C3×C9×#672
#106198802
#106198742
#106198722
#890393
D4×C3⋊D93
#886683