Finite Group Databasebeta

← C3×C3⋊D3×#67 #13049646

Subgroups — C3×C3⋊D3×#67 #13049646

Order 1296.

1140 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1140 conjugacy classes of subgroups — too many to draw. Only its 144 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×C32×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×C3×D3×C3⋊D3C211yesmaximal, characteristic, complemented
3C3×#6312C211yesmaximal, characteristic, complemented
4C33×#67C211yesmaximal, characteristic, complemented
5C3×#6292C211yesmaximal, characteristic, complemented
6C3×#6290C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
8C3×D4×C3⋊D3—31nomaximal
9C3⋊D3×#67C311yesmaximal, characteristic, direct product factor
10C3×D3×#67—34nomaximal
11C22×C34C2211yescharacteristic, complemented
12C2×C3×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
13C2×C32×C3⋊D3—22no—
14C2×C32×C3⋊D3C2211yescharacteristic
15C2×C32×C3⋊D3C2211yescharacteristic, complemented
16C2×C33×D3C2211yescharacteristic, complemented
17C3×D3×C3⋊D3—22no—
18C3×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
19C33×Dic3C2211yescharacteristic, complemented
20C22×C3×C3⋊D3—21no—
21C22×C3×C3⋊D3—31no—
22C22×C3×C3⋊D3D311yescharacteristic, complemented
23C22×C3×C3⋊D3C2×C311yescharacteristic, complemented
24C22×C32×D3—34no—
25C2×D3×C3⋊D3C2×C311yescharacteristic, complemented
26C2×C3×D32—34no—
27C33×D4—31no—
28#6312C2×C311yescharacteristic, complemented
29C3×#638—31no—
30C3×#638—34no—
31C3×#638—31no—
32C3×#638—64no—
33C3×C4⋊(C3⋊D3)—31no—
34C3×C4×C3⋊D3—31no—
35C32×#67—24no—
36C32×#67D314yescomplemented
37C32×#67C2×C311yescharacteristic, complemented
38#6292C2×C311yescharacteristic, complemented
39#6290C2×C311yescharacteristic, complemented
40Dic3×C3⋊D3C2×C311yescharacteristic, complemented
41C3×#626—34no—
42C3×#625—34no—
43C3×D3×Dic3—34no—
44C2×C34—21no—
45C2×C34C2311yescharacteristic
46C3×C3⋊(C3⋊D3)—21no—
47C32×C3⋊D3—21no—
48C32×C3⋊D3D412yescomplemented
49C33×D3—21no—
50D4×C3⋊D3—31no—
51C2×C3×#67—91no—
52C3×D3×D4—94no—
53D3×#67—34no—
54C22×C33—21no—
55C22×C33—24no—
56C22×C33—24no—
57C22×C33—31no—
58C22×C33—44no—
59C22×C33C2×D311yescharacteristic, complemented
60C22×C33C2×D314yescomplemented
61C22×C33C22×C311yescharacteristic, complemented
62C2×C3⋊(C3⋊D3)C22×C311yescharacteristic, complemented
63C2×C3×C3⋊D3—24no—
64C2×C3×C3⋊D3—21no—
65C2×C3×C3⋊D3—21no—
66C2×C3×C3⋊D3—22no—
67C2×C3×C3⋊D3—22no—
68C2×C3×C3⋊D3—31no—
69C2×C3×C3⋊D3—34no—
70C2×C3×C3⋊D3—62no—
71C2×C3×C3⋊D3—64no—
72C2×C3×C3⋊D3C2×D311yescharacteristic
73C2×C3×C3⋊D3C2×D311yescharacteristic, complemented
74C2×C3×C3⋊D3C22×C311yescharacteristic
75C2×C3×C3⋊D3C22×C311yescharacteristic, complemented
76C2×C32×D3—24no—
77C2×C32×D3—34no—
78C2×C32×D3—34no—
79C2×C32×D3—68no—
80C2×C32×D3C2×D314yescomplemented
81C2×C32×D3C22×C311yescharacteristic, complemented
82D3×C3⋊D3—22no—
83C3×D32—68no—
84C33×C4—31no—
85C3⋊(C3⋊Dic3)C22×C311yescharacteristic, complemented
86C3×C3⋊Dic3—31no—
87C3×C3⋊Dic3—34no—
88C3×C3⋊Dic3—64no—
89C32×Dic3—24no—
90C32×Dic3C2×D314yescomplemented
91C32×Dic3C22×C311yescharacteristic, complemented
92C34C2×D411yescharacteristic, complemented, Sylow
93C23×C32—91no—
94C22×C3⋊D3—31no—
95C22×C3⋊D3C3×D311yescharacteristic, complemented
96C22×C3×D3—34no—
97C22×C3×D3—34no—
98C22×C3×D3—64no—
99C22×C3×D3—94no—
100C22×C3×D3—91no—
101C2×D32—34no—
102C32×D4—31no—
103C32×D4—34no—
104C32×D4—64no—
105#638—31no—
106#638—31no—
107#638—34no—
108#638—64no—
109C4⋊(C3⋊D3)—31no—
110C4×C3⋊D3—31no—
111C3×#67—24no—
112C3×#67—94no—
113C3×#67—94no—
114C3×#67—91no—
115C3×#67—91no—
116C3×#67—91no—
117C3×#67—184no—
118C3×#67C3×D314yescomplemented
119C3×#67C3⋊D311yescharacteristic, direct product factor
120C2×C3×Dic3—91no—
121C3×D12—94no—
122C3×C4×D3—94no—
123#626—34no—
124#625—34no—
125D3×Dic3—34no—
126C2×C33—22no—
127C2×C33—24no—
128C2×C33—21no—
129C2×C33—21no—
130C2×C33—24no—
131C2×C33—24no—
132C2×C33—21no—
133C2×C33—44no—
134C2×C33—44no—
135C2×C33—44no—
136C2×C33—48no—
137C2×C33—61no—
138C2×C33C22×D311yescharacteristic
139C2×C33C22×D314yes—
140C2×C33C23×C311yescharacteristic
141C3⋊(C3⋊D3)—21no—
142C3×C3⋊D3—22no—
143C3×C3⋊D3—21no—
144C3×C3⋊D3—21no—
145C3×C3⋊D3—22no—
146C3×C3⋊D3—61no—
147C3×C3⋊D3—64no—
148C3×C3⋊D3—124no—
149C3×C3⋊D3#6712yesdirect product factor
150C3×C3⋊D3C3×D412yescomplemented
151C32×D3—21no—
152C32×D3—24no—
153C32×D3—38no—
154C32×D3—44no—
155C32×D3—64no—
156C2×C3×D4—271no—
157C2×#67—91no—
158D3×D4—94no—
159C22×C32—24no—
160C22×C32—24no—
161C22×C32—24no—
162C22×C32—24no—
163C22×C32—24no—
164C22×C32—31no—
165C22×C32—34no—
166C22×C32—44no—
167C22×C32—412no—
168C22×C32—44no—
169C22×C32—64no—
170C22×C32—91no—
171C22×C32—91no—
172C22×C32—182no—
173C22×C32D3214yescomplemented
174C22×C32C2×C3×D314yescomplemented
175C22×C32C2×C3×D311yescharacteristic, complemented
176C22×C32C2×C3⋊D311yescharacteristic, complemented
177C2×C3⋊D3—22no—
178C2×C3⋊D3—31no—
179C2×C3⋊D3—34no—
180C2×C3⋊D3—64no—
181C2×C3⋊D3—62no—
182C2×C3⋊D3C2×C3×D311yescharacteristic
183C2×C3⋊D3C2×C3×D311yescharacteristic, complemented
184C2×C3×D3—24no—
185C2×C3×D3—34no—
186C2×C3×D3—34no—
187C2×C3×D3—34no—
188C2×C3×D3—34no—
189C2×C3×D3—64no—
190C2×C3×D3—64no—
191C2×C3×D3—68no—
192C2×C3×D3—68no—
193C2×C3×D3—616no—
194C2×C3×D3—94no—
195C2×C3×D3—91no—
196C2×C3×D3—184no—
197C2×C3×D3—182no—
198C2×C3×D3—188no—
199C2×C3×D3C2×C3×D314yescomplemented
200C2×C3×D3C2×C3⋊D311yescharacteristic, complemented
201D32—68no—
202C32×C4—31no—
203C32×C4—34no—
204C32×C4—64no—
205C3⋊Dic3—31no—
206C3⋊Dic3—34no—
207C3⋊Dic3—64no—
208C3×Dic3—24no—
209C3×Dic3—94no—
210C3×Dic3—91no—
211C3×Dic3—184no—
212C3×Dic3C2×C3×D314yescomplemented
213C3×Dic3C2×C3⋊D311yescharacteristic, complemented
214C33—24no—
215C33—24no—
216C33—21no—
217C33—44no—
218C33D3×D414yescomplemented
219C33C2×#6711yescharacteristic, complemented
220C33C2×C3×D411yescharacteristic, complemented
221C23×C3—91no—
222C23×C3—91no—
223C23×C3—181no—
224C23×C3—271no—
225C22×D3—34no—
226C22×D3—94no—
227C22×D3—91no—
228C3×D4—34no—
229C3×D4—31no—
230C3×D4—64no—
231C3×D4—271no—
232C3×D4—271no—
233C3×D4—271no—
234C2×C3×C4—271no—
235#67—94no—
236#67—94no—
237#67—91no—
238#67—91no—
239#67—91no—
240#67—184no—
241#67C3×C3⋊D311yescharacteristic, direct product factor
242C2×Dic3—91no—
243D12—94no—
244C4×D3—94no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32—24no—
248C2×C32—24no—
249C2×C32—28no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32—24no—
255C2×C32—44no—
256C2×C32—412no—
257C2×C32—44no—
258C2×C32—44no—
259C2×C32—44no—
260C2×C32—44no—
261C2×C32—48no—
262C2×C32—48no—
263C2×C32—424no—
264C2×C32—44no—
265C2×C32—61no—
266C2×C32—64no—
267C2×C32—92no—
268C2×C32—124no—
269C2×C32—181no—
270C2×C32C2×D3214yes—
271C2×C32C22×C3×D314yes—
272C2×C32C22×C3×D311yescharacteristic
273C2×C32C22×C3⋊D311yescharacteristic
274C3⋊D3—21no—
275C3⋊D3—61no—
276C3⋊D3—64no—
277C3⋊D3—124no—
278C3⋊D3C3×#6712yesdirect product factor
279C3×D3—24no—
280C3×D3—21no—
281C3×D3—38no—
282C3×D3—38no—
283C3×D3—44no—
284C3×D3—68no—
285C3×D3—68no—
286C3×D3—64no—
287C3×D3—64no—
288C3×D3—184no—
289C3×D3—181no—
290C3×D3—364no—
291C2×D4—271noSylow
292C22×C3—21no—
293C22×C3—24no—
294C22×C3—24no—
295C22×C3—34no—
296C22×C3—31no—
297C22×C3—44no—
298C22×C3—64no—
299C22×C3—91no—
300C22×C3—91no—
301C22×C3—91no—
302C22×C3—91no—
303C22×C3—181no—
304C22×C3—181no—
305C22×C3—182no—
306C22×C3—182no—
307C22×C3—184no—
308C22×C3—271no—
309C22×C3—542no—
310C22×C3C3×D3214yescomplemented
311C22×C3D3×C3⋊D311yescharacteristic, complemented
312C22×C3C2×C3×C3⋊D311yescharacteristic, complemented
313C2×D3—34no—
314C2×D3—34no—
315C2×D3—68no—
316C2×D3—94no—
317C2×D3—91no—
318C2×D3—184no—
319C2×D3—182no—
320C2×D3—188no—
321C2×D3C2×C3×C3⋊D311yescharacteristic, complemented
322C3×C4—34no—
323C3×C4—31no—
324C3×C4—64no—
325C3×C4—271no—
326Dic3—94no—
327Dic3—91no—
328Dic3—184no—
329Dic3C2×C3×C3⋊D311yescharacteristic, complemented
330C32—24no—
331C32—24no—
332C32—24no—
333C32—24no—
334C32—24no—
335C32—44no—
336C32—412no—
337C32—44no—
338C32D3×#6714yescomplemented
339C32C3×D3×D414yescomplemented
340C32C2×C3×#6711yescharacteristic, complemented
341C32D4×C3⋊D311yescharacteristic, complemented
342C23—91no—
343C23—271no—
344D4—31no—
345D4—271no—
346D4—271no—
347D4—271no—
348C2×C4—271no—
349C2×C3—22no—
350C2×C3—24no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—24no—
354C2×C3—24no—
355C2×C3—21no—
356C2×C3—44no—
357C2×C3—44no—
358C2×C3—48no—
359C2×C3—44no—
360C2×C3—64no—
361C2×C3—61no—
362C2×C3—92no—
363C2×C3—92no—
364C2×C3—124no—
365C2×C3—182no—
366C2×C3—182no—
367C2×C3—181no—
368C2×C3—181no—
369C2×C3—541no—
370C2×C3C2×C3×D3214yes—
371C2×C3C2×D3×C3⋊D311yescharacteristic
372C2×C3C22×C3×C3⋊D311yescharacteristic
373D3—21no—
374D3—38no—
375D3—64no—
376D3—184no—
377D3—181no—
378D3—364no—
379C22—31no—
380C22—91no—
381C22—91no—
382C22—182no—
383C22—271no—
384C22—542no—
385C22C3×D3×C3⋊D311yescharacteristic, complemented
386C4—31no—
387C4—271no—
388C3—24no—
389C3—21no—
390C3—24no—
391C3—44no—
392C3C3×D3×#6714yescomplemented
393C3C3⋊D3×#6711yescharacteristic, direct product factor
394C3C3×D4×C3⋊D311yescharacteristic, complemented
395C2—21no—
396C2—61no—
397C2—92no—
398C2—181no—
399C2—541no—
400C2C2×C3×D3×C3⋊D311yescharacteristic
subgroupG/H
centerC2×C3C2×D3×C3⋊D3
commutator, cosocleC2×C33C23×C3
FrattiniC2C2×C3×D3×C3⋊D3
FittingC22×C34C22
socleC2×C34C23
hypercenterC22×C3D3×C3⋊D3
hypocenterC33C2×C3×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C32×C3⋊D3C2semidirect
2. C2×C3×D3×C3⋊D3C2semidirect
3. C3×#6312C2semidirect
4. C33×#67C2semidirect
5. C3×#6292C2semidirect
6. C3×#6290C2semidirect
7. C3×Dic3×C3⋊D3C2semidirect
9. C3⋊D3×#67C3direct (N × H)
11. C22×C34C22semidirect
12. C2×C3×C3⋊(C3⋊D3)C22semidirect
15. C2×C32×C3⋊D3C22semidirect
16. C2×C33×D3C22semidirect
18. C3×C3⋊(C3⋊Dic3)C22semidirect
19. C33×Dic3C22semidirect
22. C22×C3×C3⋊D3D3semidirect
23. C22×C3×C3⋊D3C2×C3semidirect
25. C2×D3×C3⋊D3C2×C3semidirect
28. #6312C2×C3semidirect
36. C32×#67D3semidirect
37. C32×#67C2×C3semidirect
38. #6292C2×C3semidirect
39. #6290C2×C3semidirect
40. Dic3×C3⋊D3C2×C3semidirect
48. C32×C3⋊D3D4semidirect
59. C22×C33C2×D3semidirect
60. C22×C33C2×D3semidirect
61. C22×C33C22×C3semidirect
62. C2×C3⋊(C3⋊D3)C22×C3semidirect
73. C2×C3×C3⋊D3C2×D3semidirect
75. C2×C3×C3⋊D3C22×C3semidirect
80. C2×C32×D3C2×D3semidirect
81. C2×C32×D3C22×C3semidirect
85. C3⋊(C3⋊Dic3)C22×C3semidirect
90. C32×Dic3C2×D3semidirect
91. C32×Dic3C22×C3semidirect
92. C34C2×D4semidirect
95. C22×C3⋊D3C3×D3semidirect
118. C3×#67C3×D3semidirect
119. C3×#67C3⋊D3direct (N × H)
149. C3×C3⋊D3#67direct (N × H)
150. C3×C3⋊D3C3×D4semidirect
173. C22×C32D32semidirect
174. C22×C32C2×C3×D3semidirect
175. C22×C32C2×C3×D3semidirect
176. C22×C32C2×C3⋊D3semidirect
183. C2×C3⋊D3C2×C3×D3semidirect
199. C2×C3×D3C2×C3×D3semidirect
200. C2×C3×D3C2×C3⋊D3semidirect
212. C3×Dic3C2×C3×D3semidirect
213. C3×Dic3C2×C3⋊D3semidirect
218. C33D3×D4semidirect
219. C33C2×#67semidirect
220. C33C2×C3×D4semidirect
241. #67C3×C3⋊D3direct (N × H)
278. C3⋊D3C3×#67direct (N × H)
310. C22×C3C3×D32semidirect
311. C22×C3D3×C3⋊D3semidirect
312. C22×C3C2×C3×C3⋊D3semidirect
321. C2×D3C2×C3×C3⋊D3semidirect
329. Dic3C2×C3×C3⋊D3semidirect
338. C32D3×#67semidirect
339. C32C3×D3×D4semidirect
340. C32C2×C3×#67semidirect
341. C32D4×C3⋊D3semidirect
385. C22C3×D3×C3⋊D3semidirect
392. C3C3×D3×#67semidirect
393. C3C3⋊D3×#67direct (N × H)
394. C3C3×D4×C3⋊D3semidirect

Maximal subgroups

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

subgroupindex
C22×C32×C3⋊D32
C2×C3×D3×C3⋊D32
C3×#63122
C33×#672
C3×#62922
C3×#62902
C3×Dic3×C3⋊D32
C3×D4×C3⋊D33
C3⋊D3×#673
C3×D3×#673