Finite Group Databasebeta

← #13049720

Subgroups — #13049720

Order 1296.

1220 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 111 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⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×C3⋊#1113C211yesmaximal, characteristic, complemented
3C3×#6312C211yesmaximal, characteristic, complemented
4C32×#638C211yesmaximal, characteristic, complemented
5#10620157C211yesmaximal, characteristic, complemented
6#10620156C211yesmaximal, characteristic, complemented
7#10620154C211yesmaximal, characteristic, complemented
8#89049—34nomaximal
9#89039—31nomaximal
10C3⋊D3×#67—31nomaximal
11C22×C34C2211yescharacteristic, complemented
12C2×C3×C3⋊(C3⋊D3)—22no—
13C2×C3×C3⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
15C2×C3×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
16C2×C32×C3⋊D3C2211yescharacteristic, complemented
17C3⋊#1113—22no—
18C3×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
19C32×C3⋊Dic3C2211yescharacteristic, complemented
20C22×C3⋊(C3⋊D3)D311yescharacteristic, complemented
21C22×C3×C3⋊D3—31no—
22C22×C3×C3⋊D3—34no—
23C22×C3×C3⋊D3—64no—
24C2×#1113—34no—
25C2×D3×C3⋊D3—31no—
26C2×D3×C3⋊D3—31no—
27#6312D311yescharacteristic, complemented
28C3×#638—31no—
29C3×#638—34no—
30C3×#638—64no—
31C3×#638D314yescomplemented
32C32×#67—31no—
33C32×#67—31no—
34C32×#67—61no—
35#6295—34no—
36#6295—34no—
37#6294—34no—
38#6292—31no—
39#6292—31no—
40#6291—31no—
41#6290—31no—
42#6289—31no—
43Dic3×C3⋊D3—31no—
44C2×C34—21no—
45C2×C34C2311yescharacteristic
46C3×C3⋊(C3⋊D3)—21no—
47C3×C3⋊(C3⋊D3)—21no—
48C3×C3⋊(C3⋊D3)D412yescomplemented
49C32×C3⋊D3—21no—
50D4×C3⋊D3—91no—
51#3811—94no—
52D3×#67—94no—
53D3×#67—91no—
54C22×C33—24no—
55C22×C33—21no—
56C22×C33—24no—
57C22×C33—44no—
58C22×C33C2×D311yescharacteristic, complemented
59C22×C33C2×D311yescharacteristic, complemented
60C22×C33C2×D314yescomplemented
61C2×C3⋊(C3⋊D3)—22no—
62C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
63C2×C3⋊(C3⋊D3)C2×D311yescharacteristic, complemented
64C2×C3⋊(C3⋊D3)C2×D311yescharacteristic, complemented
65C2×C3×C3⋊D3—31no—
66C2×C3×C3⋊D3—34no—
67C2×C3×C3⋊D3—31no—
68C2×C3×C3⋊D3—31no—
69C2×C3×C3⋊D3—34no—
70C2×C3×C3⋊D3—34no—
71C2×C3×C3⋊D3—62no—
72C2×C3×C3⋊D3—68no—
73C2×C3×C3⋊D3—64no—
74C2×C3×C3⋊D3—64no—
75C2×C3×C3⋊D3—616no—
76C2×C3×C3⋊D3—64no—
77C2×C3×C3⋊D3C2×D314yescomplemented
78C2×C32×D3—31no—
79C2×C32×D3—31no—
80C2×C32×D3—61no—
81#1113—68no—
82D3×C3⋊D3—62no—
83D3×C3⋊D3—62no—
84C3⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
85C3×C3⋊Dic3—31no—
86C3×C3⋊Dic3—34no—
87C3×C3⋊Dic3—64no—
88C3×C3⋊Dic3C2×D314yescomplemented
89C32×Dic3—31no—
90C32×Dic3—31no—
91C32×Dic3—61no—
92C34C2×D411yescharacteristic, complemented, Sylow
93C22×C3⋊D3—31no—
94C22×C3⋊D3—34no—
95C22×C3⋊D3—64no—
96C22×C3⋊D3—91no—
97C22×C3×D3—94no—
98C22×C3×D3—91no—
99C22×C3×D3—184no—
100C2×D32—94no—
101C2×D32—94no—
102C2×D32—91no—
103C32×D4—91no—
104#638—31no—
105#638—34no—
106#638—64no—
107#638—91no—
108#638C3⋊D311yescharacteristic, complemented
109C4⋊(C3⋊D3)—91no—
110C4×C3⋊D3—91no—
111C3×#67—34no—
112C3×#67—34no—
113C3×#67—64no—
114C3×#67—94no—
115C3×#67—91no—
116C3×#67—184no—
117#626—94no—
118#626—94no—
119#626—94no—
120#626—91no—
121#625—94no—
122#625—91no—
123#624—94no—
124D3×Dic3—94no—
125D3×Dic3—91no—
126C2×C33—21no—
127C2×C33—28no—
128C2×C33—24no—
129C2×C33—21no—
130C2×C33—21no—
131C2×C33—24no—
132C2×C33—24no—
133C2×C33—44no—
134C2×C33—41no—
135C2×C33—44no—
136C2×C33—48no—
137C2×C33C22×D311yescharacteristic
138C2×C33C22×D311yescharacteristic
139C2×C33C22×D314yes—
140C3⋊(C3⋊D3)—21no—
141C3⋊(C3⋊D3)—21no—
142C3⋊(C3⋊D3)#6712yescomplemented
143C3×C3⋊D3—24no—
144C3×C3⋊D3—32no—
145C3×C3⋊D3—38no—
146C3×C3⋊D3—68no—
147C3×C3⋊D3—68no—
148C3×C3⋊D3—64no—
149C3×C3⋊D3—61no—
150C3×C3⋊D3—61no—
151C3×C3⋊D3—64no—
152C3×C3⋊D3—124no—
153C32×D3—61no—
154C32×D3—61no—
155C32×D3—121no—
156C2×#67—271no—
157D3×D4—271no—
158D3×D4—274no—
159C22×C32—24no—
160C22×C32—24no—
161C22×C32—24no—
162C22×C32—24no—
163C22×C32—24no—
164C22×C32—44no—
165C22×C32—412no—
166C22×C32—44no—
167C22×C32—91no—
168C22×C32D3214yescomplemented
169C22×C32D3214yescomplemented
170C22×C32D3211yescharacteristic, complemented
171C22×C32C2×C3⋊D311yescharacteristic, complemented
172C2×C3⋊D3—31no—
173C2×C3⋊D3—31no—
174C2×C3⋊D3—31no—
175C2×C3⋊D3—34no—
176C2×C3⋊D3—34no—
177C2×C3⋊D3—34no—
178C2×C3⋊D3—64no—
179C2×C3⋊D3—64no—
180C2×C3⋊D3—64no—
181C2×C3⋊D3—62no—
182C2×C3⋊D3—68no—
183C2×C3⋊D3—616no—
184C2×C3⋊D3—182no—
185C2×C3⋊D3C2×C3⋊D311yescharacteristic, complemented
186C2×C3×D3—34no—
187C2×C3×D3—34no—
188C2×C3×D3—64no—
189C2×C3×D3—94no—
190C2×C3×D3—91no—
191C2×C3×D3—94no—
192C2×C3×D3—94no—
193C2×C3×D3—91no—
194C2×C3×D3—91no—
195C2×C3×D3—184no—
196C2×C3×D3—184no—
197C2×C3×D3—184no—
198C2×C3×D3—188no—
199C2×C3×D3—182no—
200C2×C3×D3—1816no—
201D32—188no—
202D32—188no—
203D32—182no—
204C32×C4—91no—
205C3⋊Dic3—31no—
206C3⋊Dic3—34no—
207C3⋊Dic3—64no—
208C3⋊Dic3C2×C3⋊D311yescharacteristic, complemented
209C3×Dic3—34no—
210C3×Dic3—34no—
211C3×Dic3—64no—
212C3×Dic3—94no—
213C3×Dic3—91no—
214C3×Dic3—184no—
215C33—21no—
216C33—24no—
217C33—24no—
218C33—44no—
219C33D3×D411yescharacteristic, complemented
220C33D3×D414yescomplemented
221C33C2×#6711yescharacteristic, complemented
222C23×C3—271no—
223C22×D3—91no—
224C22×D3—94no—
225C22×D3—184no—
226C22×D3—271no—
227C22×D3—271no—
228C22×D3—274no—
229C3×D4—94no—
230C3×D4—271no—
231#67—31no—
232#67—31no—
233#67—61no—
234#67—91no—
235#67—94no—
236#67—184no—
237#67—271no—
238#67—271no—
239#67—271no—
240#67—274no—
241C2×Dic3—271no—
242D12—274no—
243D12—271no—
244C4×D3—271no—
245C4×D3—274no—
246C2×C32—28no—
247C2×C32—24no—
248C2×C32—21no—
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—28no—
252C2×C32—24no—
253C2×C32—21no—
254C2×C32—24no—
255C2×C32—24no—
256C2×C32—24no—
257C2×C32—44no—
258C2×C32—412no—
259C2×C32—44no—
260C2×C32—44no—
261C2×C32—44no—
262C2×C32—48no—
263C2×C32—424no—
264C2×C32—48no—
265C2×C32—44no—
266C2×C32—181no—
267C2×C32C2×D3214yes—
268C2×C32C2×D3214yes—
269C2×C32C2×D3211yescharacteristic
270C2×C32C22×C3⋊D311yescharacteristic
271C3⋊D3—21no—
272C3⋊D3—32no—
273C3⋊D3—38no—
274C3⋊D3—68no—
275C3⋊D3—68no—
276C3⋊D3—64no—
277C3⋊D3—61no—
278C3⋊D3—61no—
279C3⋊D3—64no—
280C3⋊D3—124no—
281C3×D3—64no—
282C3×D3—64no—
283C3×D3—98no—
284C3×D3—92no—
285C3×D3—124no—
286C3×D3—188no—
287C3×D3—188no—
288C3×D3—184no—
289C3×D3—184no—
290C3×D3—181no—
291C3×D3—181no—
292C3×D3—364no—
293C2×D4—811noSylow
294C22×C3—24no—
295C22×C3—21no—
296C22×C3—24no—
297C22×C3—44no—
298C22×C3—94no—
299C22×C3—271no—
300C22×C3—271no—
301C22×C3—271no—
302C22×C3—542no—
303C22×C3D3×C3⋊D311yescharacteristic, complemented
304C22×C3D3×C3⋊D311yescharacteristic, complemented
305C22×C3#111314yescomplemented
306C2×D3—31no—
307C2×D3—31no—
308C2×D3—61no—
309C2×D3—91no—
310C2×D3—94no—
311C2×D3—91no—
312C2×D3—91no—
313C2×D3—94no—
314C2×D3—94no—
315C2×D3—184no—
316C2×D3—184no—
317C2×D3—184no—
318C2×D3—182no—
319C2×D3—188no—
320C2×D3—1816no—
321C2×D3—542no—
322C2×D3—542no—
323C2×D3—548no—
324C3×C4—94no—
325C3×C4—271no—
326Dic3—31no—
327Dic3—31no—
328Dic3—61no—
329Dic3—91no—
330Dic3—94no—
331Dic3—184no—
332C32—24no—
333C32—24no—
334C32—24no—
335C32—24no—
336C32—24no—
337C32—44no—
338C32—412no—
339C32—44no—
340C32D3×#6714yescomplemented
341C32D3×#6711yescharacteristic, complemented
342C32#381114yescomplemented
343C32D4×C3⋊D311yescharacteristic, complemented
344C23—271no—
345C23—811no—
346D4—91no—
347D4—271no—
348D4—811no—
349D4—811no—
350C2×C4—811no—
351C2×C3—28no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—24no—
355C2×C3—21no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—41no—
359C2×C3—44no—
360C2×C3—48no—
361C2×C3—44no—
362C2×C3—184no—
363C2×C3—272no—
364C2×C3—541no—
365C2×C3—541no—
366C2×C3C2×D3×C3⋊D311yescharacteristic
367C2×C3C2×D3×C3⋊D311yescharacteristic
368C2×C3C2×#111314yes—
369D3—61no—
370D3—61no—
371D3—92no—
372D3—98no—
373D3—121no—
374D3—188no—
375D3—188no—
376D3—181no—
377D3—181no—
378D3—184no—
379D3—184no—
380D3—364no—
381C22—91no—
382C22—271no—
383C22—271no—
384C22—271no—
385C22—542no—
386C22—1622no—
387C22C3⋊#111311yescharacteristic, complemented
388C4—91no—
389C4—271no—
390C3—21no—
391C3—24no—
392C3—24no—
393C3—44no—
394C3C3⋊D3×#6711yescharacteristic, complemented
395C3#8903911yescharacteristic, complemented
396C3#8904914yescomplemented
397C2—21no—
398C2—181no—
399C2—272no—
400C2—541no—
401C2—541no—
402C2C2×C3⋊#111311yescharacteristic
subgroupG/H
center, FrattiniC2C2×C3⋊#1113
commutator, socle, cosocleC2×C34C23
FittingC22×C34C22
hypercenterC22C3⋊#1113
hypocenterC34C2×D4
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×C3⋊(C3⋊D3)C2semidirect
2. C2×C3⋊#1113C2semidirect
3. C3×#6312C2semidirect
4. C32×#638C2semidirect
5. #10620157C2semidirect
6. #10620156C2semidirect
7. #10620154C2semidirect
11. C22×C34C22semidirect
14. C2×C3×C3⋊(C3⋊D3)C22semidirect
15. C2×C3×C3⋊(C3⋊D3)C22semidirect
16. C2×C32×C3⋊D3C22semidirect
18. C3×C3⋊(C3⋊Dic3)C22semidirect
19. C32×C3⋊Dic3C22semidirect
20. C22×C3⋊(C3⋊D3)D3semidirect
27. #6312D3semidirect
31. C3×#638D3semidirect
48. C3×C3⋊(C3⋊D3)D4semidirect
58. C22×C33C2×D3semidirect
59. C22×C33C2×D3semidirect
60. C22×C33C2×D3semidirect
63. C2×C3⋊(C3⋊D3)C2×D3semidirect
64. C2×C3⋊(C3⋊D3)C2×D3semidirect
77. C2×C3×C3⋊D3C2×D3semidirect
84. C3⋊(C3⋊Dic3)C2×D3semidirect
88. C3×C3⋊Dic3C2×D3semidirect
92. C34C2×D4semidirect
108. #638C3⋊D3semidirect
142. C3⋊(C3⋊D3)#67semidirect
168. C22×C32D32semidirect
169. C22×C32D32semidirect
170. C22×C32D32semidirect
171. C22×C32C2×C3⋊D3semidirect
185. C2×C3⋊D3C2×C3⋊D3semidirect
208. C3⋊Dic3C2×C3⋊D3semidirect
219. C33D3×D4semidirect
220. C33D3×D4semidirect
221. C33C2×#67semidirect
303. C22×C3D3×C3⋊D3semidirect
304. C22×C3D3×C3⋊D3semidirect
305. C22×C3#1113semidirect
340. C32D3×#67semidirect
341. C32D3×#67semidirect
342. C32#3811semidirect
343. C32D4×C3⋊D3semidirect
387. C22C3⋊#1113semidirect
394. C3C3⋊D3×#67semidirect
395. C3#89039semidirect
396. C3#89049semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊(C3⋊D3)2
C2×C3⋊#11132
C3×#63122
C32×#6382
#106201572
#106201562
#106201542
#890493
#890393
C3⋊D3×#673