Finite Group Databasebeta

← #13048732

Subgroups — #13048732

Order 1296.

673 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 673 conjugacy classes of subgroups — too many to draw. Only its 86 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⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C2×((C3⋊D3)⋊C3)⋊D3C211yesmaximal, characteristic, complemented
3#10619994C211yesmaximal, characteristic, complemented
4#10619977C211yesmaximal, characteristic, complemented
5#10619886C211yesmaximal, characteristic, complemented
6#10619884C211yesmaximal, characteristic, complemented
7((C3⋊D3)⋊C3)⋊Dic3C211yesmaximal, characteristic, complemented
8#89049—31nomaximal
9D3×#638—31nomaximal
10#88676—33nomaximal
11C22×C3×H(3,𝔽3)C2211yescharacteristic, complemented
12C2×C3⋊#272C2211yescharacteristic, complemented
13C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
14C2×C3×(C3⋊D3)⋊C3—22no—
15C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
17((C3⋊D3)⋊C3)⋊D3—22no—
18C3⋊#1084C2211yescharacteristic, complemented
19C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
20C22×C3×C3⋊D3D311yescharacteristic, complemented
21C22×C32×D3—31no—
22C2×#1113—31no—
23C2×D3×C3⋊D3—31no—
24#6312D311yescharacteristic, complemented
25C3×#638—31no—
26C3×#638—31no—
27C3×#638—31no—
28C3×#638—61no—
29#6295—31no—
30#6295—31no—
31#6294—31no—
32#6291—31no—
33#6290—31no—
34D3×C3⋊Dic3—31no—
35C22×(C3⋊D3)⋊C3D313yescomplemented
36C2×((C3⋊D3)⋊C3)⋊C2—33no—
37#6235—33no—
38#6235—63no—
39#638⋊C3—33no—
40#6200—33no—
41#6198—33no—
42((C3⋊D3)⋊C3)⋊C4—33no—
43C2×C3×H(3,𝔽3)—21no—
44C2×C3×H(3,𝔽3)C2311yescharacteristic
45C3⋊#272—21no—
46C3⋊((C3⋊D3)⋊C3)—21no—
47C3×(C3⋊D3)⋊C3—21no—
48C3×(C3⋊D3)⋊C3D412yescomplemented
49C2×#638—91no—
50#3811—91no—
51D3×#67—93no—
52D3×#67—91no—
53D3×#67—91no—
54C22×C33—21no—
55C22×C33C2×D311yescharacteristic, complemented
56C22×C33C2×D311yescharacteristic, complemented
57C2×C3⋊(C3⋊D3)C2×D311yescharacteristic, complemented
58C2×C3×C3⋊D3—22no—
59C2×C3×C3⋊D3—31no—
60C2×C3×C3⋊D3—31no—
61C2×C3×C3⋊D3—31no—
62C2×C3×C3⋊D3—61no—
63C2×C3×C3⋊D3C2×D311yescharacteristic
64C2×C3×C3⋊D3C2×D311yescharacteristic, complemented
65C2×C32×D3—31no—
66C2×C32×D3—31no—
67C2×C32×D3—62no—
68#1113—62no—
69D3×C3⋊D3—62no—
70C3⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
71C3×C3⋊Dic3—31no—
72C3×C3⋊Dic3—31no—
73C3×C3⋊Dic3—31no—
74C3×C3⋊Dic3—61no—
75C22×H(3,𝔽3)—23no—
76C22×H(3,𝔽3)C2×D313yescomplemented
77C2×#272—33no—
78C2×#272—63no—
79C2×(C3⋊D3)⋊C3—26no—
80C2×(C3⋊D3)⋊C3—33no—
81C2×(C3⋊D3)⋊C3C2×D313yes—
82C2×(C3⋊D3)⋊C3C2×D313yescomplemented
83((C3⋊D3)⋊C3)⋊C2—66no—
84#1084—33no—
85#1084—63no—
86(C3⋊Dic3)⋊C3—33no—
87C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
88C23×C32—91no—
89C22×C3⋊D3—91no—
90C22×C3⋊D3C3⋊D311yescharacteristic, complemented
91C22×C3×D3—31no—
92C22×C3×D3—33no—
93C22×C3×D3—91no—
94C2×D32—93no—
95C2×D32—91no—
96C2×D32—91no—
97C2×D32—91no—
98#638—31no—
99#638—31no—
100#638—31no—
101#638—33no—
102#638—61no—
103#638—91no—
104#638—91no—
105#638—91no—
106#638—181no—
107C2×C3⋊Dic3—91no—
108C3×#67—91no—
109C3×#67—91no—
110C3×#67—93no—
111C3×#67—91no—
112C3×#67—91no—
113C3×#67—183no—
114C3×#67—181no—
115C3×#67—181no—
116#626—93no—
117#626—91no—
118#626—91no—
119#626—91no—
120#626—91no—
121#625—93no—
122#625—91no—
123#625—91no—
124#624—91no—
125D3×Dic3—93no—
126D3×Dic3—91no—
127D3×Dic3—91no—
128C2×C33—21no—
129C2×C33—21no—
130C2×C33—21no—
131C2×C33—41no—
132C2×C33C22×D311yescharacteristic
133C2×C33C22×D311yescharacteristic
134C3⋊(C3⋊D3)—21no—
135C3×C3⋊D3—21no—
136C3×C3⋊D3—61no—
137C3×C3⋊D3—61no—
138C3×C3⋊D3—61no—
139C3×C3⋊D3—121no—
140C3×C3⋊D3#6712yescomplemented
141C32×D3—32no—
142C32×D3—61no—
143C2×H(3,𝔽3)—23no—
144C2×H(3,𝔽3)—23no—
145C2×H(3,𝔽3)—43no—
146C2×H(3,𝔽3)C22×D313yes—
147#272—63no—
148#272—123no—
149(C3⋊D3)⋊C3—23no—
150(C3⋊D3)⋊C3—63no—
151(C3⋊D3)⋊C3#6716yescomplemented
152C2×#67—271no—
153C2×#67—273no—
154D3×D4—271no—
155D3×D4—271no—
156C22×C32—21no—
157C22×C32—23no—
158C22×C32—31no—
159C22×C32—31no—
160C22×C32—61no—
161C22×C32—61no—
162C22×C32—61no—
163C22×C32—91no—
164C22×C32—91no—
165C22×C32—121no—
166C22×C32—182no—
167C22×C32D3211yescharacteristic
168C22×C32D3213yescomplemented
169C22×C32C2×C3⋊D311yescharacteristic, complemented
170C2×C3⋊D3—22no—
171C2×C3⋊D3—31no—
172C2×C3⋊D3—31no—
173C2×C3⋊D3—31no—
174C2×C3⋊D3—33no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—91no—
177C2×C3⋊D3—181no—
178C2×C3⋊D3—182no—
179C2×C3⋊D3C2×C3⋊D311yescharacteristic
180C2×C3⋊D3C2×C3⋊D311yescharacteristic, complemented
181C2×C3×D3—33no—
182C2×C3×D3—31no—
183C2×C3×D3—33no—
184C2×C3×D3—31no—
185C2×C3×D3—66no—
186C2×C3×D3—62no—
187C2×C3×D3—91no—
188C2×C3×D3—91no—
189C2×C3×D3—91no—
190C2×C3×D3—93no—
191C2×C3×D3—91no—
192C2×C3×D3—91no—
193C2×C3×D3—91no—
194C2×C3×D3—182no—
195C2×C3×D3—183no—
196C2×C3×D3—181no—
197C2×C3×D3—181no—
198D32—186no—
199D32—182no—
200D32—182no—
201D32—182no—
202C3⋊Dic3—31no—
203C3⋊Dic3—31no—
204C3⋊Dic3—31no—
205C3⋊Dic3—33no—
206C3⋊Dic3—61no—
207C3⋊Dic3—91no—
208C3⋊Dic3—181no—
209C3×Dic3—91no—
210C3×Dic3—93no—
211C3×Dic3—91no—
212C3×Dic3—91no—
213C3×Dic3—91no—
214C3×Dic3—183no—
215C3×Dic3—181no—
216C3×Dic3—181no—
217C33—21no—
218C33D3×D411yescharacteristic, complemented
219C33C2×#6711yescharacteristic, complemented
220H(3,𝔽3)—23no—
221H(3,𝔽3)C2×#6713yescomplemented
222C23×C3—93no—
223C23×C3—91no—
224C22×D3—31no—
225C22×D3—91no—
226C22×D3—271no—
227C22×D3—271no—
228C22×D3—271no—
229C22×D3—273no—
230C3×D4—271no—
231C3×D4—271no—
232#67—91no—
233#67—93no—
234#67—91no—
235#67—91no—
236#67—91no—
237#67—181no—
238#67—181no—
239#67—183no—
240#67—273no—
241#67—271no—
242#67—271no—
243#67—271no—
244#67—271no—
245#67—273no—
246#67—273no—
247C2×Dic3—271no—
248C2×Dic3—273no—
249D12—271no—
250D12—271no—
251C4×D3—271no—
252C4×D3—271no—
253C2×C32—21no—
254C2×C32—23no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—23no—
258C2×C32—31no—
259C2×C32—31no—
260C2×C32—43no—
261C2×C32—41no—
262C2×C32—61no—
263C2×C32—61no—
264C2×C32—61no—
265C2×C32—61no—
266C2×C32—61no—
267C2×C32—92no—
268C2×C32—121no—
269C2×C32—121no—
270C2×C32—121no—
271C2×C32—121no—
272C2×C32—122no—
273C2×C32—181no—
274C2×C32C2×D3211yescharacteristic
275C2×C32C2×D3213yes—
276C2×C32C22×C3⋊D311yescharacteristic
277C3⋊D3—21no—
278C3⋊D3—61no—
279C3⋊D3—61no—
280C3⋊D3—61no—
281C3⋊D3—63no—
282C3⋊D3—121no—
283C3⋊D3—181no—
284C3⋊D3—361no—
285C3⋊D3#63812yescomplemented
286C3×D3—32no—
287C3×D3—36no—
288C3×D3—61no—
289C3×D3—63no—
290C3×D3—92no—
291C3×D3—181no—
292C3×D3—181no—
293C3×D3—181no—
294C3×D3—183no—
295C3×D3—181no—
296C3×D3—181no—
297C3×D3—363no—
298C3×D3—361no—
299C3×D3—361no—
300C2×D4—811noSylow
301C22×C3—21no—
302C22×C3—33no—
303C22×C3—31no—
304C22×C3—61no—
305C22×C3—63no—
306C22×C3—93no—
307C22×C3—93no—
308C22×C3—91no—
309C22×C3—91no—
310C22×C3—182no—
311C22×C3—186no—
312C22×C3—271no—
313C22×C3—271no—
314C22×C3((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
315C22×C3D3×C3⋊D311yescharacteristic
316C2×D3—31no—
317C2×D3—31no—
318C2×D3—62no—
319C2×D3—91no—
320C2×D3—91no—
321C2×D3—91no—
322C2×D3—93no—
323C2×D3—91no—
324C2×D3—91no—
325C2×D3—91no—
326C2×D3—182no—
327C2×D3—181no—
328C2×D3—181no—
329C2×D3—183no—
330C2×D3—273no—
331C2×D3—542no—
332C2×D3—542no—
333C2×D3—542no—
334C2×D3—546no—
335C3×C4—271no—
336C3×C4—271no—
337Dic3—91no—
338Dic3—93no—
339Dic3—91no—
340Dic3—91no—
341Dic3—91no—
342Dic3—181no—
343Dic3—181no—
344Dic3—183no—
345Dic3—273no—
346C32—21no—
347C32—23no—
348C32—31no—
349C32—31no—
350C32—61no—
351C32—61no—
352C32—61no—
353C32—121no—
354C32D3×#6711yescharacteristic
355C32D3×#6713yescomplemented
356C32C2×#63811yescharacteristic, complemented
357C23—91no—
358C23—811no—
359D4—271no—
360D4—271no—
361D4—811no—
362D4—811no—
363C2×C4—811no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—33no—
368C2×C3—31no—
369C2×C3—41no—
370C2×C3—61no—
371C2×C3—63no—
372C2×C3—61no—
373C2×C3—63no—
374C2×C3—96no—
375C2×C3—92no—
376C2×C3—121no—
377C2×C3—123no—
378C2×C3—183no—
379C2×C3—181no—
380C2×C3—541no—
381C2×C3—541no—
382C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
383C2×C3C2×D3×C3⋊D311yescharacteristic
384D3—32no—
385D3—61no—
386D3—92no—
387D3—181no—
388D3—181no—
389D3—183no—
390D3—181no—
391D3—181no—
392D3—181no—
393D3—361no—
394D3—361no—
395D3—363no—
396D3—543no—
397C22—91no—
398C22—91no—
399C22—182no—
400C22—271no—
401C22—271no—
402C22—1622no—
403C22((C3⋊D3)⋊C3)⋊D311yescharacteristic, complemented
404C4—271no—
405C4—271no—
406C3—21no—
407C3—31no—
408C3—33no—
409C3—61no—
410C3—63no—
411C3#8867611yescharacteristic, complemented
412C3D3×#63811yescharacteristic
413C2—21no—
414C2—92no—
415C2—181no—
416C2—541no—
417C2—541no—
418C2C2×((C3⋊D3)⋊C3)⋊D311yescharacteristic
subgroupG/H
centerC2C2×((C3⋊D3)⋊C3)⋊D3
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×D3×C3⋊D3
FittingC22×C3×H(3,𝔽3)C22
socleC2×C32C2×D32
hypercenterC22((C3⋊D3)⋊C3)⋊D3
hypocenterC3×H(3,𝔽3)C2×D4
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×(C3⋊D3)⋊C3C2semidirect
2. C2×((C3⋊D3)⋊C3)⋊D3C2semidirect
3. #10619994C2semidirect
4. #10619977C2semidirect
5. #10619886C2semidirect
6. #10619884C2semidirect
7. ((C3⋊D3)⋊C3)⋊Dic3C2semidirect
11. C22×C3×H(3,𝔽3)C22semidirect
12. C2×C3⋊#272C22semidirect
13. C2×C3⋊((C3⋊D3)⋊C3)C22semidirect
16. C2×C3×(C3⋊D3)⋊C3C22semidirect
18. C3⋊#1084C22semidirect
19. C3⋊((C3⋊Dic3)⋊C3)C22semidirect
20. C22×C3×C3⋊D3D3semidirect
24. #6312D3semidirect
35. C22×(C3⋊D3)⋊C3D3semidirect
48. C3×(C3⋊D3)⋊C3D4semidirect
55. C22×C33C2×D3semidirect
56. C22×C33C2×D3semidirect
57. C2×C3⋊(C3⋊D3)C2×D3semidirect
64. C2×C3×C3⋊D3C2×D3semidirect
70. C3⋊(C3⋊Dic3)C2×D3semidirect
76. C22×H(3,𝔽3)C2×D3semidirect
82. C2×(C3⋊D3)⋊C3C2×D3semidirect
87. C3×H(3,𝔽3)C2×D4semidirect
90. C22×C3⋊D3C3⋊D3semidirect
140. C3×C3⋊D3#67semidirect
151. (C3⋊D3)⋊C3#67semidirect
168. C22×C32D32semidirect
169. C22×C32C2×C3⋊D3semidirect
180. C2×C3⋊D3C2×C3⋊D3semidirect
218. C33D3×D4semidirect
219. C33C2×#67semidirect
221. H(3,𝔽3)C2×#67semidirect
285. C3⋊D3#638semidirect
314. C22×C3((C3⋊D3)⋊C3)⋊C2semidirect
355. C32D3×#67semidirect
356. C32C2×#638semidirect
403. C22((C3⋊D3)⋊C3)⋊D3semidirect
411. C3#88676semidirect

Maximal subgroups

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

subgroupindex
C22×C3×(C3⋊D3)⋊C32
C2×((C3⋊D3)⋊C3)⋊D32
#106199942
#106199772
#106198862
#106198842
((C3⋊D3)⋊C3)⋊Dic32
#890493
D3×#6383
#886763