Finite Group Databasebeta

← #13048780

Subgroups — #13048780

Order 1296.

675 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 675 conjugacy classes of subgroups — too many to draw. Only its 63 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)⋊C3)C211yesmaximal, characteristic, complemented
2C2×#66432C211yesmaximal, characteristic, complemented
3C3×#6235C211yesmaximal, characteristic, complemented
4#10619977C211yesmaximal, characteristic, complemented
5#10619904C211yesmaximal, characteristic, complemented
6#10619902C211yesmaximal, characteristic, complemented
7#10619901C211yesmaximal, characteristic, complemented
8#89039—31nomaximal
9C3⋊D3×#67—31nomaximal
10#88676—31nomaximal
11C22×C3×H(3,𝔽3)C2211yescharacteristic, complemented
12C2×C3×#272C2211yescharacteristic, complemented
13C2×C3⋊((C3⋊D3)⋊C3)—22no—
14C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
16C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
17#66432—22no—
18C3×#1084C2211yescharacteristic, complemented
19C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
20C22×C3⋊(C3⋊D3)D311yescharacteristic, complemented
21C22×C3×C3⋊D3—31no—
22C2×D3×C3⋊D3—31no—
23C2×D3×C3⋊D3—31no—
24#6312D311yescharacteristic, complemented
25C3×#638—31no—
26C32×#67—31no—
27C32×#67—31no—
28C32×#67—61no—
29#6292—31no—
30#6292—31no—
31#6291—31no—
32#6290—31no—
33#6289—31no—
34Dic3×C3⋊D3—31no—
35C22×(C3⋊D3)⋊C3—31no—
36C22×(C3⋊D3)⋊C3—61no—
37C2×((C3⋊D3)⋊C3)⋊C2—31no—
38#6235D311yescharacteristic, complemented
39#638⋊C3—31no—
40#638⋊C3—61no—
41#6200—31no—
42#6198—31no—
43((C3⋊D3)⋊C3)⋊C4—31no—
44C2×C3×H(3,𝔽3)—21no—
45C2×C3×H(3,𝔽3)C2311yescharacteristic
46C3×#272—21no—
47C3⋊((C3⋊D3)⋊C3)—21no—
48C3⋊((C3⋊D3)⋊C3)—21no—
49C3⋊((C3⋊D3)⋊C3)D412yescomplemented
50D4×C3⋊D3—91no—
51#3811—91no—
52#3811—93no—
53D3×#67—91no—
54D3×#67—93no—
55C22×C33—21no—
56C22×C33C2×D311yescharacteristic, complemented
57C22×C33C2×D311yescharacteristic, complemented
58C2×C3⋊(C3⋊D3)—22no—
59C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
60C2×C3⋊(C3⋊D3)C2×D311yescharacteristic, complemented
61C2×C3⋊(C3⋊D3)C2×D311yescharacteristic, complemented
62C2×C3×C3⋊D3—31no—
63C2×C3×C3⋊D3—31no—
64C2×C3×C3⋊D3—31no—
65C2×C3×C3⋊D3—62no—
66C2×C32×D3—31no—
67C2×C32×D3—31no—
68C2×C32×D3—61no—
69D3×C3⋊D3—62no—
70D3×C3⋊D3—62no—
71C3⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
72C3×C3⋊Dic3—31no—
73C32×Dic3—31no—
74C32×Dic3—31no—
75C32×Dic3—61no—
76C22×H(3,𝔽3)—21no—
77C22×H(3,𝔽3)—21no—
78C22×H(3,𝔽3)—41no—
79C22×H(3,𝔽3)C2×D311yescharacteristic, complemented
80C2×#272C2×D311yescharacteristic, complemented
81C2×(C3⋊D3)⋊C3—31no—
82C2×(C3⋊D3)⋊C3—31no—
83C2×(C3⋊D3)⋊C3—31no—
84C2×(C3⋊D3)⋊C3—62no—
85C2×(C3⋊D3)⋊C3—61no—
86C2×(C3⋊D3)⋊C3—61no—
87C2×(C3⋊D3)⋊C3—64no—
88C2×(C3⋊D3)⋊C3—61no—
89((C3⋊D3)⋊C3)⋊C2—62no—
90#1084C2×D311yescharacteristic, complemented
91(C3⋊Dic3)⋊C3—31no—
92(C3⋊Dic3)⋊C3—61no—
93C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
94C22×C3⋊D3—31no—
95C22×C3⋊D3—31no—
96C22×C3⋊D3—61no—
97C22×C3⋊D3—93no—
98C22×C3⋊D3—91no—
99C22×C3×D3—93no—
100C22×C3×D3—91no—
101C2×D32—91no—
102C2×D32—91no—
103C2×D32—93no—
104C2×D32—93no—
105C32×D4—91no—
106#638—31no—
107#638—31no—
108#638—61no—
109#638—93no—
110#638—91no—
111C4⋊(C3⋊D3)—91no—
112C4×C3⋊D3—91no—
113C3×#67—33no—
114C3×#67—33no—
115C3×#67—31no—
116C3×#67—31no—
117C3×#67—61no—
118C3×#67—63no—
119C3×#67—93no—
120C3×#67—91no—
121#626—91no—
122#626—91no—
123#626—93no—
124#626—93no—
125#626—91no—
126#626—93no—
127#625—91no—
128#625—93no—
129#624—91no—
130#624—93no—
131D3×Dic3—91no—
132D3×Dic3—93no—
133C2×C33—21no—
134C2×C33—21no—
135C2×C33—21no—
136C2×C33—41no—
137C2×C33C22×D311yescharacteristic
138C2×C33C22×D311yescharacteristic
139C3⋊(C3⋊D3)—21no—
140C3⋊(C3⋊D3)—21no—
141C3⋊(C3⋊D3)#6712yescomplemented
142C3×C3⋊D3—32no—
143C3×C3⋊D3—61no—
144C3×C3⋊D3—61no—
145C32×D3—61no—
146C32×D3—61no—
147C32×D3—121no—
148C2×H(3,𝔽3)—22no—
149C2×H(3,𝔽3)—21no—
150C2×H(3,𝔽3)—21no—
151C2×H(3,𝔽3)—21no—
152C2×H(3,𝔽3)—41no—
153C2×H(3,𝔽3)—41no—
154C2×H(3,𝔽3)—42no—
155C2×H(3,𝔽3)C22×D311yescharacteristic
156#272—21no—
157(C3⋊D3)⋊C3—32no—
158(C3⋊D3)⋊C3—62no—
159(C3⋊D3)⋊C3—62no—
160(C3⋊D3)⋊C3—61no—
161(C3⋊D3)⋊C3—61no—
162(C3⋊D3)⋊C3—121no—
163C2×#67—271no—
164D3×D4—271no—
165D3×D4—273no—
166D3×D4—271no—
167C22×C32—21no—
168C22×C32—21no—
169C22×C32—21no—
170C22×C32—33no—
171C22×C32—33no—
172C22×C32—41no—
173C22×C32—63no—
174C22×C32—91no—
175C22×C32D3211yescharacteristic
176C22×C32D3211yescharacteristic, complemented
177C22×C32D3211yescharacteristic, complemented
178C2×C3⋊D3—31no—
179C2×C3⋊D3—31no—
180C2×C3⋊D3—31no—
181C2×C3⋊D3—31no—
182C2×C3⋊D3—31no—
183C2×C3⋊D3—31no—
184C2×C3⋊D3—61no—
185C2×C3⋊D3—61no—
186C2×C3⋊D3—61no—
187C2×C3⋊D3—62no—
188C2×C3⋊D3—62no—
189C2×C3⋊D3—64no—
190C2×C3⋊D3—93no—
191C2×C3⋊D3—93no—
192C2×C3⋊D3—93no—
193C2×C3⋊D3—186no—
194C2×C3⋊D3—182no—
195C2×C3×D3—33no—
196C2×C3×D3—33no—
197C2×C3×D3—31no—
198C2×C3×D3—31no—
199C2×C3×D3—61no—
200C2×C3×D3—63no—
201C2×C3×D3—93no—
202C2×C3×D3—91no—
203C2×C3×D3—93no—
204C2×C3×D3—93no—
205C2×C3×D3—91no—
206C2×C3×D3—91no—
207C2×C3×D3—182no—
208C2×C3×D3—186no—
209D32—182no—
210D32—186no—
211D32—182no—
212D32—186no—
213C32×C4—91no—
214C3⋊Dic3—31no—
215C3⋊Dic3—31no—
216C3⋊Dic3—61no—
217C3⋊Dic3—93no—
218C3×Dic3—33no—
219C3×Dic3—33no—
220C3×Dic3—31no—
221C3×Dic3—31no—
222C3×Dic3—61no—
223C3×Dic3—63no—
224C3×Dic3—93no—
225C3×Dic3—91no—
226C33—21no—
227C33D3×D411yescharacteristic, complemented
228C33C2×#6711yescharacteristic, complemented
229H(3,𝔽3)—21no—
230H(3,𝔽3)—21no—
231H(3,𝔽3)—41no—
232H(3,𝔽3)D3×D411yescharacteristic, complemented
233C23×C3—271no—
234C22×D3—93no—
235C22×D3—91no—
236C22×D3—271no—
237C22×D3—271no—
238C22×D3—271no—
239C22×D3—273no—
240C22×D3—271no—
241C22×D3—541no—
242C3×D4—93no—
243C3×D4—91no—
244C3×D4—271no—
245#67—31no—
246#67—31no—
247#67—61no—
248#67—93no—
249#67—91no—
250#67—271no—
251#67—271no—
252#67—271no—
253#67—273no—
254#67—271no—
255#67—271no—
256#67—541no—
257C2×Dic3—271no—
258D12—273no—
259D12—271no—
260D12—271no—
261C4×D3—271no—
262C4×D3—273no—
263C4×D3—271no—
264C2×C32—21no—
265C2×C32—22no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—33no—
272C2×C32—33no—
273C2×C32—41no—
274C2×C32—41no—
275C2×C32—42no—
276C2×C32—41no—
277C2×C32—63no—
278C2×C32—63no—
279C2×C32—63no—
280C2×C32—123no—
281C2×C32—181no—
282C2×C32C2×D3211yescharacteristic
283C2×C32C2×D3211yescharacteristic
284C2×C32C2×D3211yescharacteristic
285C3⋊D3—32no—
286C3⋊D3—32no—
287C3⋊D3—62no—
288C3⋊D3—61no—
289C3⋊D3—61no—
290C3⋊D3—62no—
291C3⋊D3—61no—
292C3⋊D3—61no—
293C3⋊D3—96no—
294C3⋊D3—121no—
295C3⋊D3—183no—
296C3⋊D3—183no—
297C3×D3—63no—
298C3×D3—63no—
299C3×D3—61no—
300C3×D3—61no—
301C3×D3—96no—
302C3×D3—92no—
303C3×D3—121no—
304C3×D3—123no—
305C3×D3—183no—
306C3×D3—183no—
307C3×D3—181no—
308C3×D3—181no—
309C2×D4—811noSylow
310C22×C3—31no—
311C22×C3—31no—
312C22×C3—61no—
313C22×C3—61no—
314C22×C3—61no—
315C22×C3—93no—
316C22×C3—91no—
317C22×C3—121no—
318C22×C3—271no—
319C22×C3—271no—
320C22×C3—271no—
321C22×C3—542no—
322C22×C3((C3⋊D3)⋊C3)⋊C213yescomplemented
323C22×C3#111311yescharacteristic
324C2×D3—31no—
325C2×D3—31no—
326C2×D3—61no—
327C2×D3—93no—
328C2×D3—91no—
329C2×D3—93no—
330C2×D3—93no—
331C2×D3—91no—
332C2×D3—91no—
333C2×D3—182no—
334C2×D3—186no—
335C2×D3—271no—
336C2×D3—271no—
337C2×D3—271no—
338C2×D3—541no—
339C2×D3—541no—
340C2×D3—541no—
341C2×D3—542no—
342C2×D3—544no—
343C2×D3—542no—
344C2×D3—542no—
345C2×D3—546no—
346C2×D3—542no—
347C3×C4—93no—
348C3×C4—91no—
349C3×C4—271no—
350Dic3—31no—
351Dic3—31no—
352Dic3—61no—
353Dic3—93no—
354Dic3—91no—
355Dic3—271no—
356Dic3—541no—
357C32—21no—
358C32—21no—
359C32—21no—
360C32—33no—
361C32—33no—
362C32—41no—
363C32—63no—
364C32D3×#6711yescharacteristic
365C32D3×#6711yescharacteristic, complemented
366C32#381111yescharacteristic, complemented
367C23—271no—
368C23—811no—
369D4—91no—
370D4—271no—
371D4—811no—
372D4—811no—
373C2×C4—811no—
374C2×C3—23no—
375C2×C3—21no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—61no—
379C2×C3—61no—
380C2×C3—62no—
381C2×C3—61no—
382C2×C3—61no—
383C2×C3—61no—
384C2×C3—121no—
385C2×C3—121no—
386C2×C3—122no—
387C2×C3—121no—
388C2×C3—183no—
389C2×C3—181no—
390C2×C3—272no—
391C2×C3—541no—
392C2×C3—541no—
393C2×C3C2×((C3⋊D3)⋊C3)⋊C213yes—
394C2×C3C2×#111311yescharacteristic
395D3—61no—
396D3—61no—
397D3—96no—
398D3—92no—
399D3—121no—
400D3—183no—
401D3—183no—
402D3—181no—
403D3—181no—
404D3—272no—
405D3—542no—
406D3—541no—
407D3—541no—
408D3—542no—
409D3—1081no—
410C22—91no—
411C22—271no—
412C22—271no—
413C22—271no—
414C22—542no—
415C22—1622no—
416C22#6643211yescharacteristic, complemented
417C4—91no—
418C4—271no—
419C3—31no—
420C3—31no—
421C3—61no—
422C3—61no—
423C3—61no—
424C3—121no—
425C3#8867613yescomplemented
426C3#8904911yescharacteristic
427C2—21no—
428C2—181no—
429C2—272no—
430C2—541no—
431C2—541no—
432C2C2×#6643211yescharacteristic
subgroupG/H
centerC2C2×#66432
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×#1113
FittingC22×C3×H(3,𝔽3)C22
socleC2×C32C2×D32
hypercenterC22#66432
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)⋊C3)C2semidirect
2. C2×#66432C2semidirect
3. C3×#6235C2semidirect
4. #10619977C2semidirect
5. #10619904C2semidirect
6. #10619902C2semidirect
7. #10619901C2semidirect
11. C22×C3×H(3,𝔽3)C22semidirect
12. C2×C3×#272C22semidirect
15. C2×C3⋊((C3⋊D3)⋊C3)C22semidirect
16. C2×C3⋊((C3⋊D3)⋊C3)C22semidirect
18. C3×#1084C22semidirect
19. C3⋊((C3⋊Dic3)⋊C3)C22semidirect
20. C22×C3⋊(C3⋊D3)D3semidirect
24. #6312D3semidirect
38. #6235D3semidirect
49. C3⋊((C3⋊D3)⋊C3)D4semidirect
56. C22×C33C2×D3semidirect
57. C22×C33C2×D3semidirect
60. C2×C3⋊(C3⋊D3)C2×D3semidirect
61. C2×C3⋊(C3⋊D3)C2×D3semidirect
71. C3⋊(C3⋊Dic3)C2×D3semidirect
79. C22×H(3,𝔽3)C2×D3semidirect
80. C2×#272C2×D3semidirect
90. #1084C2×D3semidirect
93. C3×H(3,𝔽3)C2×D4semidirect
141. C3⋊(C3⋊D3)#67semidirect
176. C22×C32D32semidirect
177. C22×C32D32semidirect
227. C33D3×D4semidirect
228. C33C2×#67semidirect
232. H(3,𝔽3)D3×D4semidirect
322. C22×C3((C3⋊D3)⋊C3)⋊C2semidirect
365. C32D3×#67semidirect
366. C32#3811semidirect
416. C22#66432semidirect
425. 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)⋊C3)2
C2×#664322
C3×#62352
#106199772
#106199042
#106199022
#106199012
#890393
C3⋊D3×#673
#886763