Finite Group Databasebeta

← #13048731

Subgroups — #13048731

Order 1296.

697 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 697 conjugacy classes of subgroups — too many to draw. Only its 78 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×((C3⋊D3)⋊C3)⋊D3C211yesmaximal, characteristic, complemented
3#10619994C211yesmaximal, characteristic, complemented
4C3×#638⋊C3C211yesmaximal, characteristic, complemented
5#10619887C211yesmaximal, characteristic, complemented
6#10619884C211yesmaximal, characteristic, complemented
7#10619883C211yesmaximal, characteristic, complemented
8#89049—31nomaximal
9#89039—31nomaximal
10#88676—33nomaximal
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)⋊C3C2211yescharacteristic, complemented
17((C3⋊D3)⋊C3)⋊D3—22no—
18C3⋊#1084C2211yescharacteristic, complemented
19C3×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
20C22×C3⋊(C3⋊D3)D311yescharacteristic, complemented
21C22×C3×C3⋊D3—31no—
22C2×#1113—31no—
23C2×D3×C3⋊D3—31no—
24C3×#638—31no—
25C3×#638—31no—
26C3×#638—61no—
27C3×#638D311yescharacteristic, complemented
28C32×#67—31no—
29#6295—31no—
30#6295—31no—
31#6294—31no—
32#6292—31no—
33#6291—31no—
34#6289—31no—
35C22×(C3⋊D3)⋊C3—33no—
36C2×((C3⋊D3)⋊C3)⋊C2—33no—
37#6235—33no—
38#6235—63no—
39#638⋊C3D313yescomplemented
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)D412yescomplemented
48C3×(C3⋊D3)⋊C3—21no—
49D4×C3⋊D3—91no—
50#3811—93no—
51#3811—91no—
52D3×#67—91no—
53D3×#67—91no—
54C22×C33—21no—
55C22×C33C2×D311yescharacteristic, complemented
56C22×C33C2×D311yescharacteristic, complemented
57C2×C3⋊(C3⋊D3)—22no—
58C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
59C2×C3⋊(C3⋊D3)C2×D311yescharacteristic, complemented
60C2×C3×C3⋊D3—31no—
61C2×C3×C3⋊D3—31no—
62C2×C3×C3⋊D3—31no—
63C2×C3×C3⋊D3—31no—
64C2×C3×C3⋊D3—62no—
65C2×C3×C3⋊D3—61no—
66C2×C3×C3⋊D3C2×D311yescharacteristic, complemented
67C2×C32×D3—31no—
68#1113—62no—
69D3×C3⋊D3—62no—
70C3×C3⋊Dic3—31no—
71C3×C3⋊Dic3—31no—
72C3×C3⋊Dic3—61no—
73C3×C3⋊Dic3C2×D311yescharacteristic, complemented
74C32×Dic3—31no—
75C22×H(3,𝔽3)—23no—
76C22×H(3,𝔽3)C2×D313yescomplemented
77C2×#272—33no—
78C2×#272—63no—
79C2×(C3⋊D3)⋊C3—33no—
80C2×(C3⋊D3)⋊C3—33no—
81C2×(C3⋊D3)⋊C3—66no—
82C2×(C3⋊D3)⋊C3C2×D313yescomplemented
83((C3⋊D3)⋊C3)⋊C2—66no—
84#1084—33no—
85#1084—63no—
86(C3⋊Dic3)⋊C3C2×D313yescomplemented
87C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
88C22×C3⋊D3—31no—
89C22×C3⋊D3—33no—
90C22×C3⋊D3—91no—
91C22×C3⋊D3—91no—
92C22×C3⋊D3—181no—
93C22×C3×D3—91no—
94C22×C3×D3—91no—
95C22×C3×D3—181no—
96C2×D32—93no—
97C2×D32—91no—
98C2×D32—91no—
99C2×D32—91no—
100C32×D4—91no—
101#638—31no—
102#638—31no—
103#638—61no—
104#638—91no—
105#638C3⋊D311yescharacteristic, complemented
106C4⋊(C3⋊D3)—91no—
107C4×C3⋊D3—91no—
108C3×#67—31no—
109C3×#67—33no—
110C3×#67—91no—
111C3×#67—91no—
112C3×#67—93no—
113C3×#67—91no—
114C3×#67—183no—
115C3×#67—181no—
116#626—93no—
117#626—91no—
118#626—93no—
119#626—91no—
120#626—91no—
121#626—91no—
122#625—91no—
123#625—91no—
124#624—93no—
125#624—91no—
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)#6712yescomplemented
136C3×C3⋊D3—21no—
137C3×C3⋊D3—32no—
138C3×C3⋊D3—61no—
139C3×C3⋊D3—61no—
140C3×C3⋊D3—61no—
141C3×C3⋊D3—121no—
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—36no—
151(C3⋊D3)⋊C3—63no—
152C2×#67—271no—
153D3×D4—271no—
154D3×D4—271no—
155D3×D4—273no—
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—121no—
165C22×C32D3211yescharacteristic
166C22×C32D3213yescomplemented
167C22×C32C2×C3⋊D311yescharacteristic, complemented
168C2×C3⋊D3—31no—
169C2×C3⋊D3—31no—
170C2×C3⋊D3—31no—
171C2×C3⋊D3—31no—
172C2×C3⋊D3—33no—
173C2×C3⋊D3—33no—
174C2×C3⋊D3—61no—
175C2×C3⋊D3—62no—
176C2×C3⋊D3—66no—
177C2×C3⋊D3—91no—
178C2×C3⋊D3—91no—
179C2×C3⋊D3—181no—
180C2×C3⋊D3—181no—
181C2×C3⋊D3—182no—
182C2×C3⋊D3—184no—
183C2×C3⋊D3—182no—
184C2×C3⋊D3C2×C3⋊D311yescharacteristic, complemented
185C2×C3×D3—33no—
186C2×C3×D3—31no—
187C2×C3×D3—91no—
188C2×C3×D3—91no—
189C2×C3×D3—91no—
190C2×C3×D3—91no—
191C2×C3×D3—93no—
192C2×C3×D3—91no—
193C2×C3×D3—91no—
194C2×C3×D3—91no—
195C2×C3×D3—181no—
196C2×C3×D3—181no—
197C2×C3×D3—183no—
198C2×C3×D3—181no—
199C2×C3×D3—182no—
200C2×C3×D3—182no—
201C2×C3×D3—184no—
202D32—186no—
203D32—182no—
204D32—182no—
205D32—182no—
206C32×C4—91no—
207C3⋊Dic3—31no—
208C3⋊Dic3—31no—
209C3⋊Dic3—61no—
210C3⋊Dic3C2×C3⋊D311yescharacteristic, complemented
211C3×Dic3—33no—
212C3×Dic3—31no—
213C3×Dic3—91no—
214C3×Dic3—93no—
215C3×Dic3—91no—
216C3×Dic3—91no—
217C3×Dic3—183no—
218C3×Dic3—181no—
219C33—21no—
220C33D3×D411yescharacteristic, complemented
221C33C2×#6711yescharacteristic, complemented
222H(3,𝔽3)—23no—
223H(3,𝔽3)D3×D413yescomplemented
224C23×C3—271no—
225C22×D3—91no—
226C22×D3—91no—
227C22×D3—181no—
228C22×D3—273no—
229C22×D3—271no—
230C22×D3—271no—
231C22×D3—271no—
232C22×D3—273no—
233C3×D4—93no—
234C3×D4—91no—
235C3×D4—271no—
236#67—31no—
237#67—91no—
238#67—93no—
239#67—91no—
240#67—91no—
241#67—181no—
242#67—183no—
243#67—271no—
244#67—271no—
245#67—271no—
246#67—271no—
247#67—273no—
248C2×Dic3—271no—
249D12—271no—
250D12—273no—
251D12—271no—
252C4×D3—271no—
253C4×D3—271no—
254C4×D3—273no—
255C2×C32—21no—
256C2×C32—23no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—23no—
260C2×C32—31no—
261C2×C32—31no—
262C2×C32—43no—
263C2×C32—41no—
264C2×C32—61no—
265C2×C32—61no—
266C2×C32—61no—
267C2×C32—62no—
268C2×C32—61no—
269C2×C32—62no—
270C2×C32—61no—
271C2×C32—121no—
272C2×C32—121no—
273C2×C32—122no—
274C2×C32—181no—
275C2×C32C2×D3211yescharacteristic
276C2×C32C2×D3213yes—
277C2×C32C22×C3⋊D311yescharacteristic
278C3⋊D3—21no—
279C3⋊D3—32no—
280C3⋊D3—36no—
281C3⋊D3—61no—
282C3⋊D3—61no—
283C3⋊D3—61no—
284C3⋊D3—63no—
285C3⋊D3—92no—
286C3⋊D3—121no—
287C3⋊D3—182no—
288C3⋊D3—181no—
289C3⋊D3—182no—
290C3×D3—61no—
291C3×D3—63no—
292C3×D3—92no—
293C3×D3—92no—
294C3×D3—181no—
295C3×D3—182no—
296C3×D3—181no—
297C3×D3—182no—
298C3×D3—181no—
299C3×D3—183no—
300C3×D3—181no—
301C3×D3—181no—
302C3×D3—363no—
303C3×D3—361no—
304C2×D4—811noSylow
305C22×C3—21no—
306C22×C3—31no—
307C22×C3—33no—
308C22×C3—61no—
309C22×C3—63no—
310C22×C3—93no—
311C22×C3—91no—
312C22×C3—271no—
313C22×C3—271no—
314C22×C3—271no—
315C22×C3—542no—
316C22×C3((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
317C22×C3D3×C3⋊D311yescharacteristic
318C2×D3—31no—
319C2×D3—91no—
320C2×D3—91no—
321C2×D3—91no—
322C2×D3—93no—
323C2×D3—91no—
324C2×D3—91no—
325C2×D3—91no—
326C2×D3—91no—
327C2×D3—181no—
328C2×D3—181no—
329C2×D3—181no—
330C2×D3—183no—
331C2×D3—182no—
332C2×D3—182no—
333C2×D3—184no—
334C2×D3—273no—
335C2×D3—273no—
336C2×D3—546no—
337C2×D3—542no—
338C2×D3—542no—
339C2×D3—542no—
340C2×D3—546no—
341C3×C4—93no—
342C3×C4—91no—
343C3×C4—271no—
344Dic3—31no—
345Dic3—91no—
346Dic3—93no—
347Dic3—91no—
348Dic3—91no—
349Dic3—181no—
350Dic3—183no—
351C32—21no—
352C32—23no—
353C32—31no—
354C32—31no—
355C32—61no—
356C32—61no—
357C32—61no—
358C32—121no—
359C32D3×#6711yescharacteristic
360C32D3×#6713yescomplemented
361C32D4×C3⋊D311yescharacteristic, complemented
362C23—271no—
363C23—811no—
364D4—91no—
365D4—271no—
366D4—811no—
367D4—811no—
368C2×C4—811no—
369C2×C3—22no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—31no—
374C2×C3—33no—
375C2×C3—63no—
376C2×C3—61no—
377C2×C3—61no—
378C2×C3—63no—
379C2×C3—121no—
380C2×C3—123no—
381C2×C3—183no—
382C2×C3—181no—
383C2×C3—272no—
384C2×C3—541no—
385C2×C3—541no—
386C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
387C2×C3C2×D3×C3⋊D311yescharacteristic
388D3—61no—
389D3—92no—
390D3—92no—
391D3—181no—
392D3—182no—
393D3—182no—
394D3—181no—
395D3—183no—
396D3—181no—
397D3—181no—
398D3—181no—
399D3—276no—
400D3—361no—
401D3—363no—
402D3—543no—
403C22—91no—
404C22—271no—
405C22—271no—
406C22—271no—
407C22—542no—
408C22—1622no—
409C22((C3⋊D3)⋊C3)⋊D311yescharacteristic, complemented
410C4—91no—
411C4—271no—
412C3—21no—
413C3—31no—
414C3—33no—
415C3—61no—
416C3—63no—
417C3#8867611yescharacteristic, complemented
418C3C3⋊D3×#6711yescharacteristic
419C2—21no—
420C2—181no—
421C2—272no—
422C2—541no—
423C2—541no—
424C2C2×((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)⋊C3)C2semidirect
2. C2×((C3⋊D3)⋊C3)⋊D3C2semidirect
3. #10619994C2semidirect
4. C3×#638⋊C3C2semidirect
5. #10619887C2semidirect
6. #10619884C2semidirect
7. #10619883C2semidirect
11. C22×C3×H(3,𝔽3)C22semidirect
12. C2×C3⋊#272C22semidirect
15. C2×C3⋊((C3⋊D3)⋊C3)C22semidirect
16. C2×C3×(C3⋊D3)⋊C3C22semidirect
18. C3⋊#1084C22semidirect
19. C3×(C3⋊Dic3)⋊C3C22semidirect
20. C22×C3⋊(C3⋊D3)D3semidirect
27. C3×#638D3semidirect
39. #638⋊C3D3semidirect
47. C3⋊((C3⋊D3)⋊C3)D4semidirect
55. C22×C33C2×D3semidirect
56. C22×C33C2×D3semidirect
59. C2×C3⋊(C3⋊D3)C2×D3semidirect
66. C2×C3×C3⋊D3C2×D3semidirect
73. C3×C3⋊Dic3C2×D3semidirect
76. C22×H(3,𝔽3)C2×D3semidirect
82. C2×(C3⋊D3)⋊C3C2×D3semidirect
86. (C3⋊Dic3)⋊C3C2×D3semidirect
87. C3×H(3,𝔽3)C2×D4semidirect
105. #638C3⋊D3semidirect
135. C3⋊(C3⋊D3)#67semidirect
166. C22×C32D32semidirect
167. C22×C32C2×C3⋊D3semidirect
184. C2×C3⋊D3C2×C3⋊D3semidirect
210. C3⋊Dic3C2×C3⋊D3semidirect
220. C33D3×D4semidirect
221. C33C2×#67semidirect
223. H(3,𝔽3)D3×D4semidirect
316. C22×C3((C3⋊D3)⋊C3)⋊C2semidirect
360. C32D3×#67semidirect
361. C32D4×C3⋊D3semidirect
409. C22((C3⋊D3)⋊C3)⋊D3semidirect
417. 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×((C3⋊D3)⋊C3)⋊D32
#106199942
C3×#638⋊C32
#106198872
#106198842
#106198832
#890493
#890393
#886763