Finite Group Databasebeta

← #13048340

Subgroups — #13048340

Order 1296.

526 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C2×((C3⋊D3)⋊C3)⋊D3C211yesmaximal, characteristic, complemented
2C2×C3×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic, complemented
3#10619904C211yesmaximal, characteristic, complemented
4#10619893C211yesmaximal, characteristic, complemented
5#10619886C211yesmaximal, characteristic, complemented
6#6292⋊C3C211yesmaximal, characteristic, complemented
7Dic3×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
8#88953—31nomaximal
9D3×#626—31nomaximal
10#88654—31nomaximal
11C2×C3⋊#272C2211yescharacteristic, complemented
12C2×C3×#272C2211yescharacteristic, complemented
13C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
14C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
16((C3⋊D3)⋊C3)⋊D3—22no—
17C3×((C3⋊D3)⋊C3)⋊C2—22no—
18C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
19Dic3×H(3,𝔽3)C2211yescharacteristic, complemented
20C2×#1113—31no—
21C2×D3×C3⋊D3—31no—
22C2×C3×D32—31no—
23C2×C3×D32—31no—
24#6295—31no—
25#6292D311yescharacteristic, complemented
26#6291—31no—
27#6290—31no—
28#6290—31no—
29Dic3×C3⋊D3D311yescharacteristic, complemented
30C3×#626—31no—
31C3×#626—31no—
32C3×#626—31no—
33C3×#626—61no—
34C3×D3×Dic3—31no—
35C2×((C3⋊D3)⋊C3)⋊C2—31no—
36C2×((C3⋊D3)⋊C3)⋊C2—61no—
37C2×((C3⋊D3)⋊C3)⋊C2D311yescharacteristic, complemented
38C4⋊#272—31no—
39C4⋊((C3⋊D3)⋊C3)—31no—
40C4×(C3⋊D3)⋊C3—31no—
41#6200—31no—
42#6200—31no—
43#6200—61no—
44C2×C3×H(3,𝔽3)C2311yescharacteristic
45C3⋊#272—21no—
46C3×#272—21no—
47C3⋊((C3⋊D3)⋊C3)—21no—
48C3×(C3⋊D3)⋊C3—21no—
49C3×(C3⋊D3)⋊C3D412yescomplemented
50D3×#67—91no—
51D3×#67—91no—
52C2×#626—91no—
53#3802—91no—
54D3×D12—91no—
55C2×C3⋊(C3⋊D3)C2×D311yescharacteristic, complemented
56C2×C3×C3⋊D3—31no—
57C2×C3×C3⋊D3—31no—
58C2×C3×C3⋊D3—31no—
59C2×C3×C3⋊D3—61no—
60C2×C3×C3⋊D3C2×D311yescharacteristic
61C2×C3×C3⋊D3C2×D311yescharacteristic, complemented
62C2×C32×D3—31no—
63C2×C32×D3—31no—
64C2×C32×D3—31no—
65C2×C32×D3—31no—
66C2×C32×D3—61no—
67#1113—62no—
68D3×C3⋊D3—62no—
69C3×D32—62no—
70C3×D32—62no—
71C3⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
72C3×C3⋊Dic3—31no—
73C32×Dic3—21no—
74C32×Dic3C2×D311yescharacteristic, complemented
75C32×Dic3C2×D311yescharacteristic, complemented
76C2×#272—31no—
77C2×#272—61no—
78C2×#272—61no—
79C2×#272—121no—
80C2×#272C2×D311yescharacteristic, complemented
81C2×(C3⋊D3)⋊C3—21no—
82C2×(C3⋊D3)⋊C3—21no—
83C2×(C3⋊D3)⋊C3—31no—
84C2×(C3⋊D3)⋊C3—61no—
85C2×(C3⋊D3)⋊C3C2×D311yescharacteristic
86C2×(C3⋊D3)⋊C3C2×D311yescharacteristic, complemented
87((C3⋊D3)⋊C3)⋊C2—22no—
88((C3⋊D3)⋊C3)⋊C2—64no—
89((C3⋊D3)⋊C3)⋊C2—62no—
90C4×H(3,𝔽3)—31no—
91(C3⋊Dic3)⋊C3—31no—
92(C3⋊Dic3)⋊C3—61no—
93C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
94C22×C3⋊D3—91no—
95C22×C3×D3—91no—
96C22×C3×D3—91no—
97C22×C3×D3—91no—
98C2×D32—31no—
99C2×D32—31no—
100C2×D32—91no—
101C2×D32—91no—
102C2×D32—91no—
103C2×D32—91no—
104C2×D32—181no—
105#638—91no—
106#638—91no—
107C4⋊(C3⋊D3)—31no—
108C4×C3⋊D3—31no—
109C3×#67—91no—
110C3×#67—91no—
111C2×C3×Dic3—91no—
112C3×D12—91no—
113C3×D12—91no—
114C3×D12—91no—
115C3×D12—181no—
116C3×C4×D3—91no—
117#626—31no—
118#626—31no—
119#626—31no—
120#626—61no—
121#626—91no—
122#626—91no—
123#626—91no—
124#626—91no—
125#626—91no—
126#626—181no—
127#625—91no—
128#625—91no—
129#625—91no—
130#625—91no—
131#625—181no—
132#625—181no—
133D3×Dic3—31no—
134D3×Dic3—91no—
135C2×C33—21no—
136C2×C33C22×D311yescharacteristic
137C2×C33C22×D311yescharacteristic
138C3⋊(C3⋊D3)—21no—
139C3×C3⋊D3—21no—
140C3×C3⋊D3—61no—
141C3×C3⋊D3—61no—
142C3×C3⋊D3—61no—
143C3×C3⋊D3—121no—
144C3×C3⋊D3D1212yescomplemented
145C32×D3—32no—
146C32×D3—61no—
147C32×D3—61no—
148C32×D3—61no—
149C32×D3—121no—
150C2×H(3,𝔽3)—21no—
151C2×H(3,𝔽3)—21no—
152C2×H(3,𝔽3)—41no—
153C2×H(3,𝔽3)C22×D311yescharacteristic
154#272—21no—
155#272—62no—
156#272—61no—
157#272—121no—
158#272—122no—
159(C3⋊D3)⋊C3—21no—
160(C3⋊D3)⋊C3—22no—
161(C3⋊D3)⋊C3—41no—
162(C3⋊D3)⋊C3—62no—
163(C3⋊D3)⋊C3—61no—
164(C3⋊D3)⋊C3#6712yescomplemented
165C2×#67—271no—
166D3×D4—271no—
167D3×D4—271no—
168C2×D12—271no—
169C22×C32—91no—
170C22×C32—91no—
171C22×C32—91no—
172C2×C3⋊D3—31no—
173C2×C3⋊D3—31no—
174C2×C3⋊D3—31no—
175C2×C3⋊D3—31no—
176C2×C3⋊D3—61no—
177C2×C3⋊D3—61no—
178C2×C3⋊D3—91no—
179C2×C3⋊D3—181no—
180C2×C3⋊D3—182no—
181C2×C3⋊D3D3211yescharacteristic
182C2×C3⋊D3D3211yescharacteristic, complemented
183C2×C3×D3—31no—
184C2×C3×D3—31no—
185C2×C3×D3—31no—
186C2×C3×D3—31no—
187C2×C3×D3—31no—
188C2×C3×D3—31no—
189C2×C3×D3—31no—
190C2×C3×D3—31no—
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193C2×C3×D3—61no—
194C2×C3×D3—61no—
195C2×C3×D3—61no—
196C2×C3×D3—61no—
197C2×C3×D3—91no—
198C2×C3×D3—91no—
199C2×C3×D3—91no—
200C2×C3×D3—91no—
201C2×C3×D3—91no—
202C2×C3×D3—91no—
203C2×C3×D3—91no—
204C2×C3×D3—121no—
205C2×C3×D3—181no—
206C2×C3×D3—181no—
207C2×C3×D3—181no—
208C2×C3×D3—181no—
209C2×C3×D3—182no—
210C2×C3×D3—182no—
211C2×C3×D3—182no—
212C2×C3×D3—361no—
213D32—62no—
214D32—62no—
215D32—182no—
216D32—182no—
217D32—184no—
218D32—182no—
219D32—182no—
220C32×C4—31no—
221C32×C4—31no—
222C32×C4—61no—
223C3⋊Dic3—31no—
224C3⋊Dic3—31no—
225C3⋊Dic3—61no—
226C3⋊Dic3—91no—
227C3⋊Dic3—181no—
228C3×Dic3—31no—
229C3×Dic3—31no—
230C3×Dic3—61no—
231C3×Dic3—91no—
232C3×Dic3—91no—
233C3×Dic3—181no—
234C3×Dic3D3211yescharacteristic
235C33—21no—
236C33C2×D1211yescharacteristic, complemented
237C33D3×D411yescharacteristic, complemented
238H(3,𝔽3)—21no—
239H(3,𝔽3)—21no—
240H(3,𝔽3)—41no—
241H(3,𝔽3)C2×#6711yescharacteristic, complemented
242C23×C3—271no—
243C22×D3—91no—
244C22×D3—91no—
245C22×D3—91no—
246C22×D3—271no—
247C22×D3—271no—
248C22×D3—271no—
249C22×D3—271no—
250C22×D3—541no—
251C3×D4—271no—
252C3×D4—271no—
253C2×C3×C4—271no—
254#67—91no—
255#67—91no—
256#67—271no—
257#67—271no—
258#67—271no—
259#67—271no—
260#67—271no—
261#67—271no—
262#67—541no—
263#67—541no—
264C2×Dic3—91no—
265D12—91no—
266D12—91no—
267D12—91no—
268D12—181no—
269D12—271no—
270D12—271no—
271D12—271no—
272C4×D3—91no—
273C4×D3—271no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—31no—
278C2×C32—31no—
279C2×C32—41no—
280C2×C32—61no—
281C2×C32—61no—
282C2×C32—61no—
283C2×C32—92no—
284C2×C32—121no—
285C2×C32—181no—
286C2×C32—181no—
287C2×C32C2×D3211yescharacteristic
288C2×C32C2×D3211yescharacteristic
289C2×C32C2×D3211yescharacteristic
290C3⋊D3—21no—
291C3⋊D3—61no—
292C3⋊D3—61no—
293C3⋊D3—61no—
294C3⋊D3—62no—
295C3⋊D3—61no—
296C3⋊D3—121no—
297C3⋊D3—181no—
298C3⋊D3—361no—
299C3⋊D3#62612yescomplemented
300C3×D3—32no—
301C3×D3—32no—
302C3×D3—62no—
303C3×D3—61no—
304C3×D3—61no—
305C3×D3—61no—
306C3×D3—61no—
307C3×D3—61no—
308C3×D3—61no—
309C3×D3—92no—
310C3×D3—121no—
311C3×D3—121no—
312C3×D3—121no—
313C3×D3—121no—
314C3×D3—121no—
315C3×D3—122no—
316C3×D3—181no—
317C3×D3—182no—
318C3×D3—181no—
319C3×D3—181no—
320C3×D3—181no—
321C3×D3—181no—
322C3×D3—181no—
323C3×D3—361no—
324C3×D3—361no—
325C3×D3—362no—
326C3×D3—361no—
327C2×D4—811noSylow
328C22×C3—91no—
329C22×C3—91no—
330C22×C3—91no—
331C22×C3—91no—
332C22×C3—91no—
333C22×C3—91no—
334C22×C3—181no—
335C22×C3—181no—
336C22×C3—181no—
337C22×C3—271no—
338C22×C3—271no—
339C22×C3—542no—
340C2×D3—31no—
341C2×D3—31no—
342C2×D3—31no—
343C2×D3—31no—
344C2×D3—61no—
345C2×D3—91no—
346C2×D3—91no—
347C2×D3—91no—
348C2×D3—91no—
349C2×D3—91no—
350C2×D3—91no—
351C2×D3—91no—
352C2×D3—181no—
353C2×D3—181no—
354C2×D3—181no—
355C2×D3—181no—
356C2×D3—182no—
357C2×D3—182no—
358C2×D3—182no—
359C2×D3—271no—
360C2×D3—361no—
361C2×D3—541no—
362C2×D3—542no—
363C2×D3—542no—
364C2×D3—542no—
365C2×D3—542no—
366C2×D3—544no—
367C3×C4—31no—
368C3×C4—91no—
369C3×C4—91no—
370C3×C4—181no—
371C3×C4—271no—
372Dic3—91no—
373Dic3—91no—
374Dic3—181no—
375Dic3—271no—
376Dic3—541no—
377Dic3((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
378C32—21no—
379C32—21no—
380C32—21no—
381C32—31no—
382C32—31no—
383C32—41no—
384C32—61no—
385C32—61no—
386C32—61no—
387C32—121no—
388C32D3×D1211yescharacteristic
389C32C2×#62611yescharacteristic, complemented
390C32D3×#6711yescharacteristic, complemented
391C23—271no—
392C23—811no—
393D4—271no—
394D4—271no—
395D4—811no—
396D4—811no—
397C2×C4—271no—
398C2×C3—21no—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—61no—
402C2×C3—61no—
403C2×C3—61no—
404C2×C3—92no—
405C2×C3—92no—
406C2×C3—121no—
407C2×C3—182no—
408C2×C3—181no—
409C2×C3—181no—
410C2×C3—181no—
411C2×C3—181no—
412C2×C3—361no—
413C2×C3—361no—
414C2×C3—541no—
415C2×C3—541no—
416C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
417C2×C3D3311yescharacteristic
418D3—32no—
419D3—61no—
420D3—61no—
421D3—61no—
422D3—92no—
423D3—121no—
424D3—181no—
425D3—181no—
426D3—181no—
427D3—182no—
428D3—181no—
429D3—181no—
430D3—181no—
431D3—361no—
432D3—361no—
433D3—361no—
434D3—362no—
435D3—541no—
436D3—542no—
437C22—91no—
438C22—91no—
439C22—91no—
440C22—271no—
441C22—271no—
442C22—542no—
443C22—1622no—
444C4—31no—
445C4—271no—
446C3—21no—
447C3—31no—
448C3—31no—
449C3—61no—
450C3—61no—
451C3—61no—
452C3—121no—
453C3#8865411yescharacteristic, complemented
454C3D3×#62611yescharacteristic
455C2—92no—
456C2—181no—
457C2—181no—
458C2—541no—
459C2—541no—
460C2D3×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2D3×((C3⋊D3)⋊C3)⋊C2
commutator, Fitting, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3D33
socleC2×C32C2×D32
hypocenterC3×H(3,𝔽3)C2×D4
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×((C3⋊D3)⋊C3)⋊D3C2semidirect
2. C2×C3×((C3⋊D3)⋊C3)⋊C2C2semidirect
3. #10619904C2semidirect
4. #10619893C2semidirect
5. #10619886C2semidirect
6. #6292⋊C3C2semidirect
7. Dic3×(C3⋊D3)⋊C3C2semidirect
11. C2×C3⋊#272C22semidirect
12. C2×C3×#272C22semidirect
13. C2×C3⋊((C3⋊D3)⋊C3)C22semidirect
15. C2×C3×(C3⋊D3)⋊C3C22semidirect
18. C3⋊((C3⋊Dic3)⋊C3)C22semidirect
19. Dic3×H(3,𝔽3)C22semidirect
25. #6292D3semidirect
29. Dic3×C3⋊D3D3semidirect
37. C2×((C3⋊D3)⋊C3)⋊C2D3semidirect
49. C3×(C3⋊D3)⋊C3D4semidirect
55. C2×C3⋊(C3⋊D3)C2×D3semidirect
61. C2×C3×C3⋊D3C2×D3semidirect
71. C3⋊(C3⋊Dic3)C2×D3semidirect
74. C32×Dic3C2×D3semidirect
75. C32×Dic3C2×D3semidirect
80. C2×#272C2×D3semidirect
86. C2×(C3⋊D3)⋊C3C2×D3semidirect
93. C3×H(3,𝔽3)C2×D4semidirect
144. C3×C3⋊D3D12semidirect
164. (C3⋊D3)⋊C3#67semidirect
182. C2×C3⋊D3D32semidirect
236. C33C2×D12semidirect
237. C33D3×D4semidirect
241. H(3,𝔽3)C2×#67semidirect
299. C3⋊D3#626semidirect
377. Dic3((C3⋊D3)⋊C3)⋊C2semidirect
389. C32C2×#626semidirect
390. C32D3×#67semidirect
453. C3#88654semidirect

Maximal subgroups

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

subgroupindex
C2×((C3⋊D3)⋊C3)⋊D32
C2×C3×((C3⋊D3)⋊C3)⋊C22
#106199042
#106198932
#106198862
#6292⋊C32
Dic3×(C3⋊D3)⋊C32
#889533
D3×#6263
#886543