Finite Group Databasebeta

← #13048342

Subgroups — #13048342

Order 1296.

520 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 55 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×#66432C211yesmaximal, characteristic, complemented
2C2×D3×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3#10619891C211yesmaximal, characteristic, complemented
4#10619884C211yesmaximal, characteristic, complemented
5#10619883C211yesmaximal, characteristic, complemented
6C3×#6200C211yesmaximal, characteristic, complemented
7#6291⋊C3C211yesmaximal, characteristic, complemented
8#88954—31nomaximal
9#88952—31nomaximal
10#88676—31nomaximal
11C2×C3×#272C2211yescharacteristic, complemented
12C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
14C2×D3×H(3,𝔽3)C2211yescharacteristic, complemented
15C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
16#66432—22no—
17D3×(C3⋊D3)⋊C3—22no—
18C3⋊#1084C2211yescharacteristic, complemented
19C3×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
20C2×D3×C3⋊D3—31no—
21C2×D3×C3⋊D3—31no—
22C2×D3×C3⋊D3D311yescharacteristic, complemented
23C2×C3×D32—31no—
24#6295—31no—
25#6294—31no—
26#6291—31no—
27#6291D311yescharacteristic, complemented
28#6289—31no—
29C3×#626—31no—
30C3×#626—31no—
31C3×#625—31no—
32C3×#625—31no—
33C3×#625—31no—
34C3×#625—61no—
35C22×(C3⋊D3)⋊C3—31no—
36C2×((C3⋊D3)⋊C3)⋊C2—31no—
37#6235—31no—
38#638⋊C3—31no—
39#6200D311yescharacteristic, complemented
40#6198—31no—
41#6198—61no—
42((C3⋊D3)⋊C3)⋊C4—31no—
43((C3⋊D3)⋊C3)⋊C4—61no—
44C2×C3×H(3,𝔽3)C2311yescharacteristic
45C3×#272—21no—
46C3⋊((C3⋊D3)⋊C3)—21no—
47C3⋊((C3⋊D3)⋊C3)D412yescomplemented
48D3×H(3,𝔽3)—21no—
49C3×(C3⋊D3)⋊C3—21no—
50#3811—91no—
51#3811—91no—
52D3×#67—91no—
53D3×#67—91no—
54#3802—91no—
55C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
56C2×C3⋊(C3⋊D3)C2×D311yescharacteristic, complemented
57C2×C3×C3⋊D3—31no—
58C2×C3×C3⋊D3—31no—
59C2×C3×C3⋊D3C2×D311yescharacteristic, complemented
60C2×C32×D3—21no—
61C2×C32×D3—31no—
62C2×C32×D3—31no—
63C2×C32×D3—31no—
64C2×C32×D3—61no—
65C2×C32×D3C2×D311yescharacteristic, complemented
66C2×C32×D3C2×D311yescharacteristic, complemented
67D3×C3⋊D3—22no—
68D3×C3⋊D3—62no—
69D3×C3⋊D3—62no—
70C3×D32—62no—
71C3×C3⋊Dic3—31no—
72C3×C3⋊Dic3—31no—
73C3×C3⋊Dic3—61no—
74C3×C3⋊Dic3C2×D311yescharacteristic, complemented
75C32×Dic3—31no—
76C22×H(3,𝔽3)—31no—
77C2×#272C2×D311yescharacteristic, complemented
78C2×(C3⋊D3)⋊C3—21no—
79C2×(C3⋊D3)⋊C3—31no—
80C2×(C3⋊D3)⋊C3—31no—
81C2×(C3⋊D3)⋊C3—62no—
82C2×(C3⋊D3)⋊C3—61no—
83C2×(C3⋊D3)⋊C3—61no—
84C2×(C3⋊D3)⋊C3C2×D311yescharacteristic, complemented
85((C3⋊D3)⋊C3)⋊C2—62no—
86#1084—31no—
87#1084—61no—
88#1084—61no—
89#1084—121no—
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3C2×D311yescharacteristic, complemented
92C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
93C22×C3⋊D3—31no—
94C22×C3⋊D3—91no—
95C22×C3×D3—91no—
96C22×C3×D3—91no—
97C2×D32—31no—
98C2×D32—91no—
99C2×D32—91no—
100C2×D32—91no—
101C2×D32—91no—
102C2×D32—91no—
103C2×D32—181no—
104C2×D32—181no—
105#638—31no—
106#638—91no—
107C4×C3⋊D3—91no—
108C3×#67—91no—
109C3×#67—91no—
110C3×#67—91no—
111C3×#67—91no—
112C3×#67—91no—
113C3×#67—91no—
114C3×#67—181no—
115C3×D12—91no—
116C3×D12—91no—
117#626—31no—
118#626—31no—
119#626—91no—
120#626—91no—
121#626—91no—
122#626—91no—
123#626—181no—
124#625—31no—
125#625—31no—
126#625—31no—
127#625—61no—
128#625—91no—
129#624—91no—
130#624—91no—
131#624—181no—
132D3×Dic3—91no—
133D3×Dic3—91no—
134C2×C33—21no—
135C2×C33C22×D311yescharacteristic
136C2×C33C22×D311yescharacteristic
137C3⋊(C3⋊D3)—21no—
138C3⋊(C3⋊D3)#6712yescomplemented
139C3×C3⋊D3—21no—
140C3×C3⋊D3—32no—
141C3×C3⋊D3—61no—
142C32×D3—21no—
143C32×D3—21no—
144C32×D3—41no—
145C32×D3—61no—
146C32×D3—61no—
147C32×D3—61no—
148C32×D3—121no—
149C2×H(3,𝔽3)—21no—
150C2×H(3,𝔽3)—21no—
151C2×H(3,𝔽3)—41no—
152C2×H(3,𝔽3)—61no—
153C2×H(3,𝔽3)C22×D311yescharacteristic
154#272—21no—
155(C3⋊D3)⋊C3—21no—
156(C3⋊D3)⋊C3—32no—
157(C3⋊D3)⋊C3—41no—
158(C3⋊D3)⋊C3—62no—
159(C3⋊D3)⋊C3—61no—
160(C3⋊D3)⋊C3—121no—
161C2×#67—271no—
162D3×D4—271no—
163D3×D4—271no—
164D3×D4—271no—
165C22×C32—31no—
166C22×C32—31no—
167C22×C32—61no—
168C22×C32—91no—
169C22×C32—91no—
170C2×C3⋊D3—31no—
171C2×C3⋊D3—31no—
172C2×C3⋊D3—31no—
173C2×C3⋊D3—31no—
174C2×C3⋊D3—61no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—62no—
177C2×C3⋊D3—91no—
178C2×C3⋊D3—91no—
179C2×C3⋊D3—181no—
180C2×C3⋊D3—181no—
181C2×C3⋊D3—182no—
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—121no—
203C2×C3×D3—181no—
204C2×C3×D3—181no—
205C2×C3×D3—182no—
206C2×C3×D3—182no—
207C2×C3×D3D3211yescharacteristic
208D32—62no—
209D32—182no—
210D32—182no—
211D32—182no—
212D32—182no—
213D32—182no—
214D32—184no—
215D32—184no—
216C32×C4—91no—
217C3⋊Dic3—31no—
218C3⋊Dic3—31no—
219C3⋊Dic3—61no—
220C3⋊Dic3D3211yescharacteristic, complemented
221C3×Dic3—31no—
222C3×Dic3—31no—
223C3×Dic3—61no—
224C3×Dic3—91no—
225C3×Dic3—91no—
226C3×Dic3—91no—
227C3×Dic3—91no—
228C3×Dic3—181no—
229C3×Dic3—181no—
230C3×Dic3—181no—
231C3×Dic3—361no—
232C33—21no—
233C33D3×D411yescharacteristic, complemented
234C33C2×#6711yescharacteristic, complemented
235H(3,𝔽3)—21no—
236H(3,𝔽3)—21no—
237H(3,𝔽3)—41no—
238H(3,𝔽3)D3×D411yescharacteristic, complemented
239C23×C3—271no—
240C22×D3—91no—
241C22×D3—91no—
242C22×D3—271no—
243C22×D3—271no—
244C22×D3—271no—
245C22×D3—271no—
246C22×D3—271no—
247C22×D3—541no—
248C3×D4—271no—
249C3×D4—271no—
250C3×D4—271no—
251#67—91no—
252#67—91no—
253#67—91no—
254#67—91no—
255#67—91no—
256#67—91no—
257#67—181no—
258#67—271no—
259#67—271no—
260#67—271no—
261#67—271no—
262#67—541no—
263C2×Dic3—271no—
264D12—91no—
265D12—91no—
266D12—271no—
267C4×D3—271no—
268C4×D3—271no—
269C4×D3—271no—
270C4×D3—541no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32—31no—
275C2×C32—31no—
276C2×C32—41no—
277C2×C32—61no—
278C2×C32—61no—
279C2×C32—61no—
280C2×C32—61no—
281C2×C32—61no—
282C2×C32—121no—
283C2×C32—121no—
284C2×C32—181no—
285C2×C32—181no—
286C2×C32C2×D3211yescharacteristic
287C2×C32C2×D3211yescharacteristic
288C2×C32C2×D3211yescharacteristic
289C3⋊D3—21no—
290C3⋊D3—32no—
291C3⋊D3—32no—
292C3⋊D3—62no—
293C3⋊D3—61no—
294C3⋊D3—61no—
295C3⋊D3—92no—
296C3⋊D3—121no—
297C3⋊D3—182no—
298C3⋊D3—181no—
299C3⋊D3—182no—
300C3×D3—21no—
301C3×D3—61no—
302C3×D3—61no—
303C3×D3—61no—
304C3×D3—61no—
305C3×D3—62no—
306C3×D3—62no—
307C3×D3—61no—
308C3×D3—61no—
309C3×D3—61no—
310C3×D3—61no—
311C3×D3—92no—
312C3×D3—92no—
313C3×D3—121no—
314C3×D3—121no—
315C3×D3—121no—
316C3×D3—121no—
317C3×D3—122no—
318C3×D3—181no—
319C3×D3—182no—
320C3×D3—181no—
321C3×D3—182no—
322C3×D3—181no—
323C2×D4—811noSylow
324C22×C3—31no—
325C22×C3—91no—
326C22×C3—91no—
327C22×C3—91no—
328C22×C3—91no—
329C22×C3—91no—
330C22×C3—91no—
331C22×C3—181no—
332C22×C3—181no—
333C22×C3—181no—
334C22×C3—271no—
335C22×C3—271no—
336C22×C3—542no—
337C2×D3—31no—
338C2×D3—31no—
339C2×D3—31no—
340C2×D3—61no—
341C2×D3—91no—
342C2×D3—91no—
343C2×D3—91no—
344C2×D3—91no—
345C2×D3—91no—
346C2×D3—181no—
347C2×D3—181no—
348C2×D3—182no—
349C2×D3—182no—
350C2×D3—271no—
351C2×D3—271no—
352C2×D3—541no—
353C2×D3—541no—
354C2×D3—542no—
355C2×D3—542no—
356C2×D3—542no—
357C2×D3—542no—
358C2×D3—542no—
359C2×D3—544no—
360C2×D3((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
361C3×C4—91no—
362C3×C4—91no—
363C3×C4—181no—
364C3×C4—271no—
365Dic3—31no—
366Dic3—91no—
367Dic3—91no—
368Dic3—91no—
369Dic3—91no—
370Dic3—181no—
371Dic3—181no—
372Dic3—181no—
373Dic3—361no—
374C32—21no—
375C32—21no—
376C32—21no—
377C32—31no—
378C32—31no—
379C32—41no—
380C32—61no—
381C32—61no—
382C32—61no—
383C32—121no—
384C32#380211yescharacteristic, complemented
385C32D3×#6711yescharacteristic
386C32D3×#6711yescharacteristic, complemented
387C23—271no—
388C23—811no—
389D4—271no—
390D4—271no—
391D4—271no—
392D4—811no—
393C2×C4—811no—
394C2×C3—21no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—61no—
398C2×C3—61no—
399C2×C3—61no—
400C2×C3—61no—
401C2×C3—121no—
402C2×C3—181no—
403C2×C3—181no—
404C2×C3—182no—
405C2×C3—181no—
406C2×C3—181no—
407C2×C3—181no—
408C2×C3—181no—
409C2×C3—272no—
410C2×C3—361no—
411C2×C3—361no—
412C2×C3—541no—
413C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
414C2×C3D3311yescharacteristic
415D3—21no—
416D3—61no—
417D3—61no—
418D3—61no—
419D3—92no—
420D3—92no—
421D3—121no—
422D3—181no—
423D3—182no—
424D3—182no—
425D3—181no—
426D3—181no—
427D3—272no—
428D3—542no—
429D3—541no—
430D3—1081no—
431C22—31no—
432C22—91no—
433C22—91no—
434C22—271no—
435C22—271no—
436C22—542no—
437C22—1622no—
438C4—91no—
439C4—271no—
440C3—21no—
441C3—31no—
442C3—31no—
443C3—61no—
444C3—61no—
445C3—61no—
446C3—121no—
447C3#8867611yescharacteristic, complemented
448C3#8895311yescharacteristic
449C2—61no—
450C2—181no—
451C2—181no—
452C2—272no—
453C2—541no—
454C2D3×((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×#66432C2semidirect
2. C2×D3×(C3⋊D3)⋊C3C2semidirect
3. #10619891C2semidirect
4. #10619884C2semidirect
5. #10619883C2semidirect
6. C3×#6200C2semidirect
7. #6291⋊C3C2semidirect
11. C2×C3×#272C22semidirect
13. C2×C3⋊((C3⋊D3)⋊C3)C22semidirect
14. C2×D3×H(3,𝔽3)C22semidirect
15. C2×C3×(C3⋊D3)⋊C3C22semidirect
18. C3⋊#1084C22semidirect
19. C3×(C3⋊Dic3)⋊C3C22semidirect
22. C2×D3×C3⋊D3D3semidirect
27. #6291D3semidirect
39. #6200D3semidirect
47. C3⋊((C3⋊D3)⋊C3)D4semidirect
56. C2×C3⋊(C3⋊D3)C2×D3semidirect
59. C2×C3×C3⋊D3C2×D3semidirect
65. C2×C32×D3C2×D3semidirect
66. C2×C32×D3C2×D3semidirect
74. C3×C3⋊Dic3C2×D3semidirect
77. C2×#272C2×D3semidirect
84. C2×(C3⋊D3)⋊C3C2×D3semidirect
91. (C3⋊Dic3)⋊C3C2×D3semidirect
92. C3×H(3,𝔽3)C2×D4semidirect
138. C3⋊(C3⋊D3)#67semidirect
182. C2×C3⋊D3D32semidirect
220. C3⋊Dic3D32semidirect
233. C33D3×D4semidirect
234. C33C2×#67semidirect
238. H(3,𝔽3)D3×D4semidirect
360. C2×D3((C3⋊D3)⋊C3)⋊C2semidirect
384. C32#3802semidirect
386. C32D3×#67semidirect
447. C3#88676semidirect

Maximal subgroups

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

subgroupindex
C2×#664322
C2×D3×(C3⋊D3)⋊C32
#106198912
#106198842
#106198832
C3×#62002
#6291⋊C32
#889543
#889523
#886763