Finite Group Databasebeta

← #13048344

Subgroups — #13048344

Order 1296.

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

normal subgroups only

This group has 524 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×((C3⋊D3)⋊C3)⋊D3C211yesmaximal, characteristic, complemented
2C2×D3×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3#10619901C211yesmaximal, characteristic, complemented
4#10619892C211yesmaximal, characteristic, complemented
5#10619886C211yesmaximal, characteristic, complemented
6#6290⋊C3C211yesmaximal, characteristic, complemented
7C3×#6198C211yesmaximal, characteristic, complemented
8#88954—31nomaximal
9#88953—31nomaximal
10#88676—31nomaximal
11C2×C3⋊#272C2211yescharacteristic, complemented
12C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×D3×H(3,𝔽3)C2211yescharacteristic, complemented
14C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
15C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
16((C3⋊D3)⋊C3)⋊D3—22no—
17D3×(C3⋊D3)⋊C3—22no—
18C3×#1084C2211yescharacteristic, complemented
19C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
20C2×#1113—31no—
21C2×D3×C3⋊D3—31no—
22C2×D3×C3⋊D3D311yescharacteristic, complemented
23C2×C3×D32—31no—
24#6295—31no—
25#6290—31no—
26#6290D311yescharacteristic, complemented
27#6289—31no—
28Dic3×C3⋊D3—31no—
29C3×#626—31no—
30C3×#626—31no—
31C3×#626—31no—
32C3×#626—31no—
33C3×#626—61no—
34C3×#625—31no—
35C22×(C3⋊D3)⋊C3—31no—
36C2×((C3⋊D3)⋊C3)⋊C2—31no—
37C2×((C3⋊D3)⋊C3)⋊C2—61no—
38#6235—31no—
39#638⋊C3—31no—
40#6200—31no—
41#6200—61no—
42#6198D311yescharacteristic, complemented
43((C3⋊D3)⋊C3)⋊C4—31no—
44C2×C3×H(3,𝔽3)C2311yescharacteristic
45C3⋊#272—21no—
46C3⋊((C3⋊D3)⋊C3)D412yescomplemented
47D3×H(3,𝔽3)—21no—
48C3×(C3⋊D3)⋊C3—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—31no—
57C2×C3×C3⋊D3—31no—
58C2×C3×C3⋊D3—31no—
59C2×C3×C3⋊D3—61no—
60C2×C3×C3⋊D3C2×D311yescharacteristic, complemented
61C2×C3×C3⋊D3C2×D311yescharacteristic, complemented
62C2×C32×D3—21no—
63C2×C32×D3—31no—
64C2×C32×D3—31no—
65C2×C32×D3C2×D311yescharacteristic, complemented
66C2×C32×D3C2×D311yescharacteristic, complemented
67#1113—62no—
68D3×C3⋊D3—22no—
69D3×C3⋊D3—62no—
70C3×D32—62no—
71C3⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
72C3×C3⋊Dic3—31no—
73C32×Dic3—31no—
74C32×Dic3—31no—
75C32×Dic3—61no—
76C22×H(3,𝔽3)—31no—
77C2×#272—31no—
78C2×#272—61no—
79C2×#272—61no—
80C2×#272—121no—
81C2×(C3⋊D3)⋊C3—21no—
82C2×(C3⋊D3)⋊C3—21no—
83C2×(C3⋊D3)⋊C3—31no—
84C2×(C3⋊D3)⋊C3—62no—
85C2×(C3⋊D3)⋊C3—61no—
86C2×(C3⋊D3)⋊C3C2×D311yescharacteristic, complemented
87C2×(C3⋊D3)⋊C3C2×D311yescharacteristic, complemented
88((C3⋊D3)⋊C3)⋊C2—62no—
89((C3⋊D3)⋊C3)⋊C2—64no—
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—91no—
96C22×C3×D3—91no—
97C22×C3×D3—91no—
98C2×D32—31no—
99C2×D32—91no—
100C2×D32—91no—
101C2×D32—91no—
102C2×D32—91no—
103C2×D32—91no—
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—31no—
120#626—31no—
121#626—61no—
122#626—91no—
123#626—91no—
124#625—31no—
125#625—91no—
126#625—91no—
127#625—91no—
128#625—181no—
129#624—91no—
130#624—91no—
131#624—181no—
132D3×Dic3—91no—
133D3×Dic3—91no—
134D3×Dic3—181no—
135C2×C33—21no—
136C2×C33C22×D311yescharacteristic
137C2×C33C22×D311yescharacteristic
138C3⋊(C3⋊D3)#6712yescomplemented
139C3×C3⋊D3—21no—
140C3×C3⋊D3—21no—
141C3×C3⋊D3—32no—
142C3×C3⋊D3—61no—
143C3×C3⋊D3—61no—
144C3×C3⋊D3—121no—
145C32×D3—21no—
146C32×D3—21no—
147C32×D3—41no—
148C32×D3—61no—
149C32×D3—61no—
150C2×H(3,𝔽3)—21no—
151C2×H(3,𝔽3)—21no—
152C2×H(3,𝔽3)—41no—
153C2×H(3,𝔽3)—61no—
154C2×H(3,𝔽3)C22×D311yescharacteristic
155#272—62no—
156#272—61no—
157#272—121no—
158#272—122no—
159(C3⋊D3)⋊C3—22no—
160(C3⋊D3)⋊C3—21no—
161(C3⋊D3)⋊C3—21no—
162(C3⋊D3)⋊C3—32no—
163(C3⋊D3)⋊C3—41no—
164(C3⋊D3)⋊C3—62no—
165C2×#67—271no—
166D3×D4—271no—
167D3×D4—271no—
168D3×D4—271no—
169C22×C32—31no—
170C22×C32—31no—
171C22×C32—61no—
172C22×C32—91no—
173C22×C32—91no—
174C2×C3⋊D3—31no—
175C2×C3⋊D3—31no—
176C2×C3⋊D3—31no—
177C2×C3⋊D3—31no—
178C2×C3⋊D3—61no—
179C2×C3⋊D3—61no—
180C2×C3⋊D3—62no—
181C2×C3⋊D3—91no—
182C2×C3⋊D3—181no—
183C2×C3⋊D3—182no—
184C2×C3⋊D3D3211yescharacteristic, complemented
185C2×C3⋊D3D3211yescharacteristic, complemented
186C2×C3×D3—31no—
187C2×C3×D3—31no—
188C2×C3×D3—31no—
189C2×C3×D3—31no—
190C2×C3×D3—31no—
191C2×C3×D3—31no—
192C2×C3×D3—61no—
193C2×C3×D3—61no—
194C2×C3×D3—61no—
195C2×C3×D3—91no—
196C2×C3×D3—91no—
197C2×C3×D3—91no—
198C2×C3×D3—91no—
199C2×C3×D3—91no—
200C2×C3×D3—91no—
201C2×C3×D3—91no—
202C2×C3×D3—181no—
203C2×C3×D3—181no—
204C2×C3×D3—181no—
205C2×C3×D3—181no—
206C2×C3×D3—182no—
207C2×C3×D3—182no—
208C2×C3×D3—361no—
209C2×C3×D3D3211yescharacteristic
210D32—62no—
211D32—182no—
212D32—182no—
213D32—182no—
214D32—184no—
215D32—182no—
216D32—182no—
217C32×C4—91no—
218C3⋊Dic3—31no—
219C3⋊Dic3—31no—
220C3⋊Dic3—61no—
221C3⋊Dic3—91no—
222C3⋊Dic3—181no—
223C3×Dic3—31no—
224C3×Dic3—31no—
225C3×Dic3—31no—
226C3×Dic3—31no—
227C3×Dic3—61no—
228C3×Dic3—61no—
229C3×Dic3—61no—
230C3×Dic3—61no—
231C3×Dic3—91no—
232C3×Dic3—91no—
233C3×Dic3—121no—
234C3×Dic3—181no—
235C33—21no—
236C33D3×D411yescharacteristic, complemented
237C33C2×#6711yescharacteristic, complemented
238H(3,𝔽3)—21no—
239H(3,𝔽3)—21no—
240H(3,𝔽3)—41no—
241H(3,𝔽3)D3×D411yescharacteristic, complemented
242C23×C3—271no—
243C22×D3—91no—
244C22×D3—91no—
245C22×D3—271no—
246C22×D3—271no—
247C22×D3—271no—
248C22×D3—271no—
249C22×D3—271no—
250C22×D3—541no—
251C3×D4—271no—
252C3×D4—271no—
253C3×D4—271no—
254#67—91no—
255#67—91no—
256#67—91no—
257#67—91no—
258#67—91no—
259#67—91no—
260#67—181no—
261#67—271no—
262#67—271no—
263#67—271no—
264#67—271no—
265#67—541no—
266C2×Dic3—271no—
267D12—91no—
268D12—91no—
269D12—271no—
270C4×D3—271no—
271C4×D3—271no—
272C4×D3—271no—
273C4×D3—541no—
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—61no—
284C2×C32—61no—
285C2×C32—121no—
286C2×C32—121no—
287C2×C32—181no—
288C2×C32—181no—
289C2×C32C2×D3211yescharacteristic
290C2×C32C2×D3211yescharacteristic
291C2×C32C2×D3211yescharacteristic
292C3⋊D3—21no—
293C3⋊D3—21no—
294C3⋊D3—32no—
295C3⋊D3—32no—
296C3⋊D3—62no—
297C3⋊D3—61no—
298C3⋊D3—61no—
299C3⋊D3—92no—
300C3⋊D3—121no—
301C3⋊D3—182no—
302C3×D3—21no—
303C3×D3—61no—
304C3×D3—61no—
305C3×D3—62no—
306C3×D3—61no—
307C3×D3—61no—
308C3×D3—61no—
309C3×D3—61no—
310C3×D3—92no—
311C3×D3—92no—
312C3×D3—121no—
313C3×D3—121no—
314C3×D3—181no—
315C3×D3—181no—
316C3×D3—182no—
317C3×D3—182no—
318C3×D3—181no—
319C3×D3—181no—
320C3×D3—181no—
321C3×D3—361no—
322C3×D3—361no—
323C3×D3—362no—
324C2×D4—811noSylow
325C22×C3—31no—
326C22×C3—91no—
327C22×C3—91no—
328C22×C3—91no—
329C22×C3—91no—
330C22×C3—91no—
331C22×C3—91no—
332C22×C3—181no—
333C22×C3—181no—
334C22×C3—181no—
335C22×C3—271no—
336C22×C3—271no—
337C22×C3—542no—
338C2×D3—31no—
339C2×D3—31no—
340C2×D3—91no—
341C2×D3—91no—
342C2×D3—91no—
343C2×D3—91no—
344C2×D3—91no—
345C2×D3—91no—
346C2×D3—91no—
347C2×D3—181no—
348C2×D3—181no—
349C2×D3—181no—
350C2×D3—181no—
351C2×D3—182no—
352C2×D3—182no—
353C2×D3—271no—
354C2×D3—361no—
355C2×D3—541no—
356C2×D3—542no—
357C2×D3—542no—
358C2×D3—542no—
359C2×D3—542no—
360C2×D3—542no—
361C2×D3—544no—
362C2×D3((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
363C3×C4—91no—
364C3×C4—91no—
365C3×C4—181no—
366C3×C4—271no—
367Dic3—31no—
368Dic3—31no—
369Dic3—61no—
370Dic3—91no—
371Dic3—91no—
372Dic3—181no—
373Dic3—271no—
374Dic3—541no—
375C32—21no—
376C32—21no—
377C32—21no—
378C32—31no—
379C32—31no—
380C32—41no—
381C32—61no—
382C32—61no—
383C32—61no—
384C32—121no—
385C32D3×#6711yescharacteristic
386C32D3×#6711yescharacteristic, complemented
387C32#381111yescharacteristic, complemented
388C23—271no—
389C23—811no—
390D4—271no—
391D4—271no—
392D4—271no—
393D4—811no—
394C2×C4—811no—
395C2×C3—21no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—61no—
399C2×C3—61no—
400C2×C3—61no—
401C2×C3—61no—
402C2×C3—121no—
403C2×C3—181no—
404C2×C3—181no—
405C2×C3—181no—
406C2×C3—181no—
407C2×C3—182no—
408C2×C3—181no—
409C2×C3—181no—
410C2×C3—272no—
411C2×C3—361no—
412C2×C3—361no—
413C2×C3—541no—
414C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
415C2×C3D3311yescharacteristic
416D3—21no—
417D3—61no—
418D3—61no—
419D3—92no—
420D3—92no—
421D3—181no—
422D3—181no—
423D3—182no—
424D3—181no—
425D3—181no—
426D3—182no—
427D3—181no—
428D3—272no—
429D3—361no—
430D3—361no—
431D3—362no—
432D3—542no—
433C22—31no—
434C22—91no—
435C22—91no—
436C22—271no—
437C22—271no—
438C22—542no—
439C22—1622no—
440C4—91no—
441C4—271no—
442C3—21no—
443C3—31no—
444C3—31no—
445C3—61no—
446C3—61no—
447C3—61no—
448C3—121no—
449C3#8867611yescharacteristic, complemented
450C3#8895211yescharacteristic
451C2—61no—
452C2—181no—
453C2—181no—
454C2—272no—
455C2—541no—
456C2D3×((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×D3×(C3⋊D3)⋊C3C2semidirect
3. #10619901C2semidirect
4. #10619892C2semidirect
5. #10619886C2semidirect
6. #6290⋊C3C2semidirect
7. C3×#6198C2semidirect
11. C2×C3⋊#272C22semidirect
13. C2×D3×H(3,𝔽3)C22semidirect
14. C2×C3×(C3⋊D3)⋊C3C22semidirect
15. C2×C3×(C3⋊D3)⋊C3C22semidirect
18. C3×#1084C22semidirect
19. C3⋊((C3⋊Dic3)⋊C3)C22semidirect
22. C2×D3×C3⋊D3D3semidirect
26. #6290D3semidirect
42. #6198D3semidirect
46. C3⋊((C3⋊D3)⋊C3)D4semidirect
60. C2×C3×C3⋊D3C2×D3semidirect
61. C2×C3×C3⋊D3C2×D3semidirect
65. C2×C32×D3C2×D3semidirect
66. C2×C32×D3C2×D3semidirect
71. C3⋊(C3⋊Dic3)C2×D3semidirect
86. C2×(C3⋊D3)⋊C3C2×D3semidirect
87. C2×(C3⋊D3)⋊C3C2×D3semidirect
90. #1084C2×D3semidirect
93. C3×H(3,𝔽3)C2×D4semidirect
138. C3⋊(C3⋊D3)#67semidirect
184. C2×C3⋊D3D32semidirect
185. C2×C3⋊D3D32semidirect
236. C33D3×D4semidirect
237. C33C2×#67semidirect
241. H(3,𝔽3)D3×D4semidirect
362. C2×D3((C3⋊D3)⋊C3)⋊C2semidirect
386. C32D3×#67semidirect
387. C32#3811semidirect
449. C3#88676semidirect

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×D3×(C3⋊D3)⋊C32
#106199012
#106198922
#106198862
#6290⋊C32
C3×#61982
#889543
#889533
#886763