Finite Group Databasebeta

← #13048341

Subgroups — #13048341

Order 1296.

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

normal subgroups only

This group has 524 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×D3×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3#10619902C211yesmaximal, characteristic, complemented
4#10619892C211yesmaximal, characteristic, complemented
5#10619887C211yesmaximal, characteristic, complemented
6#6291⋊C3C211yesmaximal, characteristic, complemented
7C3×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
8#88955—31nomaximal
9D3×#626—31nomaximal
10#88676—31nomaximal
11C2×C3⋊#272C2211yescharacteristic, complemented
12C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
13C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
14C2×D3×H(3,𝔽3)C2211yescharacteristic, complemented
15C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic
16((C3⋊D3)⋊C3)⋊D3—22no—
17D3×(C3⋊D3)⋊C3—22no—
18C3×#1084C2211yescharacteristic, complemented
19C3×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
20C2×#1113—31no—
21C2×D3×C3⋊D3—31no—
22C2×D3×C3⋊D3D311yescharacteristic, complemented
23C2×C3×D32—31no—
24#6295—31no—
25#6292—31no—
26#6292—31no—
27#6292—31no—
28#6291D311yescharacteristic, complemented
29C3×#626—31no—
30C3×#626—31no—
31C3×#626—31no—
32C3×#626—61no—
33C3×#624—31no—
34C3×D3×Dic3—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#6198—31no—
43((C3⋊D3)⋊C3)⋊C4D311yescharacteristic, complemented
44C2×C3×H(3,𝔽3)C2311yescharacteristic
45C3⋊#272—21no—
46C3⋊((C3⋊D3)⋊C3)—21no—
47C3⋊((C3⋊D3)⋊C3)—21no—
48D3×H(3,𝔽3)—21no—
49C3×(C3⋊D3)⋊C3D412yescomplemented
50#3811—91no—
51D3×#67—91no—
52C2×#626—91no—
53D3×D12—91no—
54D3×D12—91no—
55C2×C3⋊(C3⋊D3)C2×D311yescharacteristic, complemented
56C2×C3⋊(C3⋊D3)C2×D311yescharacteristic, complemented
57C2×C3×C3⋊D3—31no—
58C2×C3×C3⋊D3—31no—
59C2×C3×C3⋊D3—31no—
60C2×C3×C3⋊D3—31no—
61C2×C3×C3⋊D3—61no—
62C2×C3×C3⋊D3C2×D311yescharacteristic
63C2×C32×D3—21no—
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⋊Dic3C2×D311yescharacteristic, complemented
72C32×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—31no—
83C2×(C3⋊D3)⋊C3—31no—
84C2×(C3⋊D3)⋊C3—61no—
85C2×(C3⋊D3)⋊C3—62no—
86C2×(C3⋊D3)⋊C3—61no—
87C2×(C3⋊D3)⋊C3C2×D311yescharacteristic
88((C3⋊D3)⋊C3)⋊C2—64no—
89((C3⋊D3)⋊C3)⋊C2—62no—
90#1084C2×D311yescharacteristic, complemented
91(C3⋊Dic3)⋊C3—21no—
92(C3⋊Dic3)⋊C3C2×D311yescharacteristic, complemented
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—
106C4⋊(C3⋊D3)—91no—
107C4⋊(C3⋊D3)—91no—
108C3×#67—91no—
109C3×#67—91no—
110C3×#67—91no—
111C3×#67—181no—
112C2×C3×Dic3—91no—
113C3×D12—91no—
114C3×D12—91no—
115C3×C4×D3—91no—
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—91no—
127#626—91no—
128#626—91no—
129#626—181no—
130#626—181no—
131#626—181no—
132#625—91no—
133#624—31no—
134D3×Dic3—31no—
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—61no—
144C3×C3⋊D3—121no—
145C3×C3⋊D3#6712yescomplemented
146C32×D3—21no—
147C32×D3—21no—
148C32×D3—32no—
149C32×D3—41no—
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—62no—
161(C3⋊D3)⋊C3—61no—
162(C3⋊D3)⋊C3—61no—
163(C3⋊D3)⋊C3—121no—
164(C3⋊D3)⋊C3D1212yescomplemented
165C2×#67—271no—
166D3×D4—271no—
167D3×D4—271no—
168C2×D12—271no—
169C22×C32—31no—
170C22×C32—31no—
171C22×C32—61no—
172C22×C32—91no—
173C2×C3⋊D3—31no—
174C2×C3⋊D3—31no—
175C2×C3⋊D3—31no—
176C2×C3⋊D3—31no—
177C2×C3⋊D3—31no—
178C2×C3⋊D3—31no—
179C2×C3⋊D3—61no—
180C2×C3⋊D3—61no—
181C2×C3⋊D3—61no—
182C2×C3⋊D3—62no—
183C2×C3⋊D3—91no—
184C2×C3⋊D3—91no—
185C2×C3⋊D3—181no—
186C2×C3⋊D3—181no—
187C2×C3⋊D3—182no—
188C2×C3⋊D3D3211yescharacteristic
189C2×C3×D3—31no—
190C2×C3×D3—31no—
191C2×C3×D3—31no—
192C2×C3×D3—31no—
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—91no—
203C2×C3×D3—181no—
204C2×C3×D3—181no—
205C2×C3×D3—181no—
206C2×C3×D3—181no—
207C2×C3×D3—181no—
208C2×C3×D3—182no—
209C2×C3×D3—182no—
210C2×C3×D3—361no—
211C2×C3×D3D3211yescharacteristic
212D32—62no—
213D32—182no—
214D32—182no—
215D32—182no—
216D32—184no—
217D32—182no—
218D32—182no—
219C32×C4—91no—
220C32×C4—91no—
221C3⋊Dic3D3211yescharacteristic, complemented
222C3×Dic3—31no—
223C3×Dic3—31no—
224C3×Dic3—31no—
225C3×Dic3—31no—
226C3×Dic3—31no—
227C3×Dic3—31no—
228C3×Dic3—61no—
229C3×Dic3—61no—
230C3×Dic3—61no—
231C3×Dic3—61no—
232C3×Dic3—61no—
233C3×Dic3—91no—
234C3×Dic3—121no—
235C33—21no—
236C33D3×D411yescharacteristic, complemented
237C33C2×#6711yescharacteristic, complemented
238H(3,𝔽3)—21no—
239H(3,𝔽3)—21no—
240H(3,𝔽3)—41no—
241H(3,𝔽3)C2×D1211yescharacteristic, 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—
253C2×C3×C4—271no—
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—271no—
263C2×Dic3—91no—
264D12—91no—
265D12—91no—
266D12—271no—
267D12—271no—
268D12—271no—
269D12—271no—
270D12—541no—
271D12—541no—
272C4×D3—91no—
273C4×D3—91no—
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—92no—
286C2×C32—121no—
287C2×C32—121no—
288C2×C32C2×D3211yescharacteristic
289C2×C32C2×D3211yescharacteristic
290C2×C32C2×D3211yescharacteristic
291C3⋊D3—61no—
292C3⋊D3—61no—
293C3⋊D3—61no—
294C3⋊D3—61no—
295C3⋊D3—62no—
296C3⋊D3—61no—
297C3⋊D3—61no—
298C3⋊D3—121no—
299C3⋊D3—121no—
300C3⋊D3—181no—
301C3⋊D3—181no—
302C3⋊D3—361no—
303C3⋊D3—361no—
304C3⋊D3#62612yescomplemented
305C3×D3—21no—
306C3×D3—32no—
307C3×D3—32no—
308C3×D3—61no—
309C3×D3—61no—
310C3×D3—62no—
311C3×D3—92no—
312C3×D3—121no—
313C3×D3—182no—
314C3×D3—181no—
315C3×D3—181no—
316C3×D3—181no—
317C3×D3—181no—
318C3×D3—181no—
319C3×D3—181no—
320C3×D3—181no—
321C3×D3—361no—
322C3×D3—361no—
323C3×D3—362no—
324C3×D3—361no—
325C3×D3—361no—
326C2×D4—811noSylow
327C22×C3—31no—
328C22×C3—91no—
329C22×C3—91no—
330C22×C3—91no—
331C22×C3—91no—
332C22×C3—181no—
333C22×C3—181no—
334C22×C3—271no—
335C22×C3—271no—
336C22×C3—271no—
337C22×C3—542no—
338C2×D3—31no—
339C2×D3—91no—
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—181no—
352C2×D3—182no—
353C2×D3—182no—
354C2×D3—271no—
355C2×D3—271no—
356C2×D3—361no—
357C2×D3—541no—
358C2×D3—541no—
359C2×D3—542no—
360C2×D3—542no—
361C2×D3—542no—
362C2×D3—542no—
363C2×D3—542no—
364C2×D3—544no—
365C2×D3((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
366C3×C4—91no—
367C3×C4—91no—
368C3×C4—91no—
369C3×C4—91no—
370C3×C4—181no—
371C3×C4—181no—
372Dic3—31no—
373Dic3—31no—
374Dic3—31no—
375Dic3—61no—
376Dic3—91no—
377C32—21no—
378C32—21no—
379C32—21no—
380C32—31no—
381C32—31no—
382C32—41no—
383C32—61no—
384C32—61no—
385C32—61no—
386C32—121no—
387C32D3×D1211yescharacteristic, complemented
388C32C2×#62611yescharacteristic, complemented
389C32D3×#6711yescharacteristic
390C23—271no—
391C23—811no—
392D4—271no—
393D4—271no—
394D4—811no—
395D4—811no—
396C2×C4—271no—
397C2×C3—21no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—61no—
401C2×C3—61no—
402C2×C3—61no—
403C2×C3—61no—
404C2×C3—92no—
405C2×C3—92no—
406C2×C3—121no—
407C2×C3—181no—
408C2×C3—181no—
409C2×C3—182no—
410C2×C3—361no—
411C2×C3—541no—
412C2×C3—541no—
413C2×C3—541no—
414C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
415C2×C3D3311yescharacteristic
416D3—21no—
417D3—32no—
418D3—92no—
419D3—181no—
420D3—181no—
421D3—182no—
422D3—181no—
423D3—181no—
424D3—181no—
425D3—181no—
426D3—181no—
427D3—361no—
428D3—361no—
429D3—361no—
430D3—361no—
431D3—362no—
432D3—541no—
433D3—541no—
434D3—542no—
435D3—1081no—
436C22—31no—
437C22—91no—
438C22—271no—
439C22—271no—
440C22—271no—
441C22—542no—
442C22—1622no—
443C4—91no—
444C4—91no—
445C3—21no—
446C3—31no—
447C3—31no—
448C3—61no—
449C3—61no—
450C3—61no—
451C3—121no—
452C3#8867611yescharacteristic, complemented
453C3D3×#62611yescharacteristic
454C2—61no—
455C2—92no—
456C2—541no—
457C2—541no—
458C2—541no—
459C2D3×((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. #10619902C2semidirect
4. #10619892C2semidirect
5. #10619887C2semidirect
6. #6291⋊C3C2semidirect
7. C3×((C3⋊D3)⋊C3)⋊C4C2semidirect
11. C2×C3⋊#272C22semidirect
12. C2×C3⋊((C3⋊D3)⋊C3)C22semidirect
13. C2×C3⋊((C3⋊D3)⋊C3)C22semidirect
14. C2×D3×H(3,𝔽3)C22semidirect
18. C3×#1084C22semidirect
19. C3×(C3⋊Dic3)⋊C3C22semidirect
22. C2×D3×C3⋊D3D3semidirect
28. #6291D3semidirect
43. ((C3⋊D3)⋊C3)⋊C4D3semidirect
49. C3×(C3⋊D3)⋊C3D4semidirect
55. C2×C3⋊(C3⋊D3)C2×D3semidirect
56. C2×C3⋊(C3⋊D3)C2×D3semidirect
65. C2×C32×D3C2×D3semidirect
66. C2×C32×D3C2×D3semidirect
71. C3×C3⋊Dic3C2×D3semidirect
90. #1084C2×D3semidirect
92. (C3⋊Dic3)⋊C3C2×D3semidirect
93. C3×H(3,𝔽3)C2×D4semidirect
145. C3×C3⋊D3#67semidirect
164. (C3⋊D3)⋊C3D12semidirect
221. C3⋊Dic3D32semidirect
236. C33D3×D4semidirect
237. C33C2×#67semidirect
241. H(3,𝔽3)C2×D12semidirect
304. C3⋊D3#626semidirect
365. C2×D3((C3⋊D3)⋊C3)⋊C2semidirect
387. C32D3×D12semidirect
388. C32C2×#626semidirect
452. 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
#106199022
#106198922
#106198872
#6291⋊C32
C3×((C3⋊D3)⋊C3)⋊C42
#889553
D3×#6263
#886763