Finite Group Databasebeta

← C2×#11722545 #13164820

Subgroups — C2×#11722545 #13164820

Order 1728.

3379 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6235C211yesmaximal, characteristic, complemented
2C22×#638⋊C3C212yesmaximal, complemented
3#11722545C2116yesmaximal, direct product factor
4C2×#88677C212yesmaximal, complemented
5C2×#88676C214yesmaximal, complemented
6C2×#88672C212yesmaximal, complemented
7C2×#88671C214yesmaximal, complemented
8C2×#64330—32nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic, complemented
10C23×#272C2211yescharacteristic, complemented
11C23×(C3⋊D3)⋊C3C2212yescomplemented
12C22×((C3⋊D3)⋊C3)⋊C2C2212yescomplemented
13C2×#6235C2214yescomplemented
14C2×#6235C2218yescomplemented
15C22×#1084C2211yescharacteristic, complemented
16C2×#638⋊C3C2214yescomplemented
17C2×#638⋊C3C2214yescomplemented
18C2×#638⋊C3C22116yescomplemented
19C22×(C3⋊Dic3)⋊C3C2212yescomplemented
20#88677C22116yescomplemented
21#88676C22132yescomplemented
22C2×#6200C2214yescomplemented
23C2×#6199C2212yescomplemented
24C2×#6198C2212yescomplemented
25#88672C22116yescomplemented
26#88671C22132yescomplemented
27C2×((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
28C2×#6196C2212yescomplemented
29C22×#638D312yescomplemented
30C22×C3×#67—32no—
31C22×C3×#67—32no—
32C22×C3×#67—61no—
33#64330—332no—
34C2×#3811—34no—
35C2×D3×#67—34no—
36C2×D3×#67—34no—
37C2×#3805—34no—
38C2×#3804—34no—
39C2×#3804—34no—
40C23×H(3,𝔽3)—42no—
41C23×H(3,𝔽3)C2312yes—
42C23×H(3,𝔽3)C2311yescharacteristic
43C23×H(3,𝔽3)C2314yescomplemented
44C22×#272—42no—
45C22×#272C2312yes—
46C22×#272C2314yescomplemented
47C22×(C3⋊D3)⋊C3—44no—
48C22×(C3⋊D3)⋊C3C2314yes—
49C22×(C3⋊D3)⋊C3C2318yescomplemented
50C2×((C3⋊D3)⋊C3)⋊C2—44no—
51C2×((C3⋊D3)⋊C3)⋊C2C2318yescomplemented
52#6235C23116yescomplemented
53C2×#1084C2312yes—
54C2×#1084C2314yescomplemented
55#638⋊C3C23116yescomplemented
56#638⋊C3C23116yescomplemented
57C2×(C3⋊Dic3)⋊C3C2314yes—
58C2×(C3⋊Dic3)⋊C3C2318yescomplemented
59#6200C23116yescomplemented
60#6199C2318yescomplemented
61#6198C2318yescomplemented
62((C3⋊D3)⋊C3)⋊C4C23116yescomplemented
63#6196C2318yescomplemented
64C2×#1004—91no—
65C2×#1004—92no—
66C24×C32—21no—
67C24×C32C2×D312yescomplemented
68C23×C3⋊D3C2×D312yescomplemented
69C23×C3×D3—32no—
70C23×C3×D3—32no—
71C23×C3×D3—61no—
72C22×D32—34no—
73C2×#638C2×D314yescomplemented
74C2×#638C2×D314yescomplemented
75C2×#638C2×D3116yescomplemented
76C22×C3⋊Dic3C2×D312yescomplemented
77C2×C3×#67—34no—
78C2×C3×#67—34no—
79C2×C3×#67—34no—
80C2×C3×#67—34no—
81C2×C3×#67—316no—
82C2×C3×#67—316no—
83C2×C3×#67—64no—
84C2×C3×#67—68no—
85C22×C3×Dic3—32no—
86C22×C3×Dic3—32no—
87C22×C3×Dic3—61no—
88#3811—332no—
89D3×#67—332no—
90D3×#67—332no—
91C2×#627—34no—
92C2×#626—34no—
93C2×#626—34no—
94C2×#625—34no—
95C2×#624—34no—
96#3805—332no—
97#3804—332no—
98#3804—332no—
99C2×D3×Dic3—34no—
100C2×D3×Dic3—34no—
101C22×H(3,𝔽3)—22no—
102C22×H(3,𝔽3)—24no—
103C22×H(3,𝔽3)—44no—
104C22×H(3,𝔽3)C2414yes—
105C22×H(3,𝔽3)C2412yes—
106C22×H(3,𝔽3)C2411yescharacteristic
107C2×#272—24no—
108C2×#272—44no—
109C2×#272C2414yes—
110C2×(C3⋊D3)⋊C3—28no—
111C2×(C3⋊D3)⋊C3—48no—
112C2×(C3⋊D3)⋊C3C2418yes—
113((C3⋊D3)⋊C3)⋊C2—48no—
114#1084C2414yes—
115(C3⋊Dic3)⋊C3C2418yes—
116C22×C3×D4—92no—
117C22×C3×D4—91no—
118C22×#67—32no—
119C22×#67—32no—
120C22×#67—61no—
121C22×#67—92no—
122C22×#67—92no—
123C22×#67—94no—
124#1004—916no—
125#1004—932no—
126C2×#237—94no—
127C2×#237—94no—
128C2×#237—94no—
129C2×D3×D4—94no—
130C2×D3×D4—94no—
131C2×D3×D4—94no—
132C2×#235—92no—
133C2×#235—94no—
134C23×C32—21no—
135C23×C32—22no—
136C23×C32—24no—
137C23×C32—44no—
138C23×C32—44no—
139C23×C32C22×D312yes—
140C23×C32C22×D312yes—
141C23×C32C22×D312yes—
142C23×C32C22×D318yescomplemented
143C22×C3⋊D3—44no—
144C22×C3⋊D3C22×D314yes—
145C22×C3⋊D3C22×D318yescomplemented
146C22×C3×D3—34no—
147C22×C3×D3—34no—
148C22×C3×D3—38no—
149C22×C3×D3—38no—
150C22×C3×D3—62no—
151C22×C3×D3—64no—
152C22×C3×D3—124no—
153C22×C3×D3—124no—
154C22×C3×D3—124no—
155C2×D32—316no—
156C2×D32—128no—
157#638C22×D3116yescomplemented
158#638C22×D3116yescomplemented
159C2×C3⋊Dic3C22×D314yes—
160C2×C3⋊Dic3C22×D318yescomplemented
161C3×#67—316no—
162C3×#67—316no—
163C3×#67—316no—
164C3×#67—316no—
165C3×#67—616no—
166C2×C3×Dic3—34no—
167C2×C3×Dic3—38no—
168C2×C3×Dic3—34no—
169C2×C3×Dic3—38no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—64no—
172#627—316no—
173#626—316no—
174#626—316no—
175#625—316no—
176#624—316no—
177D3×Dic3—316no—
178D3×Dic3—316no—
179C2×H(5,𝔽2)—271noSylow
180C2×H(3,𝔽3)—22no—
181C2×H(3,𝔽3)—24no—
182C2×H(3,𝔽3)H(5,𝔽2)12yescomplemented
183C2×H(3,𝔽3)C2511yescharacteristic
184#272—24no—
185(C3⋊D3)⋊C3—28no—
186C24×C3—32no—
187C24×C3—61no—
188C24×C3—92no—
189C24×C3—91no—
190C24×C3D3211yescharacteristic
191C23×D3—32no—
192C23×D3—32no—
193C23×D3—61no—
194C23×D3—92no—
195C23×D3—92no—
196C23×D3—94no—
197C2×C3×D4—94no—
198C2×C3×D4—94no—
199C2×C3×D4—916no—
200C2×C3×D4—94no—
201C2×C3×D4—98no—
202C22×C3×C4—92no—
203C22×C3×C4—91no—
204C2×#67—34no—
205C2×#67—34no—
206C2×#67—316no—
207C2×#67—34no—
208C2×#67—34no—
209C2×#67—316no—
210C2×#67—64no—
211C2×#67—68no—
212C2×#67—94no—
213C2×#67—94no—
214C2×#67—916no—
215C2×#67—94no—
216C2×#67—94no—
217C2×#67—916no—
218C2×#67—932no—
219C2×#67—94no—
220C22×Dic3—32no—
221C22×Dic3—32no—
222C22×Dic3—61no—
223C22×Dic3—92no—
224C22×Dic3—92no—
225C22×Dic3—94no—
226#237—932no—
227#237—932no—
228#237—932no—
229D3×D4—932no—
230D3×D4—932no—
231D3×D4—932no—
232#235—916no—
233#235—932no—
234C2×D12—92no—
235C2×D12—94no—
236C2×C4×D3—94no—
237C2×C4×D3—94no—
238C2×C4×D3—94no—
239C2×Dic6—92no—
240C2×Dic6—94no—
241C22×C32—24no—
242C22×C32—24no—
243C22×C32—24no—
244C22×C32—24no—
245C22×C32—22no—
246C22×C32—21no—
247C22×C32—24no—
248C22×C32—44no—
249C22×C32—48no—
250C22×C32—48no—
251C22×C32C23×D314yes—
252C22×C32C23×D314yes—
253C22×C32C23×D314yes—
254C22×C32C23×D312yes—
255C2×C3⋊D3—28no—
256C2×C3⋊D3—48no—
257C2×C3⋊D3C23×D318yes—
258C2×C3×D3—38no—
259C2×C3×D3—38no—
260C2×C3×D3—64no—
261C2×C3×D3—68no—
262C2×C3×D3—68no—
263C2×C3×D3—124no—
264C2×C3×D3—128no—
265C2×C3×D3—128no—
266C2×C3×D3—128no—
267D32—1216no—
268C3⋊Dic3C23×D318yes—
269C3×Dic3—38no—
270C3×Dic3—38no—
271C3×Dic3—64no—
272H(5,𝔽2)—2716no—
273C2×#41—272no—
274C2×#41—274no—
275C22×D4—91no—
276C22×D4—92no—
277C22×D4—274no—
278C22×D4—272no—
279H(3,𝔽3)C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
280C23×C3—32no—
281C23×C3—32no—
282C23×C3—32no—
283C23×C3—38no—
284C23×C3—42no—
285C23×C3—61no—
286C23×C3—62no—
287C23×C3—64no—
288C23×C3—94no—
289C23×C3—98no—
290C23×C3—92no—
291C23×C3—94no—
292C23×C3—124no—
293C23×C3—124no—
294C23×C3—362no—
295C23×C3—364no—
296C23×C3C2×D3212yes—
297C23×C3C2×D3211yescharacteristic
298C23×C3C2×D3214yes—
299C22×D3—34no—
300C22×D3—34no—
301C22×D3—38no—
302C22×D3—38no—
303C22×D3—62no—
304C22×D3—64no—
305C22×D3—94no—
306C22×D3—98no—
307C22×D3—98no—
308C22×D3—916no—
309C22×D3—124no—
310C22×D3—124no—
311C22×D3—124no—
312C22×D3—364no—
313C22×D3—364no—
314C22×D3—368no—
315C3×D4—916no—
316C3×D4—916no—
317C3×D4—916no—
318C2×C3×C4—94no—
319C2×C3×C4—98no—
320C2×C3×C4—92no—
321C2×C3×C4—94no—
322#67—316no—
323#67—316no—
324#67—316no—
325#67—316no—
326#67—616no—
327#67—916no—
328#67—916no—
329#67—916no—
330#67—916no—
331#67—916no—
332C2×Dic3—34no—
333C2×Dic3—38no—
334C2×Dic3—34no—
335C2×Dic3—38no—
336C2×Dic3—64no—
337C2×Dic3—62no—
338C2×Dic3—94no—
339C2×Dic3—98no—
340C2×Dic3—98no—
341C2×Dic3—916no—
342D12—98no—
343D12—916no—
344C4×D3—916no—
345C4×D3—916no—
346C4×D3—916no—
347Dic6—98no—
348Dic6—916no—
349C2×C32—24no—
350C2×C32—24no—
351C2×C32—24no—
352C2×C32—24no—
353C2×C32—22no—
354C2×C32—21no—
355C2×C32—44no—
356C2×C32#100414yescomplemented
357C2×C32C24×D312yes—
358C3⋊D3—28no—
359C3×D3—68no—
360C3×D3—68no—
361C3×D3—124no—
362C24—91no—
363C24—92no—
364C24—272no—
365C24((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
366#41—2716no—
367#41—2732no—
368C2×Q8—272no—
369C2×D4—94no—
370C2×D4—94no—
371C2×D4—94no—
372C2×D4—98no—
373C2×D4—916no—
374C2×D4—272no—
375C2×D4—274no—
376C2×D4—2732no—
377C2×D4—2716no—
378C22×C4—91no—
379C22×C4—92no—
380C22×C4—274no—
381C22×C4—272no—
382C22×C3—24no—
383C22×C3—22no—
384C22×C3—34no—
385C22×C3—34no—
386C22×C3—34no—
387C22×C3—32no—
388C22×C3—44no—
389C22×C3—64no—
390C22×C3—64no—
391C22×C3—64no—
392C22×C3—64no—
393C22×C3—62no—
394C22×C3—64no—
395C22×C3—61no—
396C22×C3—98no—
397C22×C3—94no—
398C22×C3—124no—
399C22×C3—128no—
400C22×C3—128no—
401C22×C3—188no—
402C22×C3—184no—
403C22×C3—364no—
404C22×C3—368no—
405C22×C3C22×D3214yes—
406C22×C3C22×D3212yes—
407C22×C3C22×D3211yescharacteristic
408C2×D3—38no—
409C2×D3—38no—
410C2×D3—64no—
411C2×D3—68no—
412C2×D3—68no—
413C2×D3—98no—
414C2×D3—124no—
415C2×D3—128no—
416C2×D3—128no—
417C2×D3—128no—
418C2×D3—188no—
419C2×D3—368no—
420C2×D3—368no—
421C2×D3—3616no—
422C3×C4—98no—
423C3×C4—94no—
424Dic3—38no—
425Dic3—38no—
426Dic3—64no—
427Dic3—98no—
428C32—21no—
429C32C2×#100412yescomplemented
430C23—42no—
431C23—92no—
432C23—94no—
433C23—94no—
434C23—98no—
435C23—278no—
436C23—362no—
437C23—364no—
438C23—1084no—
439C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
440C23C2×((C3⋊D3)⋊C3)⋊C212yes—
441C23C2×((C3⋊D3)⋊C3)⋊C214yescomplemented
442Q8—278no—
443D4—916no—
444D4—916no—
445D4—916no—
446D4—278no—
447D4—2716no—
448C2×C4—94no—
449C2×C4—98no—
450C2×C4—92no—
451C2×C4—94no—
452C2×C4—2716no—
453C2×C4—278no—
454C2×C3—24no—
455C2×C3—22no—
456C2×C3—32no—
457C2×C3—34no—
458C2×C3—64no—
459C2×C3—64no—
460C2×C3—64no—
461C2×C3—64no—
462C2×C3—61no—
463C2×C3—62no—
464C2×C3—124no—
465C2×C3—188no—
466C2×C3—184no—
467C2×C3#6433012yes—
468C2×C3C23×D3211yescharacteristic
469D3—68no—
470D3—68no—
471D3—124no—
472D3—188no—
473C22—22no—
474C22—24no—
475C22—44no—
476C22—94no—
477C22—98no—
478C22—184no—
479C22—188no—
480C22—364no—
481C22—368no—
482C22—1088no—
483C22C22×((C3⋊D3)⋊C3)⋊C212yes—
484C22C22×((C3⋊D3)⋊C3)⋊C214yes—
485C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
486C4—94no—
487C4—98no—
488C3—32no—
489C3—61no—
490C3C2×#6433011yescharacteristic
491C2—22no—
492C2—24no—
493C2—184no—
494C2—188no—
495C2#1172254512yesdirect product factor
496C2C23×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C25
FrattiniC2×C3C23×D32
FittingC24×H(3,𝔽3)C22
socleC22×C3C22×D32
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×H(5,𝔽2)
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×#6235C2semidirect
2. C22×#638⋊C3C2semidirect
3. #11722545C2direct (N × H)
4. C2×#88677C2semidirect
5. C2×#88676C2semidirect
6. C2×#88672C2semidirect
7. C2×#88671C2semidirect
9. C24×H(3,𝔽3)C22semidirect
10. C23×#272C22semidirect
11. C23×(C3⋊D3)⋊C3C22semidirect
12. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
13. C2×#6235C22semidirect
14. C2×#6235C22semidirect
15. C22×#1084C22semidirect
16. C2×#638⋊C3C22semidirect
17. C2×#638⋊C3C22semidirect
18. C2×#638⋊C3C22semidirect
19. C22×(C3⋊Dic3)⋊C3C22semidirect
20. #88677C22semidirect
21. #88676C22semidirect
22. C2×#6200C22semidirect
23. C2×#6199C22semidirect
24. C2×#6198C22semidirect
25. #88672C22semidirect
26. #88671C22semidirect
27. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
28. C2×#6196C22semidirect
29. C22×#638D3semidirect
43. C23×H(3,𝔽3)C23semidirect
46. C22×#272C23semidirect
49. C22×(C3⋊D3)⋊C3C23semidirect
51. C2×((C3⋊D3)⋊C3)⋊C2C23semidirect
52. #6235C23semidirect
54. C2×#1084C23semidirect
55. #638⋊C3C23semidirect
56. #638⋊C3C23semidirect
58. C2×(C3⋊Dic3)⋊C3C23semidirect
59. #6200C23semidirect
60. #6199C23semidirect
61. #6198C23semidirect
62. ((C3⋊D3)⋊C3)⋊C4C23semidirect
63. #6196C23semidirect
67. C24×C32C2×D3semidirect
68. C23×C3⋊D3C2×D3semidirect
73. C2×#638C2×D3semidirect
74. C2×#638C2×D3semidirect
75. C2×#638C2×D3semidirect
76. C22×C3⋊Dic3C2×D3semidirect
142. C23×C32C22×D3semidirect
145. C22×C3⋊D3C22×D3semidirect
157. #638C22×D3semidirect
158. #638C22×D3semidirect
160. C2×C3⋊Dic3C22×D3semidirect
182. C2×H(3,𝔽3)H(5,𝔽2)semidirect
279. H(3,𝔽3)C2×H(5,𝔽2)semidirect
356. C2×C32#1004semidirect
365. C24((C3⋊D3)⋊C3)⋊C2semidirect
429. C32C2×#1004semidirect
441. C23C2×((C3⋊D3)⋊C3)⋊C2semidirect
495. C2#11722545direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62352
C22×#638⋊C32
#117225452
C2×#886772
C2×#886762
C2×#886722
C2×#886712
C2×#643303