Finite Group Databasebeta

← C2×#11722506 #13164749

Subgroups — C2×#11722506 #13164749

Order 1728.

3307 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88744C211yesmaximal, characteristic, complemented
2C2×D4×(C3⋊D3)⋊C3C212yesmaximal, complemented
3C2×#88676C214yesmaximal, complemented
4C22×#6198C212yesmaximal, complemented
5C2×#88672C212yesmaximal, complemented
6#11722506C2116yesmaximal, direct product factor
7C2×#88655C211yesmaximal, characteristic, complemented
8C2×#88650C211yesmaximal, characteristic, complemented
9C2×#88648C212yesmaximal, complemented
10C2×#64314—32nomaximal
11C23×(C3⋊D3)⋊C3C2214yescomplemented
12C22×((C3⋊D3)⋊C3)⋊C2C2212yescomplemented
13C2×D4×H(3,𝔽3)C2211yescharacteristic, complemented
14C2×#6235C2212yescomplemented
15C22×#1084C2212yescomplemented
16#88744C2218yescomplemented
17C2×C4×#272C2211yescharacteristic, complemented
18C2×#6229C2211yescharacteristic, complemented
19C2×#638⋊C3C2214yescomplemented
20D4×(C3⋊D3)⋊C3C22116yescomplemented
21C2×C4⋊((C3⋊D3)⋊C3)C2212yescomplemented
22C2×C4×(C3⋊D3)⋊C3C2212yescomplemented
23#88676C22132yescomplemented
24C2×#6200C2212yescomplemented
25C2×#6198C2212yescomplemented
26C2×#6198C2211yescharacteristic, complemented
27C2×#6198C2214yescomplemented
28C2×#6198C22116yescomplemented
29#88672C22116yescomplemented
30C2×((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
31C2×#6196C2211yescharacteristic, complemented
32#88655C2218yescomplemented
33#88650C2218yescomplemented
34#88648C22116yescomplemented
35C2×D4×C3⋊D3D312yescomplemented
36C2×C3×#237—32no—
37C2×C3×#237—61no—
38C2×C3×D3×D4—32no—
39C2×#3811—34no—
40C2×D3×#67—34no—
41C22×#626—34no—
42C2×#3804—34no—
43#64314—332no—
44C2×D3×D12—32no—
45C2×#3799—32no—
46C2×#3798—32no—
47C2×#3796—32no—
48C23×H(3,𝔽3)C2312yes—
49C22×#272C2311yescharacteristic
50C22×(C3⋊D3)⋊C3—48no—
51C22×(C3⋊D3)⋊C3C2314yes—
52C22×(C3⋊D3)⋊C3C2312yes—
53C22×(C3⋊D3)⋊C3C23116yescomplemented
54C2×((C3⋊D3)⋊C3)⋊C2—44no—
55C2×((C3⋊D3)⋊C3)⋊C2C2318yescomplemented
56D4×H(3,𝔽3)C2314yescomplemented
57C2×C4×H(3,𝔽3)C2311yescharacteristic
58#6235C2318yescomplemented
59C2×#1084C2311yescharacteristic
60C2×#1084C2312yes—
61C2×#1084C2318yescomplemented
62C4×#272C2314yescomplemented
63#6229C2314yescomplemented
64#638⋊C3C23116yescomplemented
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C4⋊((C3⋊D3)⋊C3)C2318yescomplemented
67C4×(C3⋊D3)⋊C3C2318yescomplemented
68#6200C2318yescomplemented
69#6198C23116yescomplemented
70#6198C2314yescomplemented
71#6198C2318yescomplemented
72((C3⋊D3)⋊C3)⋊C4C23116yescomplemented
73#6196C2314yescomplemented
74C2×#1009—91no—
75C2×#1004—92no—
76C23×C3⋊D3C2×D314yescomplemented
77C23×C3×D3—34no—
78C22×D32—34no—
79C2×C32×D4—21no—
80C2×C32×D4C2×D312yescomplemented
81C2×#638C2×D314yescomplemented
82D4×C3⋊D3C2×D3116yescomplemented
83C2×C4⋊(C3⋊D3)C2×D312yescomplemented
84C2×C4×C3⋊D3C2×D312yescomplemented
85C2×C3×#67—34no—
86C2×C3×#67—34no—
87C2×C3×#67—62no—
88C22×C3×Dic3—34no—
89C22×C3×Dic3—62no—
90C3×#237—316no—
91C3×#237—68no—
92C3×D3×D4—316no—
93C2×C3×D12—32no—
94C2×C3×C4×D3—32no—
95C2×C3×C4×D3—32no—
96C2×C3×C4×D3—61no—
97C2×C3×Dic6—32no—
98C2×C3×Dic6—61no—
99#3811—332no—
100D3×#67—332no—
101C2×#627—32no—
102C2×#626—34no—
103C2×#626—32no—
104C2×#626—32no—
105C2×#626—34no—
106C2×#626—34no—
107C2×#626—332no—
108C2×#625—32no—
109C2×#624—34no—
110#3804—332no—
111C2×D3×Dic3—34no—
112D3×D12—316no—
113#3799—316no—
114#3798—316no—
115#3796—316no—
116C22×H(3,𝔽3)—24no—
117C22×H(3,𝔽3)C2414yes—
118C22×H(3,𝔽3)C2411yescharacteristic
119C2×#272—22no—
120C2×#272C2412yes—
121C2×(C3⋊D3)⋊C3—28no—
122C2×(C3⋊D3)⋊C3—24no—
123C2×(C3⋊D3)⋊C3—416no—
124C2×(C3⋊D3)⋊C3C2418yes—
125C2×(C3⋊D3)⋊C3C2414yes—
126((C3⋊D3)⋊C3)⋊C2—48no—
127C4×H(3,𝔽3)C2412yes—
128#1084C2412yes—
129#1084C2414yes—
130(C3⋊Dic3)⋊C3C2414yes—
131C2×C3×#41—91no—
132C22×C3×D4—92no—
133C22×#67—94no—
134C22×#67—94no—
135#1009—916no—
136C2×#239—92no—
137#1004—932no—
138C2×#237—32no—
139C2×#237—61no—
140C2×#237—92no—
141C2×#237—94no—
142C2×D3×D4—32no—
143C2×D3×D4—92no—
144C2×D3×D4—92no—
145C2×D3×D4—94no—
146C2×D3×D4—94no—
147C2×#235—92no—
148C2×#235—91no—
149C2×#235—92no—
150C2×#235—92no—
151C22×D12—91no—
152C22×D12—92no—
153C23×C32—22no—
154C23×C32C22×D314yes—
155C22×C3⋊D3—48no—
156C22×C3⋊D3C22×D314yes—
157C22×C3⋊D3C22×D312yes—
158C22×C3⋊D3C22×D3116yescomplemented
159C22×C3×D3—32no—
160C22×C3×D3—34no—
161C22×C3×D3—32no—
162C22×C3×D3—316no—
163C22×C3×D3—61no—
164C22×C3×D3—128no—
165C2×D32—316no—
166C2×D32—128no—
167C32×D4—24no—
168C32×D4C22×D318yescomplemented
169C2×C32×C4—21no—
170C2×C32×C4C22×D312yes—
171#638C22×D3116yescomplemented
172C2×C3⋊Dic3C22×D312yes—
173C4⋊(C3⋊D3)C22×D318yescomplemented
174C4×C3⋊D3C22×D318yescomplemented
175C3×#67—316no—
176C3×#67—316no—
177C3×#67—68no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—34no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—316no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—68no—
185C3×D12—38no—
186C3×C4×D3—38no—
187C3×C4×D3—38no—
188C3×C4×D3—64no—
189C3×Dic6—38no—
190C3×Dic6—64no—
191#627—38no—
192#626—316no—
193#626—38no—
194#626—316no—
195#626—316no—
196#626—38no—
197#625—38no—
198#624—316no—
199D3×Dic3—316no—
200C2×H(5,𝔽2)—271noSylow
201C2×H(3,𝔽3)—24no—
202C2×H(3,𝔽3)H(5,𝔽2)12yescomplemented
203C2×H(3,𝔽3)C2511yescharacteristic
204#272—22no—
205(C3⋊D3)⋊C3—28no—
206(C3⋊D3)⋊C3—24no—
207C24×C3—94no—
208C23×D3—34no—
209C23×D3—94no—
210C23×D3—92no—
211C23×D3—94no—
212C3×#41—98no—
213C2×C3×Q8—91no—
214C2×C3×D4—32no—
215C2×C3×D4—61no—
216C2×C3×D4—92no—
217C2×C3×D4—916no—
218C2×C3×D4—94no—
219C2×C3×D4—92no—
220C2×C3×D4D3211yescharacteristic
221C22×C3×C4—92no—
222C22×C3×C4—92no—
223C22×C3×C4—91no—
224C2×#67—34no—
225C2×#67—34no—
226C2×#67—62no—
227C2×#67—94no—
228C2×#67—92no—
229C2×#67—94no—
230C2×#67—92no—
231C2×#67—94no—
232C2×#67—94no—
233C2×#67—932no—
234C2×#67—932no—
235C2×#67—92no—
236C22×Dic3—34no—
237C22×Dic3—62no—
238C22×Dic3—94no—
239#239—916no—
240#237—316no—
241#237—68no—
242#237—916no—
243#237—932no—
244D3×D4—316no—
245D3×D4—916no—
246D3×D4—916no—
247D3×D4—932no—
248D3×D4—932no—
249#235—916no—
250#235—98no—
251#235—916no—
252#235—916no—
253C2×D12—32no—
254C2×D12—92no—
255C2×D12—98no—
256C2×D12—91no—
257C2×D12—92no—
258C2×D12—94no—
259C2×D12—916no—
260C2×D12—92no—
261C2×C4×D3—32no—
262C2×C4×D3—32no—
263C2×C4×D3—61no—
264C2×C4×D3—92no—
265C2×C4×D3—94no—
266C2×C4×D3—94no—
267C2×Dic6—32no—
268C2×Dic6—61no—
269C2×Dic6—91no—
270C2×Dic6—92no—
271C22×C32—28no—
272C22×C32—21no—
273C22×C32—24no—
274C22×C32—44no—
275C22×C32C23×D318yes—
276C22×C32C23×D312yes—
277C2×C3⋊D3—28no—
278C2×C3⋊D3—24no—
279C2×C3⋊D3—416no—
280C2×C3⋊D3C23×D318yes—
281C2×C3⋊D3C23×D314yes—
282C2×C3×D3—38no—
283C2×C3×D3—34no—
284C2×C3×D3—34no—
285C2×C3×D3—62no—
286C2×C3×D3—68no—
287C2×C3×D3—64no—
288C2×C3×D3—64no—
289C2×C3×D3—122no—
290C2×C3×D3—1216no—
291D32—1216no—
292C32×C4—22no—
293C32×C4C23×D314yes—
294C3⋊Dic3C23×D314yes—
295C3×Dic3—34no—
296C3×Dic3—38no—
297C3×Dic3—34no—
298C3×Dic3—64no—
299C3×Dic3—62no—
300H(5,𝔽2)—2716no—
301C2×#41—91no—
302C2×#41—271no—
303C2×#41—272no—
304C2×#41—272no—
305C22×D4—92no—
306C22×D4—271no—
307C22×D4—272no—
308C22×D4—274no—
309H(3,𝔽3)C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
310C23×C3—34no—
311C23×C3—62no—
312C23×C3—94no—
313C23×C3—92no—
314C23×C3—916no—
315C23×C3—91no—
316C23×C3—368no—
317C23×C3C2×D3212yes—
318C22×D3—32no—
319C22×D3—32no—
320C22×D3—34no—
321C22×D3—316no—
322C22×D3—61no—
323C22×D3—92no—
324C22×D3—94no—
325C22×D3—916no—
326C22×D3—98no—
327C22×D3—916no—
328C22×D3—128no—
329C22×D3—368no—
330C22×D3—364no—
331C22×D3—368no—
332C3×Q8—94no—
333C3×D4—38no—
334C3×D4—64no—
335C3×D4—98no—
336C3×D4—916no—
337C3×D4—98no—
338C3×D4C2×D3214yes—
339C2×C3×C4—32no—
340C2×C3×C4—61no—
341C2×C3×C4—98no—
342C2×C3×C4—92no—
343C2×C3×C4—91no—
344C2×C3×C4—92no—
345C2×C3×C4—98no—
346C2×C3×C4—94no—
347C2×C3×C4C2×D3211yescharacteristic
348#67—316no—
349#67—316no—
350#67—68no—
351#67—916no—
352#67—98no—
353#67—916no—
354#67—98no—
355#67—916no—
356#67—916no—
357#67—98no—
358C2×Dic3—34no—
359C2×Dic3—32no—
360C2×Dic3—316no—
361C2×Dic3—32no—
362C2×Dic3—62no—
363C2×Dic3—61no—
364C2×Dic3—68no—
365C2×Dic3—92no—
366C2×Dic3—916no—
367D12—38no—
368D12—98no—
369D12—94no—
370D12—98no—
371D12—916no—
372D12—98no—
373C4×D3—38no—
374C4×D3—38no—
375C4×D3—64no—
376C4×D3—98no—
377C4×D3—916no—
378C4×D3—916no—
379Dic6—38no—
380Dic6—64no—
381Dic6—94no—
382Dic6—98no—
383C2×C32—28no—
384C2×C32—22no—
385C2×C32—21no—
386C2×C32—44no—
387C2×C32#100414yescomplemented
388C2×C32C24×D312yes—
389C3⋊D3—28no—
390C3⋊D3—24no—
391C3×D3—68no—
392C3×D3—64no—
393C3×D3—64no—
394C3×D3—122no—
395C24—94no—
396C24—272no—
397#41—98no—
398#41—278no—
399#41—2716no—
400#41—2716no—
401C2×Q8—91no—
402C2×Q8—271no—
403C2×D4—94no—
404C2×D4—92no—
405C2×D4—916no—
406C2×D4—92no—
407C2×D4—278no—
408C2×D4—271no—
409C2×D4—272no—
410C2×D4—272no—
411C2×D4—274no—
412C2×D4—2716no—
413C2×D4—2732no—
414C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
415C22×C4—92no—
416C22×C4—92no—
417C22×C4—91no—
418C22×C4—274no—
419C22×C3—24no—
420C22×C3—38no—
421C22×C3—32no—
422C22×C3—68no—
423C22×C3—64no—
424C22×C3—61no—
425C22×C3—98no—
426C22×C3—94no—
427C22×C3—92no—
428C22×C3—124no—
429C22×C3—188no—
430C22×C3—184no—
431C22×C3—182no—
432C22×C3—3616no—
433C22×C3C22×D3214yes—
434C22×C3C22×D3211yescharacteristic
435C2×D3—34no—
436C2×D3—34no—
437C2×D3—38no—
438C2×D3—62no—
439C2×D3—64no—
440C2×D3—64no—
441C2×D3—68no—
442C2×D3—94no—
443C2×D3—98no—
444C2×D3—122no—
445C2×D3—1216no—
446C2×D3—184no—
447C2×D3—188no—
448C2×D3—3616no—
449C2×D3—368no—
450C2×D3—3616no—
451C3×C4—34no—
452C3×C4—62no—
453C3×C4—94no—
454C3×C4—92no—
455C3×C4—94no—
456C3×C4C22×D3212yes—
457Dic3—38no—
458Dic3—34no—
459Dic3—34no—
460Dic3—64no—
461Dic3—62no—
462Dic3—94no—
463C32—21no—
464C32C2×#100412yescomplemented
465C23—91no—
466C23—92no—
467C23—94no—
468C23—916no—
469C23—278no—
470C23—368no—
471C23—1084no—
472C23C2×((C3⋊D3)⋊C3)⋊C212yes—
473Q8—94no—
474Q8—274no—
475D4—916no—
476D4—98no—
477D4—98no—
478D4—274no—
479D4—278no—
480D4—278no—
481D4—2716no—
482D4C2×((C3⋊D3)⋊C3)⋊C214yescomplemented
483C2×C4—98no—
484C2×C4—91no—
485C2×C4—92no—
486C2×C4—92no—
487C2×C4—98no—
488C2×C4—94no—
489C2×C4—2716no—
490C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
491C2×C3—24no—
492C2×C3—34no—
493C2×C3—32no—
494C2×C3—68no—
495C2×C3—62no—
496C2×C3—61no—
497C2×C3—124no—
498C2×C3—188no—
499C2×C3—184no—
500C2×C3—182no—
501C2×C3#6431312yes—
502C2×C3C23×D3211yescharacteristic
503D3—64no—
504D3—64no—
505D3—68no—
506D3—122no—
507D3—184no—
508D3—188no—
509C22—24no—
510C22—92no—
511C22—94no—
512C22—98no—
513C22—184no—
514C22—188no—
515C22—182no—
516C22—3616no—
517C22—1088no—
518C22C22×((C3⋊D3)⋊C3)⋊C214yes—
519C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
520C4—92no—
521C4—94no—
522C4—94no—
523C4C22×((C3⋊D3)⋊C3)⋊C212yes—
524C3—32no—
525C3—61no—
526C3C2×#6431311yescharacteristic
527C2—24no—
528C2—184no—
529C2—188no—
530C2—182no—
531C2#1172250612yesdirect product factor
532C2C23×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C25
FrattiniC2×C3C23×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3C22×D32
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×H(5,𝔽2)
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88744C2semidirect
2. C2×D4×(C3⋊D3)⋊C3C2semidirect
3. C2×#88676C2semidirect
4. C22×#6198C2semidirect
5. C2×#88672C2semidirect
6. #11722506C2direct (N × H)
7. C2×#88655C2semidirect
8. C2×#88650C2semidirect
9. C2×#88648C2semidirect
11. C23×(C3⋊D3)⋊C3C22semidirect
12. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
13. C2×D4×H(3,𝔽3)C22semidirect
14. C2×#6235C22semidirect
15. C22×#1084C22semidirect
16. #88744C22semidirect
17. C2×C4×#272C22semidirect
18. C2×#6229C22semidirect
19. C2×#638⋊C3C22semidirect
20. D4×(C3⋊D3)⋊C3C22semidirect
21. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
22. C2×C4×(C3⋊D3)⋊C3C22semidirect
23. #88676C22semidirect
24. C2×#6200C22semidirect
25. C2×#6198C22semidirect
26. C2×#6198C22semidirect
27. C2×#6198C22semidirect
28. C2×#6198C22semidirect
29. #88672C22semidirect
30. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
31. C2×#6196C22semidirect
32. #88655C22semidirect
33. #88650C22semidirect
34. #88648C22semidirect
35. C2×D4×C3⋊D3D3semidirect
53. C22×(C3⋊D3)⋊C3C23semidirect
55. C2×((C3⋊D3)⋊C3)⋊C2C23semidirect
56. D4×H(3,𝔽3)C23semidirect
58. #6235C23semidirect
61. C2×#1084C23semidirect
62. C4×#272C23semidirect
63. #6229C23semidirect
64. #638⋊C3C23semidirect
66. C4⋊((C3⋊D3)⋊C3)C23semidirect
67. C4×(C3⋊D3)⋊C3C23semidirect
68. #6200C23semidirect
69. #6198C23semidirect
70. #6198C23semidirect
71. #6198C23semidirect
72. ((C3⋊D3)⋊C3)⋊C4C23semidirect
73. #6196C23semidirect
76. C23×C3⋊D3C2×D3semidirect
80. C2×C32×D4C2×D3semidirect
81. C2×#638C2×D3semidirect
82. D4×C3⋊D3C2×D3semidirect
83. C2×C4⋊(C3⋊D3)C2×D3semidirect
84. C2×C4×C3⋊D3C2×D3semidirect
158. C22×C3⋊D3C22×D3semidirect
168. C32×D4C22×D3semidirect
171. #638C22×D3semidirect
173. C4⋊(C3⋊D3)C22×D3semidirect
174. C4×C3⋊D3C22×D3semidirect
202. C2×H(3,𝔽3)H(5,𝔽2)semidirect
309. H(3,𝔽3)C2×H(5,𝔽2)semidirect
387. C2×C32#1004semidirect
414. C2×D4((C3⋊D3)⋊C3)⋊C2semidirect
464. C32C2×#1004semidirect
482. D4C2×((C3⋊D3)⋊C3)⋊C2semidirect
531. C2#11722506direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887442
C2×D4×(C3⋊D3)⋊C32
C2×#886762
C22×#61982
C2×#886722
#117225062
C2×#886552
C2×#886502
C2×#886482
C2×#643143