Finite Group Databasebeta

← #13164744

Subgroups — #13164744

Order 1728.

3379 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×#272C211yesmaximal, characteristic, complemented
2#64361⋊C3C212yesmaximal, complemented
3#11722545C214yesmaximal, complemented
4C2×#88677C214yesmaximal, complemented
5#11722507C218yesmaximal, complemented
6#11722505C214yesmaximal, complemented
7D4×((C3⋊D3)⋊C3)⋊C2C214yesmaximal, complemented
8#11722493C212yesmaximal, complemented
9#11722491C212yesmaximal, complemented
10D3×#1004—32nomaximal
11C23×#272C2212yescomplemented
12C22×((C3⋊D3)⋊C3)⋊C2C2214yescomplemented
13C2×D4×H(3,𝔽3)C2211yescharacteristic, complemented
14C2×#6235C2212yescomplemented
15D4×#272C2214yescomplemented
16D4×#272C2214yescomplemented
17C2×C4⋊#272C2211yescharacteristic, complemented
18C2×C4×#272C2211yescharacteristic, complemented
19C2×#638⋊C3C2218yescomplemented
20#3830⋊C3C2218yescomplemented
21D4×(C3⋊D3)⋊C3C2218yescomplemented
22#3828⋊C3C2214yescomplemented
23#88677C2218yescomplemented
24#88677C2214yescomplemented
25#88677C2218yescomplemented
26#88677C2218yescomplemented
27#88676C22116yescomplemented
28C2×#6200C2218yescomplemented
29C2×#6199C2214yescomplemented
30#88672C2214yescomplemented
31#88671C22116yescomplemented
32#88655C2212yescomplemented
33#88654C2218yescomplemented
34C4×((C3⋊D3)⋊C3)⋊C2C2214yescomplemented
35#88652C2218yescomplemented
36#88651C2212yescomplemented
37#88650C2214yescomplemented
38#88649C2214yescomplemented
39#64361D312yescomplemented
40C3×#1004—32no—
41C2×C3×D3×D4—32no—
42C2×C3×D3×D4—61no—
43#64330—38no—
44C2×D3×#67—38no—
45#64314—38no—
46#64313—38no—
47D3×#237—38no—
48D32×D4—38no—
49#64307—34no—
50D3×#235—34no—
51C23×H(3,𝔽3)C2312yescomplemented
52C22×#272—44no—
53C22×#272C2314yes—
54C22×#272C2311yescharacteristic
55C22×#272C2314yescomplemented
56C22×#272C2312yescomplemented
57C22×(C3⋊D3)⋊C3C2318yescomplemented
58C2×((C3⋊D3)⋊C3)⋊C2—48no—
59C2×((C3⋊D3)⋊C3)⋊C2C2314yes—
60C2×((C3⋊D3)⋊C3)⋊C2C2318yescomplemented
61D4×H(3,𝔽3)C2314yescomplemented
62C2×C4×H(3,𝔽3)C2311yescharacteristic, complemented
63#6235C2318yescomplemented
64C2×#1084C2311yescharacteristic, complemented
65C4⋊#272C2314yescomplemented
66C4×#272C2312yes—
67C4×#272C2312yescomplemented
68#638⋊C3C23116yescomplemented
69#638⋊C3C2318yescomplemented
70C2×(C3⋊Dic3)⋊C3C2318yescomplemented
71C4⋊((C3⋊D3)⋊C3)C2314yescomplemented
72C4×(C3⋊D3)⋊C3C2318yescomplemented
73#634⋊C3C2314yescomplemented
74#6200C2318yescomplemented
75#6200C23116yescomplemented
76#6199C2314yes—
77#6199C2318yescomplemented
78#6198C2314yescomplemented
79((C3⋊D3)⋊C3)⋊C4C2318yescomplemented
80#6196C2314yescomplemented
81D3×H(5,𝔽2)—92no—
82C2×#1004—91no—
83C23×C3×D3—34no—
84C23×C3×D3—62no—
85C22×D32—38no—
86C2×C32×D4—21no—
87C2×C32×D4C2×D312yescomplemented
88C2×#638C2×D318yescomplemented
89#3830C2×D318yescomplemented
90D4×C3⋊D3C2×D318yescomplemented
91#3828C2×D314yescomplemented
92C2×C3×#67—38no—
93C2×C3×#67—34no—
94C2×C3×#67—62no—
95C3×#237—38no—
96C3×D3×D4—38no—
97C3×D3×D4—38no—
98C3×D3×D4—38no—
99C3×D3×D4—64no—
100C3×D3×D4—64no—
101C3×#235—34no—
102C2×C3×D12—32no—
103C2×C3×D12—61no—
104C2×C3×C4×D3—32no—
105C2×C3×C4×D3—61no—
106#3811—316no—
107D3×#67—38no—
108D3×#67—316no—
109D3×#67—316no—
110D3×#67—316no—
111D3×#67—316no—
112C2×#626—38no—
113C2×#625—38no—
114#3805—316no—
115#3804—38no—
116#3804—316no—
117C2×D3×Dic3—38no—
118#3802—38no—
119D3×D12—34no—
120D3×D12—38no—
121C4×D32—38no—
122#3799—34no—
123#3798—38no—
124#3796—38no—
125#3795—34no—
126#3795—38no—
127D3×Dic6—34no—
128C22×H(3,𝔽3)—42no—
129C22×H(3,𝔽3)C2414yes—
130C22×H(3,𝔽3)C2411yescharacteristic
131C2×#272—22no—
132C2×#272—28no—
133C2×#272—44no—
134C2×#272—42no—
135C2×#272C2414yes—
136C2×#272C2411yescharacteristic
137C2×#272C2411yescharacteristic
138C2×(C3⋊D3)⋊C3—48no—
139C2×(C3⋊D3)⋊C3C2418yes—
140((C3⋊D3)⋊C3)⋊C2—28no—
141((C3⋊D3)⋊C3)⋊C2—48no—
142C4×H(3,𝔽3)C2412yes—
143#1084C2412yes—
144(C3⋊Dic3)⋊C3C2418yes—
145C3×H(5,𝔽2)—92no—
146C22×C3×D4—91no—
147C22×#67—94no—
148#1009—94no—
149#1009—98no—
150D3×#41—94no—
151D3×#41—98no—
152#1004—32no—
153#1004—92no—
154#1004—92no—
155#1004—98no—
156#1004—98no—
157#1004—94no—
158#1004—98no—
159C2×#237—94no—
160C2×D3×D4—32no—
161C2×D3×D4—61no—
162C2×D3×D4—94no—
163C2×D3×D4—98no—
164C2×D3×D4—98no—
165C2×#235—92no—
166C23×C32—22no—
167C23×C32C22×D314yescomplemented
168C22×C3⋊D3C22×D318yescomplemented
169C22×C3×D3—38no—
170C22×C3×D3—32no—
171C22×C3×D3—38no—
172C22×C3×D3—38no—
173C22×C3×D3—34no—
174C22×C3×D3—61no—
175C22×C3×D3—64no—
176C22×C3×D3—62no—
177C22×C3×D3—64no—
178C22×C3×D3—128no—
179C22×C3×D3—128no—
180C2×D32—38no—
181C2×D32—316no—
182C2×D32—1216no—
183C32×D4—24no—
184C32×D4C22×D318yescomplemented
185C2×C32×C4—21no—
186C2×C32×C4C22×D312yescomplemented
187#638C22×D318yescomplemented
188#638C22×D3116yescomplemented
189C2×C3⋊Dic3C22×D318yescomplemented
190C4⋊(C3⋊D3)C22×D314yescomplemented
191C4×C3⋊D3C22×D318yescomplemented
192#634C22×D314yescomplemented
193C3×#67—38no—
194C3×#67—316no—
195C3×#67—316no—
196C3×#67—68no—
197C2×C3×Dic3—38no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—61no—
200C3×D12—34no—
201C3×D12—38no—
202C3×D12—64no—
203C3×C4×D3—38no—
204C3×C4×D3—34no—
205C3×C4×D3—34no—
206C3×C4×D3—62no—
207C3×C4×D3—62no—
208C3×Dic6—34no—
209#627—38no—
210#626—38no—
211#626—316no—
212#626—38no—
213#625—38no—
214#625—316no—
215#624—38no—
216D3×Dic3—38no—
217D3×Dic3—38no—
218D3×Dic3—316no—
219C2×H(5,𝔽2)—271noSylow
220C2×H(3,𝔽3)—21no—
221C2×H(3,𝔽3)—24no—
222C2×H(3,𝔽3)C2511yescharacteristic
223#272—21no—
224#272—24no—
225#272H(5,𝔽2)12yescomplemented
226(C3⋊D3)⋊C3—28no—
227C24×C3—92no—
228C23×D3—34no—
229C23×D3—62no—
230C23×D3—94no—
231C23×D3—98no—
232C3×#41—94no—
233C3×#41—98no—
234C2×C3×D4—32no—
235C2×C3×D4—61no—
236C2×C3×D4—98no—
237C2×C3×D4—98no—
238C2×C3×D4—92no—
239C2×C3×D4—94no—
240C2×C3×D4—94no—
241C2×C3×D4—91no—
242C2×C3×D4D3211yescharacteristic
243C22×C3×C4—91no—
244C2×#67—34no—
245C2×#67—38no—
246C2×#67—62no—
247C2×#67—98no—
248C2×#67—98no—
249C2×#67—94no—
250C2×#67—98no—
251C2×#67—98no—
252C2×#67—98no—
253C2×#67—916no—
254C2×#67—98no—
255C22×Dic3—94no—
256#239—94no—
257#239—98no—
258D3×Q8—94no—
259#237—38no—
260#237—98no—
261#237—98no—
262#237—94no—
263#237—916no—
264#237—98no—
265#237—916no—
266D3×D4—38no—
267D3×D4—38no—
268D3×D4—38no—
269D3×D4—64no—
270D3×D4—64no—
271D3×D4—98no—
272D3×D4—98no—
273D3×D4—94no—
274D3×D4—98no—
275D3×D4—916no—
276D3×D4—916no—
277D3×D4—98no—
278D3×D4—916no—
279D3×D4—916no—
280D3×D4—916no—
281#235—34no—
282#235—94no—
283#235—94no—
284#235—94no—
285#235—98no—
286#235—98no—
287#235—94no—
288#235—916no—
289C2×D12—32no—
290C2×D12—61no—
291C2×D12—92no—
292C2×D12—98no—
293C2×D12—98no—
294C2×C4×D3—32no—
295C2×C4×D3—61no—
296C2×C4×D3—94no—
297C2×C4×D3—98no—
298C2×C4×D3—98no—
299C2×Dic6—92no—
300C22×C32—21no—
301C22×C32—24no—
302C22×C32—44no—
303C22×C32—44no—
304C22×C32C23×D318yes—
305C22×C32C23×D312yes—
306C2×C3⋊D3—48no—
307C2×C3⋊D3C23×D318yes—
308C2×C3×D3—32no—
309C2×C3×D3—38no—
310C2×C3×D3—38no—
311C2×C3×D3—32no—
312C2×C3×D3—61no—
313C2×C3×D3—61no—
314C2×C3×D3—64no—
315C2×C3×D3—616no—
316C2×C3×D3—64no—
317C2×C3×D3—64no—
318C2×C3×D3—128no—
319C2×C3×D3—128no—
320C2×C3×D3—124no—
321C2×C3×D3—128no—
322C2×C3×D3—128no—
323C2×C3×D3—124no—
324D32—616no—
325D32—1216no—
326C32×C4—22no—
327C32×C4C23×D314yes—
328C3⋊Dic3C23×D318yes—
329C3×Dic3—38no—
330C3×Dic3—34no—
331C3×Dic3—62no—
332H(5,𝔽2)—92no—
333H(5,𝔽2)—274no—
334H(5,𝔽2)—278no—
335H(5,𝔽2)—272no—
336C2×#41—272no—
337C2×#41—274no—
338C22×D4—91no—
339C22×D4—274no—
340C22×D4—274no—
341H(3,𝔽3)C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
342C23×C3—34no—
343C23×C3—62no—
344C23×C3—98no—
345C23×C3—94no—
346C23×C3—91no—
347C23×C3—92no—
348C23×C3—94no—
349C23×C3—364no—
350C23×C3C2×D3212yes—
351C22×D3—32no—
352C22×D3—38no—
353C22×D3—34no—
354C22×D3—38no—
355C22×D3—38no—
356C22×D3—61no—
357C22×D3—62no—
358C22×D3—64no—
359C22×D3—64no—
360C22×D3—98no—
361C22×D3—94no—
362C22×D3—98no—
363C22×D3—98no—
364C22×D3—916no—
365C22×D3—128no—
366C22×D3—128no—
367C22×D3—3616no—
368C22×D3—368no—
369C3×Q8—94no—
370C3×D4—38no—
371C3×D4—64no—
372C3×D4—94no—
373C3×D4—98no—
374C3×D4—916no—
375C3×D4—98no—
376C3×D4—94no—
377C3×D4C2×D3214yes—
378C2×C3×C4—32no—
379C2×C3×C4—61no—
380C2×C3×C4—98no—
381C2×C3×C4—98no—
382C2×C3×C4—91no—
383C2×C3×C4—92no—
384C2×C3×C4—92no—
385C2×C3×C4C2×D3211yescharacteristic
386#67—316no—
387#67—38no—
388#67—316no—
389#67—68no—
390#67—98no—
391#67—916no—
392#67—98no—
393#67—916no—
394#67—98no—
395#67—98no—
396#67—916no—
397C2×Dic3—32no—
398C2×Dic3—38no—
399C2×Dic3—61no—
400C2×Dic3—98no—
401C2×Dic3—94no—
402C2×Dic3—98no—
403C2×Dic3—98no—
404D12—38no—
405D12—34no—
406D12—64no—
407D12—94no—
408D12—94no—
409D12—98no—
410D12—916no—
411C4×D3—34no—
412C4×D3—34no—
413C4×D3—38no—
414C4×D3—62no—
415C4×D3—62no—
416C4×D3—98no—
417C4×D3—98no—
418C4×D3—98no—
419C4×D3—98no—
420C4×D3—916no—
421Dic6—34no—
422Dic6—94no—
423Dic6—94no—
424Dic6—98no—
425C2×C32—22no—
426C2×C32—28no—
427C2×C32—22no—
428C2×C32—21no—
429C2×C32—44no—
430C2×C32C24×D312yes—
431C3⋊D3—28no—
432C3×D3—34no—
433C3×D3—62no—
434C3×D3—62no—
435C3×D3—68no—
436C3×D3—68no—
437C3×D3—62no—
438C3×D3—124no—
439C24—92no—
440C24—274no—
441#41—98no—
442#41—94no—
443#41—274no—
444#41—274no—
445#41—2716no—
446#41—274no—
447#41—278no—
448C2×Q8—272no—
449C2×D4—92no—
450C2×D4—98no—
451C2×D4—94no—
452C2×D4—94no—
453C2×D4—98no—
454C2×D4—91no—
455C2×D4—272no—
456C2×D4—278no—
457C2×D4—278no—
458C2×D4—274no—
459C2×D4—278no—
460C2×D4—2716no—
461C2×D4—278no—
462C2×D4—278no—
463C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
464C22×C4—91no—
465C22×C4—274no—
466C22×C4—274no—
467C22×C3—32no—
468C22×C3—38no—
469C22×C3—42no—
470C22×C3—61no—
471C22×C3—64no—
472C22×C3—98no—
473C22×C3—91no—
474C22×C3—94no—
475C22×C3—91no—
476C22×C3—124no—
477C22×C3—124no—
478C22×C3—188no—
479C22×C3—182no—
480C22×C3—364no—
481C22×C3—362no—
482C22×C3—368no—
483C22×C3C22×D3214yes—
484C22×C3C22×D3211yescharacteristic
485C2×D3—32no—
486C2×D3—32no—
487C2×D3—38no—
488C2×D3—38no—
489C2×D3—61no—
490C2×D3—61no—
491C2×D3—64no—
492C2×D3—64no—
493C2×D3—616no—
494C2×D3—64no—
495C2×D3—98no—
496C2×D3—128no—
497C2×D3—128no—
498C2×D3—128no—
499C2×D3—124no—
500C2×D3—128no—
501C2×D3—124no—
502C2×D3—1816no—
503C2×D3—188no—
504C2×D3—368no—
505C2×D3—368no—
506C2×D3—3616no—
507C3×C4—34no—
508C3×C4—62no—
509C3×C4—98no—
510C3×C4—92no—
511C3×C4C22×D3212yes—
512Dic3—34no—
513Dic3—38no—
514Dic3—62no—
515Dic3—98no—
516C32—21no—
517C32D3×H(5,𝔽2)12yescomplemented
518C23—91no—
519C23—94no—
520C23—98no—
521C23—94no—
522C23—92no—
523C23—274no—
524C23—278no—
525C23—364no—
526C23—1088no—
527C23C2×((C3⋊D3)⋊C3)⋊C212yescomplemented
528Q8—94no—
529Q8—274no—
530D4—98no—
531D4—916no—
532D4—98no—
533D4—94no—
534D4—94no—
535D4—274no—
536D4—278no—
537D4—2716no—
538D4C2×((C3⋊D3)⋊C3)⋊C214yescomplemented
539C2×C4—91no—
540C2×C4—98no—
541C2×C4—92no—
542C2×C4—92no—
543C2×C4—98no—
544C2×C4—274no—
545C2×C4—278no—
546C2×C4—278no—
547C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
548C2×C3—24no—
549C2×C3—21no—
550C2×C3—32no—
551C2×C3—62no—
552C2×C3—68no—
553C2×C3—62no—
554C2×C3—61no—
555C2×C3—92no—
556C2×C3—124no—
557C2×C3—188no—
558C2×C3—184no—
559C2×C3—181no—
560C2×C3C23×D3211yescharacteristic
561D3—34no—
562D3—62no—
563D3—62no—
564D3—68no—
565D3—62no—
566D3—68no—
567D3—124no—
568D3—188no—
569C22—42no—
570C22—91no—
571C22—91no—
572C22—98no—
573C22—94no—
574C22—182no—
575C22—188no—
576C22—362no—
577C22—364no—
578C22—368no—
579C22—548no—
580C22—1088no—
581C22C22×((C3⋊D3)⋊C3)⋊C214yes—
582C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
583C4—92no—
584C4—98no—
585C4C22×((C3⋊D3)⋊C3)⋊C212yes—
586C3—32no—
587C3—61no—
588C3#1059667511yescharacteristic
589C2—21no—
590C2—24no—
591C2—92no—
592C2—181no—
593C2—184no—
594C2—188no—
595C2C23×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC2C23×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C25
Frattini, socleC2×C3C23×D32
FittingC2×D4×H(3,𝔽3)C22
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×H(5,𝔽2)
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×#272C2semidirect
2. #64361⋊C3C2semidirect
3. #11722545C2semidirect
4. C2×#88677C2semidirect
5. #11722507C2semidirect
6. #11722505C2semidirect
7. D4×((C3⋊D3)⋊C3)⋊C2C2semidirect
8. #11722493C2semidirect
9. #11722491C2semidirect
11. C23×#272C22semidirect
12. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
13. C2×D4×H(3,𝔽3)C22semidirect
14. C2×#6235C22semidirect
15. D4×#272C22semidirect
16. D4×#272C22semidirect
17. C2×C4⋊#272C22semidirect
18. C2×C4×#272C22semidirect
19. C2×#638⋊C3C22semidirect
20. #3830⋊C3C22semidirect
21. D4×(C3⋊D3)⋊C3C22semidirect
22. #3828⋊C3C22semidirect
23. #88677C22semidirect
24. #88677C22semidirect
25. #88677C22semidirect
26. #88677C22semidirect
27. #88676C22semidirect
28. C2×#6200C22semidirect
29. C2×#6199C22semidirect
30. #88672C22semidirect
31. #88671C22semidirect
32. #88655C22semidirect
33. #88654C22semidirect
34. C4×((C3⋊D3)⋊C3)⋊C2C22semidirect
35. #88652C22semidirect
36. #88651C22semidirect
37. #88650C22semidirect
38. #88649C22semidirect
39. #64361D3semidirect
51. C23×H(3,𝔽3)C23semidirect
55. C22×#272C23semidirect
56. C22×#272C23semidirect
57. C22×(C3⋊D3)⋊C3C23semidirect
60. C2×((C3⋊D3)⋊C3)⋊C2C23semidirect
61. D4×H(3,𝔽3)C23semidirect
62. C2×C4×H(3,𝔽3)C23semidirect
63. #6235C23semidirect
64. C2×#1084C23semidirect
65. C4⋊#272C23semidirect
67. C4×#272C23semidirect
68. #638⋊C3C23semidirect
69. #638⋊C3C23semidirect
70. C2×(C3⋊Dic3)⋊C3C23semidirect
71. C4⋊((C3⋊D3)⋊C3)C23semidirect
72. C4×(C3⋊D3)⋊C3C23semidirect
73. #634⋊C3C23semidirect
74. #6200C23semidirect
75. #6200C23semidirect
77. #6199C23semidirect
78. #6198C23semidirect
79. ((C3⋊D3)⋊C3)⋊C4C23semidirect
80. #6196C23semidirect
87. C2×C32×D4C2×D3semidirect
88. C2×#638C2×D3semidirect
89. #3830C2×D3semidirect
90. D4×C3⋊D3C2×D3semidirect
91. #3828C2×D3semidirect
167. C23×C32C22×D3semidirect
168. C22×C3⋊D3C22×D3semidirect
184. C32×D4C22×D3semidirect
186. C2×C32×C4C22×D3semidirect
187. #638C22×D3semidirect
188. #638C22×D3semidirect
189. C2×C3⋊Dic3C22×D3semidirect
190. C4⋊(C3⋊D3)C22×D3semidirect
191. C4×C3⋊D3C22×D3semidirect
192. #634C22×D3semidirect
225. #272H(5,𝔽2)semidirect
341. H(3,𝔽3)C2×H(5,𝔽2)semidirect
463. C2×D4((C3⋊D3)⋊C3)⋊C2semidirect
517. C32D3×H(5,𝔽2)semidirect
527. C23C2×((C3⋊D3)⋊C3)⋊C2semidirect
538. D4C2×((C3⋊D3)⋊C3)⋊C2semidirect
547. C2×C4C2×((C3⋊D3)⋊C3)⋊C2semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#2722
#64361⋊C32
#117225452
C2×#886772
#117225072
#117225052
D4×((C3⋊D3)⋊C3)⋊C22
#117224932
#117224912
D3×#10043