Finite Group Databasebeta

← C2×#10590928 #11867877

Subgroups — C2×#10590928 #11867877

Order 1152.

1069 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64110C211yesmaximal, characteristic, complemented
2C22×C3×#204C211yesmaximal, characteristic, complemented
3C2×#63833C211yesmaximal, characteristic, complemented
4C2×#63830C211yesmaximal, characteristic
5C2×C3×#819C211yesmaximal, characteristic, complemented
6C2×#63560C211yesmaximal, characteristic, complemented
7C2×#63558C211yesmaximal, characteristic, complemented
8#10590928C218yesmaximal, direct product factor
9C2×#5177—31nomaximal
10C2×#4898—31nomaximal
11C2×#3828C2211yescharacteristic
12C22×C3×D12C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14C22×C32×C8C2211yescharacteristic
15C2×#3748C2211yescharacteristic
16#64110—22no—
17#64110C2214yescomplemented
18C2×#3743—21no—
19C2×C8×C3⋊D3—21no—
20C2×C3×#208C2211yescharacteristic
21C2×C3×#208C2211yescharacteristic
22C2×C3×#204—21no—
23C2×C3×#204—21no—
24C2×C3×#204—22no—
25C2×C3×#204C2214yescomplemented
26C2×#3719—21no—
27#63833—22no—
28#63833C2214yescomplemented
29C2×#3717—21no—
30#63830—22no—
31#63830C2214yes—
32C2×#3716—21no—
33C2×#3714—21no—
34C3×#819C2214yescomplemented
35#63560C2214yescomplemented
36#63558C2214yescomplemented
37C22×C3×SD16—31no—
38C2×#942—31no—
39C2×#932—31no—
40C2×#909—31no—
41C2×#908—31no—
42C22×#204D311yescharacteristic, complemented
43C2×#901—31no—
44C2×#901—31no—
45C2×#901—61no—
46C2×C3×#108—31no—
47C2×#836—31no—
48C2×#833—31no—
49#5177—38no—
50C2×#824—31no—
51C2×#823—31no—
52C2×#819D311yescharacteristic, complemented
53#4898—38no—
54C23×C3×D3—21no—
55C22×C32×C4C2311yescharacteristic
56C2×#638—21no—
57#3828—21no—
58#3828—21no—
59#3828D412yes—
60#3828C2312yes—
61C2×C4⋊(C3⋊D3)D411yescharacteristic
62C2×C4×C3⋊D3—21no—
63C2×#634D411yescharacteristic
64C22×C3×Dic3—21no—
65C2×C3×D12—21no—
66C2×C3×D12—21no—
67C2×C3×D12—21no—
68C2×C3×D12—21no—
69C2×C3×D12—21no—
70C2×C3×D12C2312yes—
71C2×C3×Dic6—21no—
72C2×C3×Dic6—21no—
73C2×C3×Dic6—21no—
74C2×C3×Dic6—21no—
75C2×C3×Dic6—21no—
76C2×C3×Dic6C2312yes—
77C2×C32×C8D411yescharacteristic
78C2×C32×C8D411yescharacteristic
79C2×C32×C8D412yes—
80C2×C32×C8C2312yes—
81#3748—21no—
82#3748C2312yes—
83C2×C3⋊(C3⋊C8)—21no—
84#3743—24no—
85C8×C3⋊D3—24no—
86C3×#208—21no—
87C3×#208—21no—
88C3×#208C2312yes—
89C3×#208C2312yes—
90C2×C3×C3⋊C8—21no—
91C2×C3×C3⋊C8—21no—
92C3×#204—24no—
93C3×#204—24no—
94#3719—24no—
95#3717—24no—
96#3716—24no—
97#3714—24no—
98C2×#471—91noSylow
99C22×C3×Q8—31no—
100C22×C3×D4—31no—
101C2×#235—31no—
102C2×#235—31no—
103C2×#235—61no—
104C22×D12C2×D311yescharacteristic
105C22×Dic6C2×D311yescharacteristic
106C2×C3×SD16—34no—
107C2×C3×SD16—61no—
108C2×C3×SD16—61no—
109C2×C3×SD16—62no—
110C2×C3×M16—31no—
111C2×C3×M16—31no—
112C22×C3×C8—21no—
113C22×C3×C8C2×D311yescharacteristic
114C22×C3×C8C2×D311yescharacteristic
115C2×#216—61no—
116#942—34no—
117#942—62no—
118C2×#215—61no—
119C2×#214—61no—
120#932—34no—
121#932—62no—
122C2×#213—61no—
123C2×#208—31no—
124C2×#208—31no—
125C2×#208—61no—
126C2×#208C2×D311yescharacteristic
127C2×#208C2×D311yescharacteristic
128#909—34no—
129#909—62no—
130#908—34no—
131#908—62no—
132C2×Dic12—61no—
133C2×D24—61no—
134C2×#204—21no—
135C2×#204—21no—
136C2×#204—22no—
137C2×#204—61no—
138C2×#204—61no—
139C2×#204C2×D314yescomplemented
140#901—34no—
141#901—34no—
142#901—64no—
143#901—64no—
144#901—62no—
145#901—62no—
146C2×ΓL(1,𝔽25)—62no—
147C2×ΓL(1,𝔽25)—61no—
148C2×ΓL(1,𝔽25)—61no—
149C2×D3×C8—62no—
150C2×D3×C8—61no—
151C2×D3×C8—61no—
152C3×#108—34no—
153#836—34no—
154#833—34no—
155#824—34no—
156#823—34no—
157#819C2×D314yescomplemented
158C23×C32#4111yescharacteristic
159C22×C3⋊D3—21no—
160C22×C3×D3—22no—
161C22×C3×D3—41no—
162C22×C3×D3—41no—
163C22×C3×D3—82no—
164C2×C32×C4C2×D411yescharacteristic
165C2×C32×C4C2×D411yescharacteristic
166C2×C32×C4C2×D411yescharacteristic
167C2×C32×C4C2×D411yescharacteristic
168C2×C32×C4#4111yescharacteristic
169C2×C32×C4C2411yescharacteristic
170#638—22no—
171#638—22no—
172C2×C3⋊Dic3—21no—
173C4⋊(C3⋊D3)—21no—
174C4⋊(C3⋊D3)C2×D412yes—
175C4×C3⋊D3—22no—
176C4×C3⋊D3—22no—
177#634—21no—
178#634C2×D412yes—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—41no—
181C2×C3×Dic3—41no—
182C3×D12—22no—
183C3×D12—22no—
184C3×D12—41no—
185C3×D12—41no—
186C3×Dic6—22no—
187C3×Dic6—22no—
188C3×Dic6—41no—
189C3×Dic6—41no—
190C32×C8C2×D412yes—
191C32×C8C2×D412yes—
192C3⋊(C3⋊C8)—22no—
193C3×C3⋊C8—22no—
194C3×C3⋊C8—22no—
195C2×#137—91no—
196C2×AGL(1,ℤ/8)—91no—
197C22×SD16—31no—
198C2×#131—91no—
199#471—98no—
200C2×#108—31no—
201C2×#101—91no—
202C2×#100—91no—
203C24×C3—61no—
204C23×D3—21no—
205C2×C3×Q8—32no—
206C2×C3×Q8—61no—
207C2×C3×Q8—61no—
208C2×C3×Q8—61no—
209C2×C3×Q8—61no—
210C2×C3×Q8—61no—
211C2×C3×D4—32no—
212C2×C3×D4—61no—
213C2×C3×D4—61no—
214C2×C3×D4—61no—
215C2×C3×D4—61no—
216C2×C3×D4—61no—
217C22×C3×C4—21no—
218C22×C3×C4—61no—
219C22×C3×C4C22×D311yescharacteristic
220C22×C3×C4C22×D311yescharacteristic
221C2×#67—62no—
222C2×#67—61no—
223C2×#67—61no—
224C22×Dic3—21no—
225#235—32no—
226#235—32no—
227#235—32no—
228#235—32no—
229#235—62no—
230#235—62no—
231#235—62no—
232#235—62no—
233#235—61no—
234#235—61no—
235#235—61no—
236#235—61no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12—21no—
241C2×D12—21no—
242C2×D12—31no—
243C2×D12—31no—
244C2×D12—61no—
245C2×D12C22×D312yes—
246C2×C4×D3—62no—
247C2×C4×D3—61no—
248C2×C4×D3—61no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6—31no—
255C2×Dic6—31no—
256C2×Dic6—61no—
257C2×Dic6C22×D312yes—
258C3×SD16—64no—
259C3×SD16—64no—
260C3×M16—32no—
261C3×M16—32no—
262C3×M16—61no—
263C3×M16—61no—
264C2×C3×C8—21no—
265C2×C3×C8—21no—
266C2×C3×C8—22no—
267C2×C3×C8—22no—
268C2×C3×C8—61no—
269C2×C3×C8—61no—
270C2×C3×C8D1211yescharacteristic
271C2×C3×C8D1211yescharacteristic
272C2×C3×C8D1212yes—
273C2×C3×C8#6711yescharacteristic
274C2×C3×C8#6711yescharacteristic
275C2×C3×C8#6712yes—
276C2×C3×C8C22×D312yes—
277C2×C3×C8C22×D312yes—
278#216—64no—
279#215—64no—
280#214—64no—
281#213—64no—
282#208—21no—
283#208—21no—
284#208—32no—
285#208—32no—
286#208—62no—
287#208—62no—
288#208—61no—
289#208—61no—
290#208C22×D312yes—
291#208C22×D312yes—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—62no—
295C2×C3⋊C8—61no—
296C2×C3⋊C8—61no—
297Dic12—64no—
298D24—64no—
299#204—24no—
300#204—24no—
301#204—64no—
302#204—64no—
303ΓL(1,𝔽25)—68no—
304ΓL(1,𝔽25)—64no—
305ΓL(1,𝔽25)—64no—
306D3×C8—68no—
307D3×C8—64no—
308D3×C8—64no—
309C22×C32—22no—
310C22×C32#12111yescharacteristic
311C22×C32#12111yescharacteristic
312C22×C32C2×#4111yescharacteristic
313C2×C3⋊D3—22no—
314C2×C3⋊D3—42no—
315C2×C3×D3—42no—
316C2×C3×D3—82no—
317C2×C3×D3—82no—
318C2×C3×D3—82no—
319C32×C4#12111yescharacteristic
320C32×C4#12111yescharacteristic
321C32×C4C22×D411yescharacteristic
322C32×C4C22×D411yescharacteristic
323C3⋊Dic3—22no—
324C3×Dic3—42no—
325C2×#41—91no—
326C22×Q8—31no—
327C22×D4—31no—
328#137—94no—
329#137—182no—
330AGL(1,ℤ/8)—94no—
331AGL(1,ℤ/8)—182no—
332C2×Q16—181no—
333C2×SD16—34no—
334C2×SD16—61no—
335C2×SD16—61no—
336C2×SD16—62no—
337C2×SD16—181no—
338C2×SD16—181no—
339C2×D8—181no—
340#131—94no—
341#131—182no—
342C2×M16—31no—
343C2×M16—31no—
344C2×M16—91no—
345C2×M16—181no—
346C22×C8—181no—
347C22×C8D3211yescharacteristic
348#108—34no—
349#101—94no—
350#100—94no—
351C23×C3—21no—
352C23×C3—62no—
353C23×C3—121no—
354C23×C3—121no—
355C23×C3—242no—
356C23×C3#23711yescharacteristic
357C23×C3#23911yescharacteristic
358C22×D3—22no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—62no—
362C22×D3—61no—
363C22×D3—61no—
364C22×D3—82no—
365C3×Q8—62no—
366C3×Q8—62no—
367C3×Q8—121no—
368C3×Q8—121no—
369C3×D4—62no—
370C3×D4—62no—
371C3×D4—121no—
372C3×D4—121no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—62no—
380C2×C3×C4—121no—
381C2×C3×C4—121no—
382C2×C3×C4C2×D1211yescharacteristic
383C2×C3×C4C2×D1211yescharacteristic
384C2×C3×C4D3×D411yescharacteristic
385C2×C3×C4D3×D411yescharacteristic
386C2×C3×C4D3×D411yescharacteristic
387C2×C3×C4D3×D411yescharacteristic
388C2×C3×C4#23711yescharacteristic
389C2×C3×C4#23911yescharacteristic
390C2×C3×C4C2×#6711yescharacteristic
391C2×C3×C4C2×#6711yescharacteristic
392C2×C3×C4C23×D311yescharacteristic
393C2×C3×C4C23×D311yescharacteristic
394#67—64no—
395#67—64no—
396#67—62no—
397#67—62no—
398#67—62no—
399#67—62no—
400C2×Dic3—22no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—62no—
404C2×Dic3—61no—
405C2×Dic3—61no—
406D12—22no—
407D12—22no—
408D12—32no—
409D12—32no—
410D12—41no—
411D12—41no—
412D12—62no—
413D12—62no—
414D12—61no—
415D12—61no—
416C4×D3—64no—
417C4×D3—64no—
418C4×D3—62no—
419C4×D3—62no—
420C4×D3—62no—
421C4×D3—62no—
422Dic6—22no—
423Dic6—22no—
424Dic6—32no—
425Dic6—32no—
426Dic6—41no—
427Dic6—41no—
428Dic6—62no—
429Dic6—62no—
430Dic6—61no—
431Dic6—61no—
432C3×C8—22no—
433C3×C8—22no—
434C3×C8—62no—
435C3×C8—62no—
436C3×C8C2×D1212yes—
437C3×C8C2×D1212yes—
438C3×C8C2×#6712yes—
439C3×C8C2×#6712yes—
440C3⋊C8—22no—
441C3⋊C8—22no—
442C3⋊C8—64no—
443C3⋊C8—62no—
444C3⋊C8—62no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32#47112yescomplemented
448C2×C32C2×#12111yescharacteristic
449C3⋊D3—42no—
450C3×D3—82no—
451C24—61no—
452#41—92no—
453#41—92no—
454#41—181no—
455#41—181no—
456C2×Q8—32no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×Q8—61no—
461C2×Q8—61no—
462C2×Q8—91no—
463C2×D4—32no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—61no—
468C2×D4—61no—
469C2×D4—91no—
470C2×D4—181no—
471C22×C4—61no—
472C22×C4—181no—
473C22×C4C2×D3211yescharacteristic
474Q16—184no—
475SD16—64no—
476SD16—64no—
477SD16—184no—
478SD16—184no—
479D8—184no—
480M16—32no—
481M16—32no—
482M16—61no—
483M16—61no—
484M16—92no—
485M16—181no—
486M16—184no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—181no—
490C2×C8—184no—
491C2×C8#62611yescharacteristic
492C2×C8#62611yescharacteristic
493C2×C8#62612yes—
494C2×C8C2×D3212yes—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—42no—
501C22×C3—122no—
502C22×C3—242no—
503C22×C3—242no—
504C22×C3—242no—
505C22×C3#89511yescharacteristic
506C22×C3#89511yescharacteristic
507C22×C3#93811yescharacteristic
508C22×C3#93811yescharacteristic
509C22×C3C2×#23711yescharacteristic
510C22×C3C2×#23911yescharacteristic
511C2×D3—42no—
512C2×D3—64no—
513C2×D3—62no—
514C2×D3—62no—
515C2×D3—82no—
516C2×D3—82no—
517C2×D3—82no—
518C2×D3—124no—
519C2×D3—122no—
520C2×D3—122no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—122no—
526C3×C4#89511yescharacteristic
527C3×C4#89511yescharacteristic
528C3×C4#93811yescharacteristic
529C3×C4#93811yescharacteristic
530C3×C4C22×D1211yescharacteristic
531C3×C4C2×D3×D411yescharacteristic
532C3×C4C2×D3×D411yescharacteristic
533C3×C4C22×#6711yescharacteristic
534Dic3—42no—
535Dic3—64no—
536Dic3—62no—
537Dic3—62no—
538C32C2×#47111yescharacteristic, complemented, Sylow
539C23—62no—
540C23—121no—
541C23—121no—
542C23—181no—
543C23—242no—
544C23#379611yescharacteristic
545Q8—62no—
546Q8—62no—
547Q8—92no—
548Q8—121no—
549Q8—121no—
550Q8—181no—
551D4—62no—
552D4—62no—
553D4—92no—
554D4—121no—
555D4—121no—
556D4—182no—
557D4—182no—
558D4—181no—
559C2×C4—62no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—181no—
563C2×C4—182no—
564C2×C4—182no—
565C2×C4#379611yescharacteristic
566C2×C4#380211yescharacteristic
567C2×C4#380211yescharacteristic
568C2×C4C2×#62611yescharacteristic
569C2×C4C2×#62611yescharacteristic
570C2×C4C22×D3211yescharacteristic
571C8—62no—
572C8—62no—
573C8—182no—
574C8C2×#62612yes—
575C8C2×#62612yes—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—22no—
582C2×C3—41no—
583C2×C3—41no—
584C2×C3—242no—
585C2×C3#489812yescomplemented
586C2×C3#517712yescomplemented
587C2×C3C2×#89511yescharacteristic
588C2×C3C2×#93811yescharacteristic
589D3—82no—
590D3—124no—
591D3—122no—
592D3—122no—
593C22—22no—
594C22—122no—
595C22—182no—
596C22—242no—
597C22—242no—
598C22—242no—
599C22—362no—
600C22#6391611yescharacteristic
601C22#6391611yescharacteristic
602C22C2×#379611yescharacteristic
603C4—122no—
604C4—182no—
605C4#6391611yescharacteristic
606C4#6391611yescharacteristic
607C4C2×#380211yescharacteristic
608C4C22×#62611yescharacteristic
609C3—21no—
610C3C2×#489811yescharacteristic, complemented
611C3C2×#517711yescharacteristic, complemented
612C2—21no—
613C2—21no—
614C2—242no—
615C2—362no—
616C2#1059092812yesdirect product factor
617C2C2×#6391611yescharacteristic
subgroupG/H
centerC22#63916
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC22×C32×C8C22
socleC22×C32#121
prime radicalC2×#63830C2
hypercenterC22×C8D32
hypocenterC32C2×#471
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
1. C2×#64110C2semidirect
2. C22×C3×#204C2semidirect
3. C2×#63833C2semidirect
5. C2×C3×#819C2semidirect
6. C2×#63560C2semidirect
7. C2×#63558C2semidirect
8. #10590928C2direct (N × H)
17. #64110C22semidirect
25. C2×C3×#204C22semidirect
28. #63833C22semidirect
34. C3×#819C22semidirect
35. #63560C22semidirect
36. #63558C22semidirect
42. C22×#204D3semidirect
52. C2×#819D3semidirect
139. C2×#204C2×D3semidirect
157. #819C2×D3semidirect
447. C2×C32#471semidirect
538. C32C2×#471semidirect
585. C2×C3#4898semidirect
586. C2×C3#5177semidirect
610. C3C2×#4898semidirect
611. C3C2×#5177semidirect
616. C2#10590928direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#641102
C22×C3×#2042
C2×#638332
C2×#638302
C2×C3×#8192
C2×#635602
C2×#635582
#105909282
C2×#51773
C2×#48983