Finite Group Databasebeta

← C2×#10590985 #11867893

Subgroups — C2×#10590985 #11867893

Order 1152.

1029 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#909C211yesmaximal, characteristic, complemented
2C22×#3717C211yesmaximal, characteristic, complemented
3C2×#63825C211yesmaximal, characteristic
4C2×#63814C211yesmaximal, characteristic, complemented
5C2×#63650C211yesmaximal, characteristic, complemented
6C2×C3×#822C211yesmaximal, characteristic, complemented
7C2×#63560C211yesmaximal, characteristic, complemented
8#10590985C218yesmaximal, direct product factor
9C2×#5203—31nomaximal
10C2×#4898—31nomaximal
11C22×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#235C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14C2×C32×M16C2211yescharacteristic
15C2×#3748C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C22×C3×C3⋊C8C2211yescharacteristic
18C3×#909—22no—
19C3×#909C2214yescomplemented
20C2×C3×Dic12—21no—
21C2×C3×#204—21no—
22C2×#3717—21no—
23C2×#3717—21no—
24C2×#3717—22no—
25C2×#3717—21no—
26C2×#3717C2214yescomplemented
27#63825—22no—
28#63825C2214yes—
29C2×#3714—21no—
30C2×#3711—21no—
31#63814—22no—
32#63814C2214yescomplemented
33C2×D3×C3⋊C8—21no—
34#63650C2214yescomplemented
35C3×#822C2214yescomplemented
36#63560C2214yescomplemented
37C2×C3×#137—31no—
38C2×#949—31no—
39C2×#948—31no—
40C22×#215—31no—
41C2×#909D311yescharacteristic, complemented
42C2×#908—31no—
43C22×#204—31no—
44C2×#901—31no—
45C2×C3×#108—31no—
46C2×#836—31no—
47#5203—38no—
48C2×#824—31no—
49C2×#823—31no—
50C2×#823—31no—
51C2×#823—61no—
52C2×#822D311yescharacteristic, complemented
53#4898—38no—
54C23×C3⋊D3—21no—
55C22×C32×C4C2311yescharacteristic
56C2×C4⋊(C3⋊D3)—21no—
57C2×C4⋊(C3⋊D3)—21no—
58C2×C4⋊(C3⋊D3)—21no—
59C2×C4⋊(C3⋊D3)—21no—
60C2×C4⋊(C3⋊D3)—21no—
61C2×C4⋊(C3⋊D3)C2312yes—
62C2×C3×#67—21no—
63C22×C3×Dic3—21no—
64C3×#235—21no—
65C3×#235—21no—
66C3×#235D412yes—
67C3×#235C2312yes—
68C2×C3×D12D411yescharacteristic
69C2×C3×C4×D3—21no—
70C2×C3×Dic6—21no—
71C2×C3×Dic6—21no—
72C2×C3×Dic6—21no—
73C2×C3×Dic6—21no—
74C2×C3×Dic6—21no—
75C2×C3×Dic6D411yescharacteristic
76C2×C3×Dic6C2312yes—
77C32×M16—21no—
78C32×M16C2312yes—
79C2×C32×C8—21no—
80#3748—21no—
81#3748C2312yes—
82C2×C3⋊(C3⋊C8)—21no—
83C3×#208—21no—
84C3×#208C2312yes—
85C2×C3×C3⋊C8—21no—
86C2×C3×C3⋊C8D411yescharacteristic
87C2×C3×C3⋊C8D412yes—
88C2×C3×C3⋊C8D411yescharacteristic
89C2×C3×C3⋊C8C2312yes—
90C3×Dic12—24no—
91C3×#204—24no—
92#3717—24no—
93#3717—24no—
94#3717—24no—
95#3714—24no—
96#3711—24no—
97D3×C3⋊C8—24no—
98C2×#471—91noSylow
99C2×C3×#41—31no—
100C22×C3×Q8—31no—
101C2×#235C2×D311yescharacteristic
102C22×D12—31no—
103C22×D12—31no—
104C22×D12—61no—
105C22×Dic6C2×D311yescharacteristic
106C3×#137—34no—
107C3×#137—62no—
108C2×C3×Q16—61no—
109C2×C3×SD16—61no—
110C2×C3×M16—21no—
111C2×C3×M16—31no—
112C2×C3×M16C2×D311yescharacteristic
113C2×C3×M16C2×D311yescharacteristic
114C22×C3×C8—31no—
115#949—34no—
116#949—62no—
117#948—34no—
118#948—62no—
119C2×#215—34no—
120C2×#215—61no—
121C2×#215—61no—
122C2×#215—61no—
123C2×#215—62no—
124C2×#213—61no—
125C2×#208—31no—
126C2×#208—31no—
127C2×#208—61no—
128C2×#208—61no—
129C2×#208C2×D311yescharacteristic
130C22×C3⋊C8—61no—
131C22×C3⋊C8C2×D311yescharacteristic
132#909—22no—
133#909C2×D314yescomplemented
134#908—34no—
135#908—62no—
136C2×Dic12—21no—
137C2×D24—61no—
138C2×#204—21no—
139C2×#204—34no—
140C2×#204—61no—
141C2×#204—61no—
142C2×#204—62no—
143C2×#204—61no—
144#901—34no—
145#901—62no—
146C2×ΓL(1,𝔽25)—61no—
147C2×D3×C8—61no—
148C3×#108—34no—
149#836—34no—
150#824—34no—
151#823—34no—
152#823—34no—
153#823—64no—
154#822C2×D314yescomplemented
155C23×C32#4111yescharacteristic
156C22×C3⋊D3—22no—
157C22×C3⋊D3—41no—
158C22×C3⋊D3—41no—
159C22×C3⋊D3—82no—
160C22×C3×D3—21no—
161C2×C32×C4C2×D411yescharacteristic
162C2×C32×C4C2×D411yescharacteristic
163C2×C32×C4C2×D411yescharacteristic
164C2×C32×C4C2×D411yescharacteristic
165C2×C32×C4#4111yescharacteristic
166C2×C32×C4C2411yescharacteristic
167C4⋊(C3⋊D3)—22no—
168C4⋊(C3⋊D3)—22no—
169C4⋊(C3⋊D3)—41no—
170C4⋊(C3⋊D3)—41no—
171C3×#67—22no—
172C3×#67—22no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—41no—
176C2×C3×Dic3—41no—
177C3×D12—21no—
178C3×D12C2×D412yes—
179C3×C4×D3—22no—
180C3×C4×D3—22no—
181C3×Dic6—21no—
182C3×Dic6—22no—
183C3×Dic6—22no—
184C3×Dic6—41no—
185C3×Dic6—41no—
186C3×Dic6C2×D412yes—
187C32×C8—22no—
188C3⋊(C3⋊C8)—22no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8C2×D412yes—
191C3×C3⋊C8C2×D412yes—
192C2×#137—31no—
193C2×AGL(1,ℤ/8)—91no—
194C22×SD16—91no—
195C2×#131—91no—
196#471—98no—
197C2×#108—31no—
198C2×#101—91no—
199C2×#100—91no—
200C23×D3—62no—
201C23×D3—61no—
202C23×D3—61no—
203C3×#41—32no—
204C3×#41—32no—
205C3×#41—61no—
206C3×#41—61no—
207C2×C3×Q8—31no—
208C2×C3×Q8—32no—
209C2×C3×Q8—61no—
210C2×C3×Q8—61no—
211C2×C3×Q8—61no—
212C2×C3×Q8—61no—
213C2×C3×Q8—61no—
214C2×C3×D4—31no—
215C2×C3×D4—61no—
216C22×C3×C4—21no—
217C22×C3×C4—61no—
218C22×C3×C4—61no—
219C22×C3×C4C22×D311yescharacteristic
220C22×C3×C4C22×D311yescharacteristic
221C2×#67—21no—
222C22×Dic3—21no—
223#235—21no—
224#235—21no—
225#235#6712yes—
226#235C22×D312yes—
227C2×D12—32no—
228C2×D12—32no—
229C2×D12—62no—
230C2×D12—61no—
231C2×D12—61no—
232C2×D12—61no—
233C2×D12—61no—
234C2×D12—61no—
235C2×D12—61no—
236C2×D12—61no—
237C2×D12—61no—
238C2×D12—62no—
239C2×D12—61no—
240C2×D12—61no—
241C2×D12—121no—
242C2×D12—121no—
243C2×D12—121no—
244C2×D12—121no—
245C2×D12#6711yescharacteristic
246C2×C4×D3—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6#6711yescharacteristic
253C2×Dic6C22×D312yes—
254C3×Q16—64no—
255C3×SD16—64no—
256C3×M16—21no—
257C3×M16—21no—
258C3×M16—22no—
259C3×M16—32no—
260C3×M16—41no—
261C3×M16—61no—
262C3×M16C22×D312yes—
263C3×M16C22×D312yes—
264C2×C3×C8—21no—
265C2×C3×C8—21no—
266C2×C3×C8—31no—
267C2×C3×C8—31no—
268C2×C3×C8—32no—
269C2×C3×C8—32no—
270C2×C3×C8—41no—
271C2×C3×C8—61no—
272#215—64no—
273#215—64no—
274#215—64no—
275#213—64no—
276#208—21no—
277#208—32no—
278#208—32no—
279#208—62no—
280#208—61no—
281#208—61no—
282#208—64no—
283#208—121no—
284#208C22×D312yes—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—61no—
287C2×C3⋊C8—61no—
288C2×C3⋊C8—64no—
289C2×C3⋊C8—121no—
290C2×C3⋊C8D1211yescharacteristic
291C2×C3⋊C8D1211yescharacteristic
292C2×C3⋊C8D1212yes—
293C2×C3⋊C8C22×D312yes—
294Dic12—24no—
295D24—64no—
296#204—24no—
297#204—64no—
298#204—64no—
299#204—64no—
300ΓL(1,𝔽25)—64no—
301D3×C8—64no—
302C22×C32—22no—
303C22×C32#12111yescharacteristic
304C22×C32#12111yescharacteristic
305C22×C32C2×#4111yescharacteristic
306C2×C3⋊D3—42no—
307C2×C3⋊D3—82no—
308C2×C3⋊D3—82no—
309C2×C3⋊D3—82no—
310C2×C3×D3—22no—
311C2×C3×D3—42no—
312C32×C4#12111yescharacteristic
313C32×C4#12111yescharacteristic
314C32×C4C22×D411yescharacteristic
315C32×C4C22×D411yescharacteristic
316C3×Dic3—22no—
317C3×Dic3—42no—
318C2×#41—31no—
319C22×Q8—31no—
320C22×D4—91no—
321#137—34no—
322#137—62no—
323AGL(1,ℤ/8)—94no—
324AGL(1,ℤ/8)—182no—
325C2×Q16—61no—
326C2×SD16—61no—
327C2×SD16—94no—
328C2×SD16—181no—
329C2×SD16—181no—
330C2×SD16—182no—
331C2×SD16—181no—
332C2×D8—181no—
333#131—94no—
334#131—182no—
335C2×M16—31no—
336C2×M16—91no—
337C2×M16—181no—
338C2×M16D3211yescharacteristic
339C22×C8—31no—
340C22×C8—181no—
341#108—34no—
342#101—94no—
343#100—94no—
344C23×C3—21no—
345C23×C3—61no—
346C23×C3#23711yescharacteristic
347C23×C3#23911yescharacteristic
348C22×D3—21no—
349C22×D3—64no—
350C22×D3—62no—
351C22×D3—62no—
352C22×D3—122no—
353C22×D3—122no—
354C22×D3—121no—
355C22×D3—121no—
356C22×D3—121no—
357C22×D3—121no—
358C22×D3—242no—
359C22×D3—242no—
360C22×D3—244no—
361C3×Q8—32no—
362C3×Q8—61no—
363C3×Q8—62no—
364C3×Q8—62no—
365C3×Q8—121no—
366C3×Q8—121no—
367C3×D4—32no—
368C3×D4—62no—
369C3×D4—62no—
370C3×D4—61no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—61no—
378C2×C3×C4—62no—
379C2×C3×C4—62no—
380C2×C3×C4—62no—
381C2×C3×C4—121no—
382C2×C3×C4—121no—
383C2×C3×C4C2×D1211yescharacteristic
384C2×C3×C4C2×D1211yescharacteristic
385C2×C3×C4D3×D411yescharacteristic
386C2×C3×C4D3×D411yescharacteristic
387C2×C3×C4D3×D411yescharacteristic
388C2×C3×C4D3×D411yescharacteristic
389C2×C3×C4#23711yescharacteristic
390C2×C3×C4#23911yescharacteristic
391C2×C3×C4C2×#6711yescharacteristic
392C2×C3×C4C2×#6711yescharacteristic
393C2×C3×C4C23×D311yescharacteristic
394C2×C3×C4C23×D311yescharacteristic
395#67—22no—
396#67—22no—
397C2×Dic3—21no—
398C2×Dic3—22no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401D12—21no—
402D12—62no—
403D12—62no—
404D12—62no—
405D12—62no—
406D12—122no—
407D12—122no—
408D12—122no—
409D12—122no—
410D12—121no—
411D12—121no—
412D12—121no—
413D12—121no—
414D12C2×#6712yes—
415C4×D3—22no—
416C4×D3—22no—
417Dic6—21no—
418Dic6—22no—
419Dic6—22no—
420Dic6—41no—
421Dic6—41no—
422Dic6C2×#6712yes—
423C3×C8—22no—
424C3×C8—22no—
425C3×C8—32no—
426C3×C8—32no—
427C3×C8—42no—
428C3×C8—62no—
429C3⋊C8—22no—
430C3⋊C8—62no—
431C3⋊C8—62no—
432C3⋊C8—122no—
433C3⋊C8C2×D1212yes—
434C3⋊C8C2×D1212yes—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32#47112yescomplemented
438C2×C32C2×#12111yescharacteristic
439C3⋊D3—82no—
440C3×D3—42no—
441C24—181no—
442#41—32no—
443#41—32no—
444#41—61no—
445#41—61no—
446C2×Q8—32no—
447C2×Q8—31no—
448C2×Q8—61no—
449C2×Q8—61no—
450C2×Q8—61no—
451C2×Q8—61no—
452C2×Q8—61no—
453C2×D4—31no—
454C2×D4—61no—
455C2×D4—92no—
456C2×D4—181no—
457C2×D4—181no—
458C2×D4—181no—
459C2×D4—181no—
460C2×D4—181no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4C2×D3211yescharacteristic
464Q16—64no—
465SD16—64no—
466SD16—184no—
467SD16—184no—
468SD16—184no—
469D8—184no—
470M16—21no—
471M16—32no—
472M16—61no—
473M16—92no—
474M16—184no—
475M16—181no—
476M16C2×D3212yes—
477C2×C8—21no—
478C2×C8—32no—
479C2×C8—31no—
480C2×C8—31no—
481C2×C8—32no—
482C2×C8—61no—
483C2×C8—184no—
484C2×C8—181no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—42no—
491C22×C3—62no—
492C22×C3—122no—
493C22×C3#89511yescharacteristic
494C22×C3#89511yescharacteristic
495C22×C3#93911yescharacteristic
496C22×C3#93911yescharacteristic
497C22×C3C2×#23711yescharacteristic
498C22×C3C2×#23911yescharacteristic
499C2×D3—22no—
500C2×D3—42no—
501C2×D3—124no—
502C2×D3—122no—
503C2×D3—122no—
504C2×D3—242no—
505C2×D3—242no—
506C2×D3—244no—
507C2×D3—244no—
508C2×D3—242no—
509C2×D3—242no—
510C2×D3—242no—
511C2×D3—242no—
512C2×D3—244no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—62no—
518C3×C4—122no—
519C3×C4#89511yescharacteristic
520C3×C4#89511yescharacteristic
521C3×C4#93911yescharacteristic
522C3×C4#93911yescharacteristic
523C3×C4C22×D1211yescharacteristic
524C3×C4C2×D3×D411yescharacteristic
525C3×C4C2×D3×D411yescharacteristic
526C3×C4C22×#6711yescharacteristic
527Dic3—22no—
528Dic3—42no—
529C32C2×#47111yescharacteristic, complemented, Sylow
530C23—61no—
531C23—182no—
532C23—361no—
533C23—361no—
534C23—722no—
535C23#379611yescharacteristic
536Q8—32no—
537Q8—61no—
538Q8—62no—
539Q8—62no—
540Q8—121no—
541Q8—121no—
542D4—32no—
543D4—62no—
544D4—62no—
545D4—61no—
546D4—182no—
547D4—182no—
548D4—361no—
549D4—361no—
550C2×C4—61no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4#379611yescharacteristic
557C2×C4D3×D1211yescharacteristic
558C2×C4D3×D1211yescharacteristic
559C2×C4D3×#6711yescharacteristic
560C2×C4D3×#6711yescharacteristic
561C2×C4C22×D3211yescharacteristic
562C8—22no—
563C8—32no—
564C8—32no—
565C8—62no—
566C8—182no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—22no—
572C2×C3—21no—
573C2×C3—41no—
574C2×C3—41no—
575C2×C3—122no—
576C2×C3#489812yescomplemented
577C2×C3#520312yescomplemented
578C2×C3C2×#89511yescharacteristic
579C2×C3C2×#93911yescharacteristic
580D3—42no—
581D3—244no—
582D3—242no—
583D3—242no—
584C22—22no—
585C22—62no—
586C22—122no—
587C22—362no—
588C22—722no—
589C22—722no—
590C22—722no—
591C22#6388611yescharacteristic
592C22#6388611yescharacteristic
593C22C2×#379611yescharacteristic
594C4—62no—
595C4—122no—
596C4#6388611yescharacteristic
597C4#6388611yescharacteristic
598C4C2×D3×D1211yescharacteristic
599C4C2×D3×#6711yescharacteristic
600C3—21no—
601C3C2×#489811yescharacteristic, complemented
602C3C2×#520311yescharacteristic, complemented
603C2—21no—
604C2—21no—
605C2—122no—
606C2—722no—
607C2#1059098512yesdirect product factor
608C2C2×#6388611yescharacteristic
subgroupG/H
centerC22#63886
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC22×C32#121
prime radicalC2×#63825C2
hypercenterC2×M16D32
hypocenterC32C2×#471
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×C3×#909C2semidirect
2. C22×#3717C2semidirect
4. C2×#63814C2semidirect
5. C2×#63650C2semidirect
6. C2×C3×#822C2semidirect
7. C2×#63560C2semidirect
8. #10590985C2direct (N × H)
19. C3×#909C22semidirect
26. C2×#3717C22semidirect
32. #63814C22semidirect
34. #63650C22semidirect
35. C3×#822C22semidirect
36. #63560C22semidirect
41. C2×#909D3semidirect
52. C2×#822D3semidirect
133. #909C2×D3semidirect
154. #822C2×D3semidirect
437. C2×C32#471semidirect
529. C32C2×#471semidirect
576. C2×C3#4898semidirect
577. C2×C3#5203semidirect
601. C3C2×#4898semidirect
602. C3C2×#5203semidirect
607. C2#10590985direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9092
C22×#37172
C2×#638252
C2×#638142
C2×#636502
C2×C3×#8222
C2×#635602
#105909852
C2×#52033
C2×#48983