Finite Group Databasebeta

← C2×#10590996 #11867890

Subgroups — C2×#10590996 #11867890

Order 1152.

1009 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#907C211yesmaximal, characteristic, complemented
2C2×#63831C211yesmaximal, characteristic, complemented
3C22×#3716C211yesmaximal, characteristic, complemented
4C2×#63823C211yesmaximal, characteristic
5C2×#63651C211yesmaximal, characteristic, complemented
6C2×C3×#823C211yesmaximal, characteristic, complemented
7C2×#63577C211yesmaximal, characteristic, complemented
8#10590996C218yesmaximal, direct product factor
9C2×#5141—31nomaximal
10C2×#4748—31nomaximal
11C22×#634C2211yescharacteristic
12C2×C3×#235C2211yescharacteristic
13C22×C3×D12C2211yescharacteristic
14C2×C32×M16C2211yescharacteristic
15C2×#3748C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C22×C3×C3⋊C8C2211yescharacteristic
18C3×#907—22no—
19C3×#907C2214yescomplemented
20C2×C3×ΓL(1,𝔽25)—21no—
21C2×C3×D3×C8—21no—
22C2×#3719—21no—
23#63831—22no—
24#63831C2214yescomplemented
25C2×#3716—21no—
26C2×#3716—21no—
27C2×#3716—21no—
28C2×#3716—22no—
29C2×#3716C2214yescomplemented
30C2×#3715—21no—
31#63823—22no—
32#63823C2214yes—
33C2×#3713—21no—
34#63651C2214yescomplemented
35C3×#823C2214yescomplemented
36#63577C2214yescomplemented
37C2×C3×#131—31no—
38C2×#951—31no—
39C2×#949—31no—
40C22×#214—31no—
41C2×#932—31no—
42C2×#909—31no—
43C2×#907D311yescharacteristic, complemented
44C22×#204—31no—
45C2×C3×#100—31no—
46#5141—38no—
47C2×#824—31no—
48C2×#824—31no—
49C2×#824—61no—
50C2×#823D311yescharacteristic, complemented
51C2×#822—31no—
52C2×#819—31no—
53#4748—38no—
54C23×C3×D3—21no—
55C22×C32×C4C2311yescharacteristic
56C22×C3⋊Dic3—21no—
57C2×#634—21no—
58C2×#634—21no—
59C2×#634—21no—
60C2×#634—21no—
61C2×#634—21no—
62C2×#634C2312yes—
63C2×C3×#67—21no—
64C3×#235—21no—
65C3×#235—21no—
66C3×#235D412yes—
67C3×#235C2312yes—
68C2×C3×D12—21no—
69C2×C3×D12—21no—
70C2×C3×D12—21no—
71C2×C3×D12—21no—
72C2×C3×D12—21no—
73C2×C3×D12D411yescharacteristic
74C2×C3×D12C2312yes—
75C2×C3×C4×D3—21no—
76C2×C3×Dic6D411yescharacteristic
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⋊C8D411yescharacteristic
88C2×C3×C3⋊C8D412yes—
89C2×C3×C3⋊C8C2312yes—
90C3×ΓL(1,𝔽25)—24no—
91C3×D3×C8—24no—
92#3719—24no—
93#3716—24no—
94#3716—24no—
95#3716—24no—
96#3715—24no—
97#3713—24no—
98C2×#471—91noSylow
99C2×C3×#41—31no—
100C22×C3×D4—31no—
101C2×#235C2×D311yescharacteristic
102C22×D12C2×D311yescharacteristic
103C22×Dic6—31no—
104C22×Dic6—31no—
105C22×Dic6—61no—
106C3×#131—34no—
107C3×#131—62no—
108C2×C3×M16—21no—
109C2×C3×M16—31no—
110C2×C3×M16—61no—
111C2×C3×M16C2×D311yescharacteristic
112C2×C3×M16C2×D311yescharacteristic
113C22×C3×C8—31no—
114C22×C3×C8—61no—
115#951—34no—
116#951—62no—
117#949—34no—
118#949—62no—
119C2×#216—61no—
120C2×#215—61no—
121C2×#214—34no—
122C2×#214—61no—
123C2×#214—61no—
124C2×#214—61no—
125C2×#214—61no—
126C2×#214—62no—
127#932—34no—
128#932—62no—
129C2×#213—61no—
130C2×#213—61no—
131C2×#208—31no—
132C2×#208—31no—
133C2×#208—61no—
134C2×#208C2×D311yescharacteristic
135C22×C3⋊C8C2×D311yescharacteristic
136#909—34no—
137#909—62no—
138#907—22no—
139#907C2×D314yescomplemented
140C2×Dic12—61no—
141C2×#204—34no—
142C2×#204—61no—
143C2×#204—61no—
144C2×#204—62no—
145C2×#204—61no—
146C2×ΓL(1,𝔽25)—21no—
147C2×D3×C8—21no—
148C3×#100—34no—
149#824—34no—
150#824—34no—
151#824—64no—
152#823C2×D314yescomplemented
153#822—34no—
154#819—34no—
155C23×C32#4111yescharacteristic
156C22×C3×D3—21no—
157C22×C3×D3—22no—
158C22×C3×D3—41no—
159C22×C3×D3—41no—
160C22×C3×D3—82no—
161C2×C32×C4C2×D411yescharacteristic
162C2×C32×C4C2×D411yescharacteristic
163C2×C32×C4C2×D411yescharacteristic
164C2×C32×C4C2×D411yescharacteristic
165C2×C32×C4#4111yescharacteristic
166C2×C32×C4C2411yescharacteristic
167C2×C3⋊Dic3—22no—
168C2×C3⋊Dic3—41no—
169C2×C3⋊Dic3—41no—
170#634—22no—
171#634—22no—
172#634—41no—
173#634—41no—
174C3×#67—22no—
175C3×#67—22no—
176C2×C3×Dic3—21no—
177C3×D12—21no—
178C3×D12—22no—
179C3×D12—22no—
180C3×D12—41no—
181C3×D12—41no—
182C3×D12C2×D412yes—
183C3×C4×D3—22no—
184C3×C4×D3—22no—
185C3×Dic6—21no—
186C3×Dic6C2×D412yes—
187C32×C8—22no—
188C3⋊(C3⋊C8)—22no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8C2×D412yes—
191C3×C3⋊C8C2×D412yes—
192C2×#137—91no—
193C2×AGL(1,ℤ/8)—91no—
194C22×SD16—91no—
195C2×#131—31no—
196#471—98no—
197C2×#108—91no—
198C2×#101—91no—
199C2×#100—31no—
200C24×C3—61no—
201C23×D3—21no—
202C3×#41—32no—
203C3×#41—32no—
204C3×#41—61no—
205C3×#41—61no—
206C2×C3×Q8—31no—
207C2×C3×D4—31no—
208C2×C3×D4—32no—
209C2×C3×D4—61no—
210C2×C3×D4—61no—
211C2×C3×D4—61no—
212C2×C3×D4—61no—
213C2×C3×D4—61no—
214C2×C3×D4—61no—
215C22×C3×C4—21no—
216C22×C3×C4—61no—
217C22×C3×C4C22×D311yescharacteristic
218C22×C3×C4C22×D311yescharacteristic
219C2×#67—21no—
220C22×Dic3—62no—
221C22×Dic3—61no—
222C22×Dic3—61no—
223#235—21no—
224#235—21no—
225#235D1212yes—
226#235C22×D312yes—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12D1211yescharacteristic
233C2×D12C22×D312yes—
234C2×C4×D3—21no—
235C2×Dic6—32no—
236C2×Dic6—32no—
237C2×Dic6—62no—
238C2×Dic6—61no—
239C2×Dic6—61no—
240C2×Dic6—61no—
241C2×Dic6—61no—
242C2×Dic6—61no—
243C2×Dic6—61no—
244C2×Dic6—61no—
245C2×Dic6—61no—
246C2×Dic6—62no—
247C2×Dic6—61no—
248C2×Dic6—61no—
249C2×Dic6—121no—
250C2×Dic6—121no—
251C2×Dic6—121no—
252C2×Dic6—121no—
253C2×Dic6D1211yescharacteristic
254C3×M16—21no—
255C3×M16—21no—
256C3×M16—22no—
257C3×M16—32no—
258C3×M16—41no—
259C3×M16—64no—
260C3×M16—61no—
261C3×M16C22×D312yes—
262C3×M16C22×D312yes—
263C2×C3×C8—21no—
264C2×C3×C8—21no—
265C2×C3×C8—31no—
266C2×C3×C8—31no—
267C2×C3×C8—32no—
268C2×C3×C8—32no—
269C2×C3×C8—41no—
270C2×C3×C8—64no—
271C2×C3×C8—61no—
272#216—64no—
273#215—64no—
274#214—64no—
275#214—64no—
276#214—64no—
277#214—64no—
278#213—64no—
279#213—64no—
280#208—21no—
281#208—32no—
282#208—32no—
283#208—62no—
284#208—61no—
285#208—61no—
286#208—121no—
287#208C22×D312yes—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—61no—
290C2×C3⋊C8—61no—
291C2×C3⋊C8—121no—
292C2×C3⋊C8#6711yescharacteristic
293C2×C3⋊C8#6711yescharacteristic
294C2×C3⋊C8#6712yes—
295C2×C3⋊C8C22×D312yes—
296Dic12—64no—
297#204—64no—
298#204—64no—
299#204—64no—
300ΓL(1,𝔽25)—24no—
301D3×C8—24no—
302C22×C32—22no—
303C22×C32#12111yescharacteristic
304C22×C32#12111yescharacteristic
305C22×C32C2×#4111yescharacteristic
306C2×C3×D3—22no—
307C2×C3×D3—42no—
308C2×C3×D3—42no—
309C2×C3×D3—82no—
310C2×C3×D3—82no—
311C2×C3×D3—82no—
312C32×C4#12111yescharacteristic
313C32×C4#12111yescharacteristic
314C32×C4C22×D411yescharacteristic
315C32×C4C22×D411yescharacteristic
316C3⋊Dic3—42no—
317C3×Dic3—22no—
318C2×#41—31no—
319C22×Q8—91no—
320C22×D4—31no—
321#137—94no—
322#137—182no—
323AGL(1,ℤ/8)—94no—
324AGL(1,ℤ/8)—182no—
325C2×Q16—181no—
326C2×SD16—94no—
327C2×SD16—181no—
328C2×SD16—181no—
329C2×SD16—182no—
330C2×SD16—181no—
331C2×SD16—181no—
332C2×D8—181no—
333#131—34no—
334#131—62no—
335C2×M16—31no—
336C2×M16—61no—
337C2×M16—91no—
338C2×M16D3211yescharacteristic
339C22×C8—31no—
340C22×C8—61no—
341#108—94no—
342#101—94no—
343#100—34no—
344C23×C3—21no—
345C23×C3—61no—
346C23×C3—62no—
347C23×C3—121no—
348C23×C3—121no—
349C23×C3—242no—
350C23×C3#23511yescharacteristic
351C23×C3#23511yescharacteristic
352C22×D3—21no—
353C22×D3—22no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—82no—
357C3×Q8—32no—
358C3×Q8—61no—
359C3×D4—32no—
360C3×D4—62no—
361C3×D4—62no—
362C3×D4—61no—
363C3×D4—62no—
364C3×D4—62no—
365C3×D4—121no—
366C3×D4—121no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—61no—
374C2×C3×C4—62no—
375C2×C3×C4—62no—
376C2×C3×C4C2×D1211yescharacteristic
377C2×C3×C4C2×D1211yescharacteristic
378C2×C3×C4#23511yescharacteristic
379C2×C3×C4#23511yescharacteristic
380C2×C3×C4D3×D411yescharacteristic
381C2×C3×C4D3×D411yescharacteristic
382C2×C3×C4D3×D411yescharacteristic
383C2×C3×C4D3×D411yescharacteristic
384C2×C3×C4C2×#6711yescharacteristic
385C2×C3×C4C2×#6711yescharacteristic
386C2×C3×C4C23×D311yescharacteristic
387C2×C3×C4C23×D311yescharacteristic
388#67—22no—
389#67—22no—
390C2×Dic3—21no—
391C2×Dic3—64no—
392C2×Dic3—62no—
393C2×Dic3—62no—
394C2×Dic3—122no—
395C2×Dic3—122no—
396C2×Dic3—121no—
397C2×Dic3—121no—
398C2×Dic3—121no—
399C2×Dic3—121no—
400D12—21no—
401D12—22no—
402D12—22no—
403D12—41no—
404D12—41no—
405D12C2×D1212yes—
406C4×D3—22no—
407C4×D3—22no—
408Dic6—21no—
409Dic6—62no—
410Dic6—62no—
411Dic6—62no—
412Dic6—62no—
413Dic6—122no—
414Dic6—122no—
415Dic6—122no—
416Dic6—122no—
417Dic6—121no—
418Dic6—121no—
419Dic6—121no—
420Dic6—121no—
421Dic6C2×D1212yes—
422C3×C8—22no—
423C3×C8—22no—
424C3×C8—32no—
425C3×C8—32no—
426C3×C8—42no—
427C3×C8—62no—
428C3⋊C8—22no—
429C3⋊C8—62no—
430C3⋊C8—62no—
431C3⋊C8—122no—
432C3⋊C8C2×#6712yes—
433C3⋊C8C2×#6712yes—
434C2×C32—21no—
435C2×C32—21no—
436C2×C32#47112yescomplemented
437C2×C32C2×#12111yescharacteristic
438C3×D3—42no—
439C3×D3—82no—
440C24—61no—
441#41—32no—
442#41—32no—
443#41—61no—
444#41—61no—
445C2×Q8—31no—
446C2×Q8—92no—
447C2×Q8—181no—
448C2×Q8—181no—
449C2×Q8—181no—
450C2×Q8—181no—
451C2×Q8—181no—
452C2×D4—32no—
453C2×D4—31no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—61no—
460C22×C4—61no—
461C22×C4—181no—
462C22×C4C2×D3211yescharacteristic
463Q16—184no—
464SD16—184no—
465SD16—184no—
466SD16—184no—
467SD16—184no—
468D8—184no—
469M16—21no—
470M16—32no—
471M16—64no—
472M16—61no—
473M16—92no—
474M16—181no—
475M16C2×D3212yes—
476C2×C8—21no—
477C2×C8—32no—
478C2×C8—31no—
479C2×C8—31no—
480C2×C8—32no—
481C2×C8—64no—
482C2×C8—61no—
483C2×C8—181no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—42no—
490C22×C3—62no—
491C22×C3—122no—
492C22×C3—122no—
493C22×C3—242no—
494C22×C3—242no—
495C22×C3—242no—
496C22×C3#88411yescharacteristic
497C22×C3#88411yescharacteristic
498C22×C3#93011yescharacteristic
499C22×C3#93011yescharacteristic
500C22×C3C2×#23511yescharacteristic
501C22×C3C2×#23511yescharacteristic
502C2×D3—22no—
503C2×D3—42no—
504C2×D3—42no—
505C2×D3—82no—
506C2×D3—82no—
507C2×D3—82no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—62no—
513C3×C4#88411yescharacteristic
514C3×C4#88411yescharacteristic
515C3×C4#93011yescharacteristic
516C3×C4#93011yescharacteristic
517C3×C4C22×D1211yescharacteristic
518C3×C4C2×D3×D411yescharacteristic
519C3×C4C2×D3×D411yescharacteristic
520C3×C4C22×#6711yescharacteristic
521Dic3—22no—
522Dic3—124no—
523Dic3—122no—
524Dic3—122no—
525C32C2×#47111yescharacteristic, complemented, Sylow
526C23—61no—
527C23—62no—
528C23—121no—
529C23—121no—
530C23—242no—
531C23#379811yescharacteristic
532Q8—32no—
533Q8—61no—
534Q8—182no—
535Q8—182no—
536Q8—361no—
537Q8—361no—
538D4—32no—
539D4—62no—
540D4—62no—
541D4—62no—
542D4—62no—
543D4—61no—
544D4—121no—
545D4—121no—
546C2×C4—61no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—182no—
550C2×C4—361no—
551C2×C4—361no—
552C2×C4#379811yescharacteristic
553C2×C4D3×D1211yescharacteristic
554C2×C4D3×D1211yescharacteristic
555C2×C4D3×#6711yescharacteristic
556C2×C4D3×#6711yescharacteristic
557C2×C4C22×D3211yescharacteristic
558C8—22no—
559C8—32no—
560C8—32no—
561C8—62no—
562C8—182no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—22no—
568C2×C3—21no—
569C2×C3—41no—
570C2×C3—41no—
571C2×C3—122no—
572C2×C3—242no—
573C2×C3#474812yescomplemented
574C2×C3#514112yescomplemented
575C2×C3C2×#88411yescharacteristic
576C2×C3C2×#93011yescharacteristic
577D3—42no—
578D3—82no—
579C22—22no—
580C22—62no—
581C22—122no—
582C22—122no—
583C22—242no—
584C22—242no—
585C22—242no—
586C22#6390611yescharacteristic
587C22#6390611yescharacteristic
588C22C2×#379811yescharacteristic
589C4—62no—
590C4—362no—
591C4#6390611yescharacteristic
592C4#6390611yescharacteristic
593C4C2×D3×D1211yescharacteristic
594C4C2×D3×#6711yescharacteristic
595C3—21no—
596C3C2×#474811yescharacteristic, complemented
597C3C2×#514111yescharacteristic, complemented
598C2—21no—
599C2—21no—
600C2—122no—
601C2—242no—
602C2#1059099612yesdirect product factor
603C2C2×#6390611yescharacteristic
subgroupG/H
centerC22#63906
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC22×C32#121
prime radicalC2×#63823C2
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×#907C2semidirect
2. C2×#63831C2semidirect
3. C22×#3716C2semidirect
5. C2×#63651C2semidirect
6. C2×C3×#823C2semidirect
7. C2×#63577C2semidirect
8. #10590996C2direct (N × H)
19. C3×#907C22semidirect
24. #63831C22semidirect
29. C2×#3716C22semidirect
34. #63651C22semidirect
35. C3×#823C22semidirect
36. #63577C22semidirect
43. C2×#907D3semidirect
50. C2×#823D3semidirect
139. #907C2×D3semidirect
152. #823C2×D3semidirect
436. C2×C32#471semidirect
525. C32C2×#471semidirect
573. C2×C3#4748semidirect
574. C2×C3#5141semidirect
596. C3C2×#4748semidirect
597. C3C2×#5141semidirect
602. C2#10590996direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9072
C2×#638312
C22×#37162
C2×#638232
C2×#636512
C2×C3×#8232
C2×#635772
#105909962
C2×#51413
C2×#47483