Finite Group Databasebeta

← C2×#70218 #10797113

Subgroups — C2×#70218 #10797113

Order 768.

1826 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#932C211yesmaximal, characteristic
2C22×#901C211yesmaximal, characteristic, complemented
3C22×C3×#102C211yesmaximal, characteristic, complemented
4C22×#833C211yesmaximal, characteristic, complemented
5C2×#4966C211yesmaximal, characteristic, complemented
6C2×#4552C211yesmaximal, characteristic, complemented
7C2×#4530C211yesmaximal, characteristic, complemented
8#70218C218yesmaximal, direct product factor
9C2×#1907—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C22×#235C2211yescharacteristic
12C23×C3×C8C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×#214—21no—
15C2×#932—24no—
16C2×#932—22no—
17C2×#932C418yescomplemented
18C2×#932C2214yes—
19C22×#213—21no—
20C22×#208C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C2×#901—24no—
24C2×#901—22no—
25C2×#901—24no—
26C2×#901C2214yescomplemented
27C22×ΓL(1,𝔽25)—21no—
28C22×D3×C8—21no—
29C2×C3×#102—21no—
30C2×C3×#102—21no—
31C2×C3×#102—22no—
32C2×C3×#102C2214yescomplemented
33C2×#833—24no—
34C2×#833C2214yescomplemented
35C2×#810—21no—
36#4966—22no—
37#4966C2214yescomplemented
38C2×#809—21no—
39#4552C2214yescomplemented
40#4530C2214yescomplemented
41C22×AGL(1,ℤ/8)—31no—
42C22×#131—31no—
43C2×#417—31no—
44C22×#102D311yescharacteristic, complemented
45C22×#100—31no—
46#1907—38no—
47C2×#343—31no—
48C2×#341—31no—
49C25×C3—21no—
50C22×C3×D4—21no—
51C22×C3×D4—22no—
52C22×C3×D4—21no—
53C22×C3×D4—21no—
54C22×C3×D4—21no—
55C22×C3×D4—22no—
56C22×C3×D4—22no—
57C22×C3×D4—21no—
58C22×C3×D4C2×C414yes—
59C22×C3×D4C2312yes—
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C2×#235—22no—
64C2×#235—21no—
65C2×#235—22no—
66C2×#235—21no—
67C2×#235C2×C414yes—
68C2×#235D414yes—
69C2×#235D412yes—
70C2×#235C2312yes—
71C22×D12D411yescharacteristic
72C22×C4×D3—21no—
73C22×Dic6D411yescharacteristic
74C22×C3×C8—22no—
75C22×C3×C8—22no—
76C22×C3×C8D411yescharacteristic
77C22×C3×C8D411yescharacteristic
78C22×C3×C8D412yes—
79C22×C3×C8C2312yes—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2312yes—
86C2×#214—28no—
87C2×#214—24no—
88#932—216no—
89#932—28no—
90#932C2×C4116yescomplemented
91C2×#213—28no—
92C2×#213—24no—
93C2×#208—22no—
94C2×#208—21no—
95C2×#208—22no—
96C2×#208—21no—
97C2×#208—21no—
98C2×#208—22no—
99C2×#208—22no—
100C2×#208C2×C414yes—
101C2×#208D414yes—
102C2×#208C2312yes—
103C2×#208C2312yes—
104C2×#208C2312yes—
105C22×C3⋊C8—21no—
106C22×C3⋊C8—21no—
107C22×C3⋊C8—21no—
108#901—216no—
109#901—28no—
110#901—28no—
111C2×ΓL(1,𝔽25)—28no—
112C2×ΓL(1,𝔽25)—24no—
113C2×D3×C8—28no—
114C2×D3×C8—24no—
115C3×#102—24no—
116C3×#102—24no—
117#833—28no—
118#810—24no—
119#809—24no—
120C22×#41—31no—
121C23×D4C2×D311yescharacteristic
122C2×AGL(1,ℤ/8)—38no—
123C2×AGL(1,ℤ/8)—34no—
124C2×AGL(1,ℤ/8)—64no—
125C2×AGL(1,ℤ/8)—62no—
126C22×SD16—61no—
127C22×D8—61no—
128C2×#131—34no—
129C2×#131—64no—
130C2×#131—64no—
131C2×#131—62no—
132C22×M16—31no—
133C22×M16—31no—
134C22×M16—31no—
135C22×M16—61no—
136C23×C8—61no—
137C23×C8C2×D311yescharacteristic
138C22×C4⋊C4C2×D311yescharacteristic
139#417—34no—
140#417—62no—
141C2×#103—61no—
142C2×#102—21no—
143C2×#102—21no—
144C2×#102—22no—
145C2×#102—61no—
146C2×#102C2×D314yescomplemented
147C2×#100—34no—
148C2×#100—64no—
149#343—34no—
150#341—34no—
151C24×C3—24no—
152C24×C3—22no—
153C24×C3—41no—
154C24×C3—41no—
155C24×C3—42no—
156C24×C3—82no—
157C24×C3—82no—
158C24×C3#4111yescharacteristic
159C23×D3—21no—
160C2×C3×D4—24no—
161C2×C3×D4—24no—
162C2×C3×D4—24no—
163C2×C3×D4—24no—
164C2×C3×D4—24no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—42no—
171C2×C3×D4—42no—
172C2×C3×D4—42no—
173C2×C3×D4—42no—
174C2×C3×D4—41no—
175C2×C3×D4—41no—
176C2×C3×D4—42no—
177C2×C3×D4C22×C414yes—
178C22×C3×C4—22no—
179C22×C3×C4—41no—
180C22×C3×C4—41no—
181C22×C3×C4—82no—
182C22×C3×C4C22×C412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4#4112yes—
190C22×C3×C4#4111yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C2×#67—24no—
193C2×#67—22no—
194C2×#67—24no—
195C2×#67—22no—
196C22×Dic3—21no—
197#235—24no—
198#235—24no—
199#235—24no—
200#235—24no—
201#235—24no—
202#235—22no—
203#235—22no—
204#235C22⋊C418yes—
205#235C22×C414yes—
206#235C2×D414yes—
207C2×D12—22no—
208C2×D12—21no—
209C2×D12C22⋊C414yes—
210C2×D12C2×D412yes—
211C2×C4×D3—24no—
212C2×C4×D3—24no—
213C2×C4×D3—22no—
214C2×C4×D3—22no—
215C2×Dic6—22no—
216C2×Dic6—21no—
217C2×Dic6C22⋊C414yes—
218C2×Dic6C2×D412yes—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—24no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8C2×D412yes—
225C2×C3×C8C2×D412yes—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—41no—
229C3×C4⋊C4—41no—
230#214—216no—
231#213—216no—
232#208—24no—
233#208—24no—
234#208—22no—
235#208—22no—
236#208—22no—
237#208—22no—
238#208—42no—
239#208C22×C414yes—
240#208C2×D414yes—
241C2×C3⋊C8—24no—
242C2×C3⋊C8—24no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—22no—
246C2×C3⋊C8—42no—
247ΓL(1,𝔽25)—216no—
248D3×C8—216no—
249C25—21no—
250C2×#41—34no—
251C2×#41—34no—
252C2×#41—32no—
253C2×#41—32no—
254C2×#41—62no—
255C2×#41—62no—
256C2×#41—61no—
257C2×#41—61no—
258C22×Q8—31no—
259C22×D4—22no—
260C22×D4—21no—
261C22×D4—22no—
262C22×D4—21no—
263C22×D4—22no—
264C22×D4—21no—
265C22×D4—21no—
266C22×D4—21no—
267C22×D4—31no—
268C22×D4—61no—
269C22×D4C4×D314yes—
270C22×D4C22×D312yes—
271C23×C4—21no—
272C23×C4—61no—
273C23×C4C22×D311yescharacteristic
274AGL(1,ℤ/8)—316no—
275AGL(1,ℤ/8)—616no—
276AGL(1,ℤ/8)—68no—
277C2×SD16—68no—
278C2×SD16—64no—
279C2×D8—68no—
280C2×D8—64no—
281#131—68no—
282#131—68no—
283#131—616no—
284C2×M16—34no—
285C2×M16—34no—
286C2×M16—32no—
287C2×M16—32no—
288C2×M16—32no—
289C2×M16—68no—
290C2×M16—62no—
291C2×M16—62no—
292C2×M16—61no—
293C2×M16—61no—
294C2×M16—61no—
295C2×M16—64no—
296C2×M16—62no—
297C2×M16—62no—
298C22×C8—22no—
299C22×C8—22no—
300C22×C8—68no—
301C22×C8—61no—
302C22×C8—61no—
303C22×C8—64no—
304C22×C8—61no—
305C22×C8#6711yescharacteristic
306C22×C8#6711yescharacteristic
307C22×C8#6712yes—
308C22×C8C22×D312yes—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4C22×D312yes—
315#103—64no—
316#102—24no—
317#102—24no—
318#102—64no—
319#100—68no—
320C23×C3—22no—
321C23×C3—22no—
322C23×C3—24no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—44no—
326C23×C3—44no—
327C23×C3—88no—
328C23×C3—82no—
329C23×C3—82no—
330C23×C3—88no—
331C23×C3—82no—
332C23×C3—82no—
333C23×C3—82no—
334C23×C3—82no—
335C23×C3C4×D412yes—
336C23×C3#12111yescharacteristic
337C23×C3#12111yescharacteristic
338C23×C3#12312yes—
339C23×C3C2×#4111yescharacteristic
340C22×D3—22no—
341C22×D3—24no—
342C22×D3—42no—
343C22×D3—42no—
344C3×D4—24no—
345C3×D4—24no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—44no—
349C3×D4—44no—
350C2×C3×C4—42no—
351C2×C3×C4—82no—
352C2×C3×C4—82no—
353C2×C3×C4—82no—
354C2×C3×C4C2×C22⋊C412yes—
355C2×C3×C4C2×C22⋊C412yes—
356C2×C3×C4C4×D412yes—
357C2×C3×C4C4×D412yes—
358C2×C3×C4C4×D412yes—
359C2×C3×C4#12012yes—
360C2×C3×C4#12012yes—
361C2×C3×C4#12112yes—
362C2×C3×C4#12112yes—
363C2×C3×C4#12111yescharacteristic
364C2×C3×C4#12111yescharacteristic
365C2×C3×C4#12312yes—
366C2×C3×C4C23×C411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372#67—24no—
373#67—24no—
374#67—28no—
375C2×Dic3—22no—
376C2×Dic3—24no—
377D12—22no—
378D12—24no—
379D12C2×C22⋊C414yes—
380C4×D3—24no—
381C4×D3—24no—
382C4×D3—28no—
383Dic6—22no—
384Dic6—24no—
385Dic6C2×C22⋊C414yes—
386C3×C8—22no—
387C3×C8—22no—
388C3⋊C8—24no—
389C3⋊C8—24no—
390C3⋊C8—42no—
391C24—24no—
392C24—22no—
393C24—42no—
394C24—41no—
395C24—41no—
396C24—61no—
397C24—82no—
398C24—82no—
399C24#23711yescharacteristic
400#41—38no—
401#41—34no—
402#41—34no—
403#41—64no—
404#41—64no—
405#41—64no—
406#41—64no—
407#41—64no—
408#41—62no—
409#41—62no—
410C2×Q8—34no—
411C2×Q8—32no—
412C2×Q8—62no—
413C2×Q8—61no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—34no—
425C2×D4—32no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—41no—
432C2×D4—41no—
433C2×D4—64no—
434C2×D4—64no—
435C2×D4—62no—
436C2×D4—62no—
437C2×D4—62no—
438C2×D4—61no—
439C2×D4C2×C4×D314yes—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—61no—
444C22×C4—64no—
445C22×C4—64no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—82no—
449C22×C4C2×C4×D312yes—
450C22×C4#23512yes—
451C22×C4D3×D412yes—
452C22×C4D3×D411yescharacteristic
453C22×C4D3×D411yescharacteristic
454C22×C4D3×D412yes—
455C22×C4#23711yescharacteristic
456C22×C4C2×#6711yescharacteristic
457C22×C4C2×#6711yescharacteristic
458C22×C4C23×D311yescharacteristic
459SD16—616no—
460D8—616no—
461M16—34no—
462M16—34no—
463M16—616no—
464M16—64no—
465M16—64no—
466M16—62no—
467M16—62no—
468M16—62no—
469M16—62no—
470M16—122no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—616no—
477C2×C8—64no—
478C2×C8—64no—
479C2×C8—62no—
480C2×C8—62no—
481C2×C8—62no—
482C2×C8—122no—
483C2×C8C2×#6712yes—
484C2×C8C2×#6712yes—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C22×C3—24no—
490C22×C3—24no—
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—44no—
496C22×C3—84no—
497C22×C3—84no—
498C22×C3—88no—
499C22×C3—88no—
500C22×C3—82no—
501C22×C3—82no—
502C22×C3#38612yes—
503C22×C3#38612yes—
504C22×C3#47112yes—
505C22×C3#47212yes—
506C22×C3C2×C4×D411yescharacteristic
507C22×C3C2×#12111yescharacteristic
508C22×C3C2×#12311yescharacteristic
509C2×D3—24no—
510C2×D3—48no—
511C2×D3—42no—
512C2×D3—42no—
513C3×C4—82no—
514C3×C4#38612yes—
515C3×C4#38612yes—
516C3×C4C22×C22⋊C411yescharacteristic
517C3×C4C2×C4×D411yescharacteristic
518C3×C4C2×#12011yescharacteristic
519C3×C4C2×#12111yescharacteristic
520Dic3—24no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—44no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—64no—
529C23—62no—
530C23—88no—
531C23—82no—
532C23—82no—
533C23—88no—
534C23—82no—
535C23—82no—
536C23—82no—
537C23—82no—
538C23—122no—
539C23—122no—
540C23#67⋊C412yes—
541C23#88312yes—
542C23#93811yescharacteristic
543C23#93811yescharacteristic
544C23C2×#23711yescharacteristic
545Q8—34no—
546Q8—64no—
547Q8—62no—
548D4—24no—
549D4—24no—
550D4—34no—
551D4—44no—
552D4—44no—
553D4—42no—
554D4—42no—
555D4—68no—
556D4—64no—
557D4—64no—
558D4—64no—
559D4—62no—
560C2×C4—42no—
561C2×C4—64no—
562C2×C4—62no—
563C2×C4—68no—
564C2×C4—64no—
565C2×C4—64no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4D3×C22⋊C412yes—
570C2×C4D3×C22⋊C412yes—
571C2×C4#67⋊C412yes—
572C2×C4#88312yes—
573C2×C4#88412yes—
574C2×C4#88412yes—
575C2×C4C4×#6712yes—
576C2×C4C4×#6712yes—
577C2×C4#93712yes—
578C2×C4#93712yes—
579C2×C4#93811yescharacteristic
580C2×C4#93811yescharacteristic
581C2×C4C22×C4×D311yescharacteristic
582C2×C4C2×#23511yescharacteristic
583C2×C4C2×D3×D411yescharacteristic
584C2×C4C2×D3×D411yescharacteristic
585C2×C4C2×D3×D411yescharacteristic
586C2×C4C22×#6711yescharacteristic
587C8—22no—
588C8—22no—
589C8—64no—
590C8—64no—
591C8—122no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—22no—
595C2×C3—84no—
596C2×C3#190714yescomplemented
597C2×C3C2×#38611yescharacteristic
598C2×C3C2×#47111yescharacteristic
599C2×C3C2×#47211yescharacteristic
600D3—44no—
601C22—24no—
602C22—24no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—44no—
608C22—64no—
609C22—84no—
610C22—88no—
611C22—84no—
612C22—88no—
613C22—82no—
614C22—82no—
615C22—128no—
616C22—122no—
617C22—122no—
618C22#495612yes—
619C22#495612yes—
620C22#516012yes—
621C22#517712yes—
622C22C2×#67⋊C411yescharacteristic
623C22C2×#88311yescharacteristic
624C22C2×#93811yescharacteristic
625C4—64no—
626C4—82no—
627C4#495612yes—
628C4#495612yes—
629C4C2×D3×C22⋊C411yescharacteristic
630C4C2×#88411yescharacteristic
631C4C2×C4×#6711yescharacteristic
632C4C2×#93711yescharacteristic
633C3C2×#190711yescharacteristic, complemented, Sylow
634C2—22no—
635C2—21no—
636C2—21no—
637C2—84no—
638C2—124no—
639C2#7021814yesdirect product factor
640C2C2×#495611yescharacteristic
641C2C2×#516011yescharacteristic
642C2C2×#517711yescharacteristic
subgroupG/H
centerC23#938
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC22×C3×#102C2
socleC23×C3#121
prime radicalC22×#932C2
cosocleC22×C3×C4C24
hypercenterC22×#102D3
hypocenterC3C2×#1907
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C22×#901C2semidirect
3. C22×C3×#102C2semidirect
4. C22×#833C2semidirect
5. C2×#4966C2semidirect
6. C2×#4552C2semidirect
7. C2×#4530C2semidirect
8. #70218C2direct (N × H)
17. C2×#932C4semidirect
26. C2×#901C22semidirect
32. C2×C3×#102C22semidirect
34. C2×#833C22semidirect
37. #4966C22semidirect
39. #4552C22semidirect
40. #4530C22semidirect
44. C22×#102D3semidirect
90. #932C2×C4semidirect
146. C2×#102C2×D3semidirect
596. C2×C3#1907semidirect
633. C3C2×#1907semidirect
639. C2#70218direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9322
C22×#9012
C22×C3×#1022
C22×#8332
C2×#49662
C2×#45522
C2×#45302
#702182
C2×#19073