Finite Group Databasebeta

← C2×#13071 #494654

Subgroups — C2×#13071 #494654

Order 512.

1274 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#504C211yesmaximal, characteristic, complemented
2C2×#3067C211yesmaximal, characteristic
3C2×#2145C211yesmaximal, characteristic, complemented
4C22×#359C211yesmaximal, characteristic, complemented
5C2×#1814C211yesmaximal, characteristic, complemented
6#13071C218yesmaximal, direct product factor
7C2×#1388C211yesmaximal, characteristic, complemented
8C2×#1382C211yesmaximal, characteristic, complemented
9C2×#575—21no—
10C22×#135C2211yescharacteristic
11C22×#131C2211yescharacteristic
12C2×#504—22no—
13C2×#504—22no—
14C2×#504—22no—
15C2×#504—22no—
16C2×#504—22no—
17C2×#504C2214yescomplemented
18C22×M32—21no—
19C22×M32C2211yescharacteristic
20C23×C16—21no—
21C23×C16C2211yescharacteristic
22#3067—22no—
23#3067—22no—
24#3067C414yescomplemented
25#3067C2214yes—
26C2×#473C412yescomplemented
27C2×#472C412yescomplemented
28C2×#471—22no—
29C22×#108C2211yescharacteristic
30C22×#108C2211yescharacteristic
31C2×#427C2211yescharacteristic
32C2×#421—21no—
33C2×#413—21no—
34#2145—22no—
35#2145C2214yescomplemented
36C2×#360—22no—
37C2×#359—21no—
38C2×#359—21no—
39C2×#359—22no—
40C2×#359C2214yescomplemented
41#1814—22no—
42#1814C2214yescomplemented
43C2×#344—21no—
44#1388C2214yescomplemented
45#1382C2214yescomplemented
46C22×#41—21no—
47C22×#41D411yescharacteristic
48#575—24no—
49#575—24no—
50#575—42no—
51#575—42no—
52C2×#137—22no—
53C2×#137—41no—
54C2×AGL(1,ℤ/8)—22no—
55C2×AGL(1,ℤ/8)—41no—
56C2×#135—21no—
57C2×#135—21no—
58C2×#135—21no—
59C2×#135—21no—
60C2×#135—21no—
61C2×#135—21no—
62C2×#135—21no—
63C2×#135—21no—
64C2×#135—21no—
65C2×#135—21no—
66C2×#135—42no—
67C2×#135C2×C412yes—
68C2×#135C2312yes—
69C22×Q16C2×C411yescharacteristic
70C22×SD16—21no—
71C22×D8C2×C411yescharacteristic
72C2×#131—21no—
73C2×#131—21no—
74C2×#131—21no—
75C2×#131—21no—
76C2×#131—21no—
77C2×#131—21no—
78C2×#131C2×C412yes—
79C2×#131C2×C412yes—
80C2×#131D412yes—
81C2×#131D412yes—
82C2×#131D412yes—
83C2×#131C2312yes—
84C22×M16—21no—
85C22×M16—21no—
86C22×M16—21no—
87C22×M16D411yescharacteristic
88C23×C8—21no—
89C23×C8C2311yescharacteristic
90C2×#117—21no—
91#504—28no—
92#504—28no—
93#504—28no—
94#504—28no—
95C2×M32—24no—
96C2×M32—24no—
97C2×M32—21no—
98C2×M32—21no—
99C2×M32—21no—
100C2×M32—24no—
101C2×M32D412yes—
102C2×M32D412yes—
103C2×M32C2312yes—
104C22×C16—24no—
105C22×C16—24no—
106C22×C16—21no—
107C22×C16—21no—
108C22×C16—21no—
109C22×C16—21no—
110C22×C16—21no—
111C22×C16—24no—
112C22×C16D411yescharacteristic
113C22×C16D411yescharacteristic
114C22×C16D412yes—
115C22×C16C2312yes—
116#473—24no—
117#473C2×C418yescomplemented
118#472—24no—
119#472C2×C418yescomplemented
120#471—28no—
121#471—28no—
122C2×#108—21no—
123C2×#108—21no—
124C2×#108—21no—
125C2×#108—21no—
126C2×#108—21no—
127C2×#108—21no—
128C2×#108—21no—
129C2×#108—21no—
130C2×#108C2×C412yes—
131C2×#108C2×C412yes—
132C2×#108C2312yes—
133C2×#108C2312yes—
134#427—21no—
135#427—21no—
136#427—21no—
137#427C2312yes—
138C2×#107—21no—
139C2×#106—21no—
140#421—24no—
141#421—42no—
142C2×#104—42no—
143#413—24no—
144#413—24no—
145C2×#101—22no—
146C2×#100—22no—
147#360—28no—
148#359—24no—
149#359—24no—
150#344—24no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—41no—
160C2×#41—41no—
161C2×#41—41no—
162C2×#41—41no—
163C2×#41—41no—
164C2×#41—41no—
165C2×#41—41no—
166C2×#41—41no—
167C2×#41—41no—
168C2×#41—41no—
169C2×#41C22⋊C412yes—
170C2×#41C22⋊C412yes—
171C2×#41D812yes—
172C2×#41D812yes—
173C2×#41SD1612yes—
174C2×#41C2×D412yes—
175C22×Q8—21no—
176C22×Q8SD1611yescharacteristic
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—41no—
180C22×D4SD1611yescharacteristic
181C23×C4—21no—
182C23×C4—41no—
183C23×C4C2×D411yescharacteristic
184#137—28no—
185#137—44no—
186#137—44no—
187#137—44no—
188AGL(1,ℤ/8)—28no—
189AGL(1,ℤ/8)—44no—
190AGL(1,ℤ/8)—44no—
191AGL(1,ℤ/8)—44no—
192#135—22no—
193#135—22no—
194#135—22no—
195#135—22no—
196#135—22no—
197#135—22no—
198#135—22no—
199#135—22no—
200#135—48no—
201#135—48no—
202#135—41no—
203#135—41no—
204#135—41no—
205#135—41no—
206#135—41no—
207#135—41no—
208#135—41no—
209#135—41no—
210C2×Q16—21no—
211C2×Q16—21no—
212C2×Q16—21no—
213C2×Q16—21no—
214C2×Q16—21no—
215C2×Q16—42no—
216C2×Q16C22×C412yes—
217C2×SD16—22no—
218C2×SD16—22no—
219C2×SD16—42no—
220C2×SD16—42no—
221C2×SD16—41no—
222C2×SD16—41no—
223C2×SD16—41no—
224C2×SD16—41no—
225C2×D8—21no—
226C2×D8—21no—
227C2×D8—21no—
228C2×D8—21no—
229C2×D8—21no—
230C2×D8—42no—
231C2×D8C22×C412yes—
232#131—22no—
233#131—22no—
234#131—22no—
235#131—22no—
236#131—22no—
237#131—22no—
238#131—22no—
239#131—22no—
240#131—22no—
241#131—22no—
242#131—22no—
243#131—22no—
244#131C22⋊C414yes—
245#131C22⋊C414yes—
246#131C22×C414yes—
247#131C2×D414yes—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—22no—
252C2×M16—22no—
253C2×M16—22no—
254C2×M16—22no—
255C2×M16—22no—
256C2×M16—22no—
257C2×M16—22no—
258C2×M16—22no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—41no—
262C2×M16—41no—
263C2×M16—41no—
264C2×M16—41no—
265C2×M16—41no—
266C2×M16—41no—
267C2×M16—41no—
268C2×M16—41no—
269C2×M16C22⋊C412yes—
270C2×M16C22⋊C412yes—
271C2×M16C2×D412yes—
272C22×C8—21no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8—41no—
280C22×C8—41no—
281C22×C8C22×C411yescharacteristic
282C22×C8C22×C411yescharacteristic
283C22×C8C2×D411yescharacteristic
284C22×C8C2×D411yescharacteristic
285C22×C8C2×D411yescharacteristic
286C22×C8C2×D411yescharacteristic
287C22×C8C2×D411yescharacteristic
288C22×C8C2×D411yescharacteristic
289C22×C8C2×D411yescharacteristic
290C22×C8#4111yescharacteristic
291C22×C8#4111yescharacteristic
292C22×C8#4111yescharacteristic
293C22×C8#4111yescharacteristic
294C22×C8C2411yescharacteristic
295#117—22no—
296#117—41no—
297#117—41no—
298#117—41no—
299C2×C4⋊C4—41no—
300C2×C22⋊C4—41no—
301C2×C42—41no—
302M32—216no—
303M32—22no—
304M32—22no—
305M32—22no—
306M32C2×D414yes—
307C2×C16—216no—
308C2×C16—21no—
309C2×C16—21no—
310C2×C16—22no—
311C2×C16—22no—
312C2×C16—21no—
313C2×C16—21no—
314C2×C16—22no—
315C2×C16—22no—
316C2×C16—22no—
317C2×C16—22no—
318C2×C16—22no—
319C2×C16C2×D412yes—
320C2×C16C2×D412yes—
321#108—22no—
322#108—22no—
323#108—22no—
324#108—22no—
325#108—22no—
326#108—41no—
327#108—41no—
328#108C22×C414yes—
329#107—22no—
330#107—41no—
331#106—22no—
332#106—41no—
333#104—48no—
334#101—28no—
335#100—28no—
336C24—21no—
337C24—41no—
338C24#13511yescharacteristic
339#41—22no—
340#41—22no—
341#41—22no—
342#41—22no—
343#41—22no—
344#41—22no—
345#41—22no—
346#41—22no—
347#41—22no—
348#41—22no—
349#41—22no—
350#41—22no—
351#41—42no—
352#41—42no—
353#41—42no—
354#41—42no—
355#41—42no—
356#41—42no—
357#41—42no—
358#41—42no—
359#41—81no—
360#41—81no—
361#41—81no—
362#41—81no—
363#41—81no—
364#41—81no—
365#41—81no—
366#41—81no—
367#41#10214yes—
368#41#10214yes—
369#41C2×C22⋊C414yes—
370#41C2×D814yes—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—22no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8#10212yes—
381C2×Q8#10212yes—
382C2×Q8C2×SD1612yes—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—81no—
401C2×D4—81no—
402C2×D4—81no—
403C2×D4—81no—
404C2×D4#10212yes—
405C2×D4#10212yes—
406C2×D4C2×SD1612yes—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4#12111yescharacteristic
425C22×C4#12111yescharacteristic
426C22×C4#12111yescharacteristic
427C22×C4#12111yescharacteristic
428C22×C4C2×D811yescharacteristic
429C22×C4C2×D811yescharacteristic
430C22×C4C2×SD1611yescharacteristic
431C22×C4C2×SD1611yescharacteristic
432C22×C4#13511yescharacteristic
433C22×C4#13511yescharacteristic
434C22×C4#13511yescharacteristic
435C22×C4C22×D411yescharacteristic
436Q16—22no—
437Q16—22no—
438Q16—48no—
439Q16—41no—
440Q16—41no—
441SD16—48no—
442SD16—48no—
443SD16—42no—
444SD16—42no—
445SD16—42no—
446SD16—42no—
447D8—22no—
448D8—22no—
449D8—48no—
450D8—41no—
451D8—41no—
452M16—22no—
453M16—22no—
454M16—22no—
455M16—24no—
456M16—24no—
457M16—24no—
458M16—24no—
459M16—24no—
460M16—42no—
461M16—42no—
462M16—42no—
463M16—42no—
464M16—42no—
465M16—81no—
466M16—81no—
467M16C2×C22⋊C414yes—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—81no—
480C2×C8—81no—
481C2×C8C2×C22⋊C411yescharacteristic
482C2×C8C2×C22⋊C411yescharacteristic
483C2×C8C4×D411yescharacteristic
484C2×C8C4×D411yescharacteristic
485C2×C8C4×D411yescharacteristic
486C2×C8C4×D411yescharacteristic
487C2×C8C4×D411yescharacteristic
488C2×C8C4×D411yescharacteristic
489C2×C8C4×D411yescharacteristic
490C2×C8C4×D411yescharacteristic
491C2×C8#12011yescharacteristic
492C2×C8#12011yescharacteristic
493C2×C8#12011yescharacteristic
494C2×C8#12011yescharacteristic
495C2×C8#12111yescharacteristic
496C2×C8#12111yescharacteristic
497C2×C8#12111yescharacteristic
498C2×C8#12111yescharacteristic
499C2×C8#12311yescharacteristic
500C2×C8#12311yescharacteristic
501C2×C8#12311yescharacteristic
502C2×C8#12311yescharacteristic
503C2×C8C23×C411yescharacteristic
504C2×C8C22×D411yescharacteristic
505C2×C8C22×D411yescharacteristic
506C2×C8C22×D411yescharacteristic
507C2×C8C2×#4111yescharacteristic
508C2×C8C2×#4111yescharacteristic
509C4⋊C4—42no—
510C4⋊C4—81no—
511C22⋊C4—42no—
512C22⋊C4—81no—
513C42—42no—
514C42—81no—
515C16—22no—
516C16—24no—
517C16—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—42no—
522C23—44no—
523C23—42no—
524C23—81no—
525C23—81no—
526C23—162no—
527C23#41611yescharacteristic
528C23#41611yescharacteristic
529C23#46511yescharacteristic
530C23#46511yescharacteristic
531C23#46611yescharacteristic
532C23#46611yescharacteristic
533C23C2×#13511yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—22no—
537Q8—42no—
538Q8—42no—
539Q8—81no—
540Q8—81no—
541Q8C2×#10214yes—
542D4—24no—
543D4—24no—
544D4—24no—
545D4—24no—
546D4—22no—
547D4—22no—
548D4—22no—
549D4—42no—
550D4—42no—
551D4—82no—
552D4—82no—
553D4—82no—
554D4—82no—
555D4—81no—
556D4—81no—
557D4C2×#10214yes—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—42no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—162no—
574C2×C4#38611yescharacteristic
575C2×C4#38611yescharacteristic
576C2×C4#38611yescharacteristic
577C2×C4#38611yescharacteristic
578C2×C4C2×#10211yescharacteristic
579C2×C4C2×#10211yescharacteristic
580C2×C4C2×#10211yescharacteristic
581C2×C4C2×#10211yescharacteristic
582C2×C4#41611yescharacteristic
583C2×C4#41611yescharacteristic
584C2×C4#46511yescharacteristic
585C2×C4#46511yescharacteristic
586C2×C4#46511yescharacteristic
587C2×C4#46511yescharacteristic
588C2×C4#46511yescharacteristic
589C2×C4#46511yescharacteristic
590C2×C4#46611yescharacteristic
591C2×C4#46611yescharacteristic
592C2×C4#46611yescharacteristic
593C2×C4#46611yescharacteristic
594C2×C4#46611yescharacteristic
595C2×C4#46611yescharacteristic
596C2×C4C22×C22⋊C411yescharacteristic
597C2×C4C2×#12111yescharacteristic
598C2×C4C2×#12111yescharacteristic
599C2×C4C22×D811yescharacteristic
600C2×C4C22×SD1611yescharacteristic
601C2×C4C2×#13511yescharacteristic
602C8—24no—
603C8—44no—
604C8—82no—
605C8#38611yescharacteristic
606C8#38611yescharacteristic
607C8#38611yescharacteristic
608C8#38611yescharacteristic
609C8C2×C4×D411yescharacteristic
610C8C2×C4×D411yescharacteristic
611C8C2×#12011yescharacteristic
612C8C2×#12311yescharacteristic
613C22—22no—
614C22—22no—
615C22—22no—
616C22—24no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—82no—
625C22—162no—
626C22—162no—
627C22—162no—
628C22#189311yescharacteristic
629C22#189311yescharacteristic
630C22#189311yescharacteristic
631C22#189311yescharacteristic
632C22C2×#41611yescharacteristic
633C22C2×#46511yescharacteristic
634C22C2×#46611yescharacteristic
635C4—24no—
636C4—82no—
637C4—162no—
638C4#189311yescharacteristic
639C4#189311yescharacteristic
640C4#189311yescharacteristic
641C4#189311yescharacteristic
642C4C2×#38611yescharacteristic
643C4C22×#10211yescharacteristic
644C4C2×#46511yescharacteristic
645C4C2×#46611yescharacteristic
646C2—21no—
647C2—21no—
648C2—21no—
649C2—21no—
650C2—21no—
651C2—21no—
652C2—44no—
653C2—162no—
654C2#1307112yesdirect product factor
655C2C2×#189311yescharacteristic
subgroupG/H
centerC2×C4#466
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22#1893
prime radicalC2×#3067C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#131 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#504C2semidirect
3. C2×#2145C2semidirect
4. C22×#359C2semidirect
5. C2×#1814C2semidirect
6. #13071C2direct (N × H)
7. C2×#1388C2semidirect
8. C2×#1382C2semidirect
17. C2×#504C22semidirect
24. #3067C4semidirect
26. C2×#473C4semidirect
27. C2×#472C4semidirect
35. #2145C22semidirect
40. C2×#359C22semidirect
42. #1814C22semidirect
44. #1388C22semidirect
45. #1382C22semidirect
117. #473C2×C4semidirect
119. #472C2×C4semidirect
654. C2#13071direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5042
C2×#30672
C2×#21452
C22×#3592
C2×#18142
#130712
C2×#13882
C2×#13822