Finite Group Databasebeta

← C2×#31593 #7505151

Subgroups — C2×#31593 #7505151

Order 512.

1792 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2570C211yesmaximal, characteristic, complemented
2C2×#2447C211yesmaximal, characteristic
3#31593C2116yesmaximal, direct product factor
4C2×#2097C211yesmaximal, characteristic, complemented
5C2×#2092C211yesmaximal, characteristic
6C2×#2083C211yesmaximal, characteristic, complemented
7C2×#2074C211yesmaximal, characteristic, complemented
8C2×#2068C211yesmaximal, characteristic, complemented
9C2×#2042C211yesmaximal, characteristic
10C2×#1959C211yesmaximal, characteristic
11C2×#1944C211yesmaximal, characteristic, complemented
12C2×#1928C211yesmaximal, characteristic, complemented
13C2×#1925C211yesmaximal, characteristic
14C2×#1917C211yesmaximal, characteristic, complemented
15C2×#1862C211yesmaximal, characteristic
16C2×#1804C211yesmaximal, characteristic
17C22×#124C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#97C2211yescharacteristic
33#2570—24no—
34#2570C2218yescomplemented
35#2447—24no—
36#2447C2218yes—
37C2×#400—21no—
38C2×#398—21no—
39C2×#397—21no—
40C2×#397—21no—
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#387—21no—
48C2×#387—21no—
49C2×#385—21no—
50C2×#384C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#382C2211yescharacteristic
53C2×#381—21no—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56#2097C2218yescomplemented
57#2092C2218yes—
58#2083C2218yescomplemented
59#2074C2218yescomplemented
60#2068C2218yescomplemented
61#2042C2218yes—
62#1959C2218yes—
63#1944C2218yescomplemented
64#1928C2218yescomplemented
65#1925C2218yes—
66#1917C2218yescomplemented
67#1862C2218yes—
68#1804C2218yes—
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×#124—24no—
85C2×#124—22no—
86C2×#124—42no—
87C2×#124C2314yes—
88C2×C4×D4—24no—
89C2×C4×D4—22no—
90C2×C4×D4—42no—
91C2×C4×D4C2314yes—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4D411yescharacteristic
98C22×C4⋊C4D411yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C42—21no—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C2×#107—42no—
115C2×#107—42no—
116C2×#107—42no—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#107C2314yes—
120C2×#106—24no—
121C2×#106C2314yes—
122C2×C4⋊C8—42no—
123C2×C4⋊C8C2314yes—
124C2×#103—24no—
125C2×#103—42no—
126C2×#103—42no—
127C2×#103—42no—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#102—24no—
133C2×#102—24no—
134C2×#102—42no—
135C2×#102—42no—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#97—42no—
141C2×#97C2314yes—
142#400—28no—
143#398—28no—
144#397—28no—
145#397—28no—
146#391—22no—
147#391—22no—
148#391C2314yes—
149#391C2314yes—
150#390—22no—
151#390C2314yes—
152#389—22no—
153#389—22no—
154#389—22no—
155#389C2314yes—
156#389C2314yes—
157#389C2314yes—
158#387—28no—
159#387—28no—
160#385—28no—
161#384—22no—
162#384C2314yes—
163#382—22no—
164#382—22no—
165#382C2314yes—
166#382C2314yes—
167#381—28no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4D414yes—
170C4×C4⋊C4C2314yes—
171C4×C4⋊C4C2314yes—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95D411yescharacteristic
185C2×#95D411yescharacteristic
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25—21no—
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8C2412yes—
203C22×D4—21no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4#4114yes—
210C22×D4C2412yes—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—24no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—82no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2411yescharacteristic
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8#4114yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245#124—28no—
246#124—44no—
247#124—48no—
248C4×D4—28no—
249C4×D4—44no—
250C4×D4—48no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—82no—
279C2×C4⋊C4—82no—
280C2×C4⋊C4—82no—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—82no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—82no—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2412yes—
311C2×C42C2412yes—
312C2×C42C2412yes—
313#107—44no—
314#107—44no—
315#107—44no—
316#106—28no—
317C4⋊C8—44no—
318#103—28no—
319#103—44no—
320#103—44no—
321#103—44no—
322#102—28no—
323#102—28no—
324#102—44no—
325#102—44no—
326#97—44no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95C2×D414yes—
340#95C2×D414yes—
341C24—24no—
342C24—22no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24C22×D411yescharacteristic
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—44no—
363C2×D4—28no—
364C2×D4—24no—
365C2×D4—28no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—42no—
372C2×D4—44no—
373C2×D4—42no—
374C2×D4—44no—
375C2×D4C2×#4114yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—84no—
425C22×C4—816no—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4H(5,𝔽2)12yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C42−1+412yes—
438C22×C42−1+412yes—
439C22×C4C2511yescharacteristic
440C2×C8—28no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—44no—
453C2×C8—44no—
454C2×C8—44no—
455C2×C8C2×#4114yes—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—84no—
486C42—24no—
487C42—24no—
488C42—24no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—84no—
496C23—24no—
497C23—42no—
498C23—48no—
499C23—42no—
500C23—48no—
501C23—48no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23#53612yes—
511C23#54612yes—
512C23#55012yes—
513C23#57512yes—
514C23#57512yes—
515C23#57612yes—
516C23#57812yes—
517C23C23×D411yescharacteristic
518Q8—42no—
519Q8—42no—
520Q8—42no—
521Q8—42no—
522Q8—84no—
523D4—24no—
524D4—24no—
525D4—42no—
526D4—48no—
527D4—42no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—816no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4#53612yes—
582C2×C4#53914yes—
583C2×C4#53914yes—
584C2×C4#54612yes—
585C2×C4#55012yes—
586C2×C4#55214yes—
587C2×C4#56214yes—
588C2×C4C22×#4111yescharacteristic
589C2×C4C22×#4111yescharacteristic
590C2×C4C2×H(5,𝔽2)11yescharacteristic
591C2×C4C2×H(5,𝔽2)11yescharacteristic
592C2×C4C2×2−1+411yescharacteristic
593C2×C4C2×2−1+411yescharacteristic
594C8—24no—
595C8—42no—
596C8—42no—
597C8—42no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—816no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22#269414yes—
607C22#324014yes—
608C22#324314yes—
609C22#331514yes—
610C22#332214yes—
611C22#340214yes—
612C22#340314yes—
613C22C2×#53611yescharacteristic
614C22C2×#54611yescharacteristic
615C22C2×#55011yescharacteristic
616C22C2×#57511yescharacteristic
617C22C2×#57511yescharacteristic
618C22C2×#57611yescharacteristic
619C22C2×#57811yescharacteristic
620C4—42no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4#269414yes—
634C4C2×#53911yescharacteristic
635C4C2×#53911yescharacteristic
636C4C2×#55211yescharacteristic
637C4C2×#56211yescharacteristic
638C2—84no—
639C2#3159318yesdirect product factor
640C2C2×#269411yescharacteristic
641C2C2×#324011yescharacteristic
642C2C2×#324311yescharacteristic
643C2C2×#331511yescharacteristic
644C2C2×#332211yescharacteristic
645C2C2×#340211yescharacteristic
646C2C2×#340311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2570C2semidirect
3. #31593C2direct (N × H)
4. C2×#2097C2semidirect
6. C2×#2083C2semidirect
7. C2×#2074C2semidirect
8. C2×#2068C2semidirect
11. C2×#1944C2semidirect
12. C2×#1928C2semidirect
14. C2×#1917C2semidirect
34. #2570C22semidirect
56. #2097C22semidirect
58. #2083C22semidirect
59. #2074C22semidirect
60. #2068C22semidirect
63. #1944C22semidirect
64. #1928C22semidirect
66. #1917C22semidirect
639. C2#31593direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25702
C2×#24472
#315932
C2×#20972
C2×#20922
C2×#20832
C2×#20742
C2×#20682
C2×#20422
C2×#19592
C2×#19442
C2×#19282
C2×#19252
C2×#19172
C2×#18622
C2×#18042