Finite Group Databasebeta

← C2×#31875 #7506501

Subgroups — C2×#31875 #7506501

Order 512.

1928 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2508C211yesmaximal, characteristic, complemented
2C2×#2452C211yesmaximal, characteristic
3#31875C2116yesmaximal, direct product factor
4C2×#2087C211yesmaximal, characteristic, complemented
5C2×#2088C211yesmaximal, characteristic, complemented
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2067C211yesmaximal, characteristic
8C2×#2065C211yesmaximal, characteristic
9C2×#2057C211yesmaximal, characteristic, complemented
10C2×#1987C211yesmaximal, characteristic, complemented
11C2×#1942C211yesmaximal, characteristic, complemented
12C2×#1939C211yesmaximal, characteristic, complemented
13C2×#1934C211yesmaximal, characteristic
14C2×#1925C211yesmaximal, characteristic
15C2×#1862C211yesmaximal, characteristic
16C2×#1804C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C22×C4×D4C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2508—24no—
33#2508C2218yescomplemented
34#2452—24no—
35#2452C2218yes—
36C2×#400—21no—
37C2×#398—21no—
38C2×#396—21no—
39C2×#393—21no—
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#387—21no—
49C2×#385—21no—
50C2×#382—21no—
51C2×#382C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2088C2218yescomplemented
55#2087C2218yescomplemented
56#2085C2218yescomplemented
57#2067C2218yes—
58#2065C2218yes—
59#2057C2218yescomplemented
60#1987C2218yescomplemented
61#1942C2218yescomplemented
62#1939C2218yescomplemented
63#1934C2218yes—
64#1925C2218yes—
65#1862C2218yes—
66#1804C2218yes—
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8C2314yes—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—48no—
90C2×C4×D4—24no—
91C2×C4×D4—22no—
92C2×C4×D4—42no—
93C2×C4×D4C2314yes—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4D411yescharacteristic
102C22×C4⋊C4D411yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C42—21no—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C2×#107—24no—
120C2×#107—42no—
121C2×#107—42no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106—42no—
126C2×#106C2314yes—
127C2×C4⋊C8—42no—
128C2×C4⋊C8—42no—
129C2×C4⋊C8—42no—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2314yes—
133C2×#102—42no—
134C2×#102—42no—
135C2×#102—42no—
136C2×#102Q818yescomplemented
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C4×C8—42no—
142C2×C4×C8C2314yes—
143#400—28no—
144#398—28no—
145#396—28no—
146#393—28no—
147#390—22no—
148#390—22no—
149#390C2314yes—
150#390C2314yes—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389C2314yes—
158#389C2314yes—
159#389C2314yes—
160#389C2314yes—
161#389C2314yes—
162#389C2314yes—
163#387—28no—
164#385—28no—
165#382—22no—
166#382—28no—
167#382C2314yes—
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×#95D411yescharacteristic
182C2×#95D411yescharacteristic
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192#336C2314yes—
193C25—21no—
194C22×Q8—24no—
195C22×Q8—44no—
196C22×Q8—44no—
197C22×Q8—82no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4C2×Q814yes—
205C22×D4C2412yes—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—24no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—82no—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2411yescharacteristic
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8C2×Q814yes—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C22×C8C2412yes—
240C4⋊Q8—28no—
241C4⋊Q8—44no—
242C4⋊Q8—48no—
243C22⋊Q8—432no—
244C4×D4—28no—
245C4×D4—44no—
246C4×D4—48no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
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—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—82no—
280C2×C4⋊C4—82no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×Q814yes—
285C2×C4⋊C4#4114yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—82no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—44no—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2412yes—
315C2×C42C2412yes—
316C2×C42C2412yes—
317#107—28no—
318#107—44no—
319#107—44no—
320#106—44no—
321C4⋊C8—44no—
322C4⋊C8—44no—
323C4⋊C8—44no—
324#102—44no—
325#102—44no—
326#102—44no—
327#102C2×Q8116yescomplemented
328C4×C8—44no—
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#95C2×D414yes—
339#95C2×D414yes—
340C24—24no—
341C24—22no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24C22×D411yescharacteristic
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—84no—
352C2×Q8—816no—
353C2×D4—28no—
354C2×D4—24no—
355C2×D4—28no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—42no—
362C2×D4—44no—
363C2×D4—42no—
364C2×D4—44no—
365C2×D4C22×Q814yes—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
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—48no—
392C22×C4—48no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
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—84no—
413C22×C4—816no—
414C22×C4#13514yes—
415C22×C4AGL(1,ℤ/8)14yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×Q812yes—
424C22×C4C2×#4112yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C42−1+412yes—
428C22×C42−1+412yes—
429C22×C4C2511yescharacteristic
430C2×C8—28no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—44no—
443C2×C8—44no—
444C2×C8—44no—
445C2×C8C22×Q814yes—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C4⋊C4C22×Q814yes—
475C4⋊C4C2×#4114yes—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—84no—
481C42—24no—
482C42—24no—
483C42—24no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—48no—
491C23—24no—
492C23—48no—
493C23—48no—
494C23—42no—
495C23—42no—
496C23—48no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—84no—
502C23—84no—
503C23—84no—
504C23—84no—
505C23#53712yes—
506C23#54612yes—
507C23D4×Q812yes—
508C23C2×#13512yes—
509C23C2×AGL(1,ℤ/8)12yes—
510C23#57612yes—
511C23#57812yes—
512C23C23×D411yescharacteristic
513Q8—816no—
514D4—24no—
515D4—24no—
516D4—42no—
517D4—48no—
518D4—42no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
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—48no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
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—816no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4#53712yes—
571C2×C4#54114yes—
572C2×C4#54414yes—
573C2×C4#54612yes—
574C2×C4D4×Q812yes—
575C2×C4#55414yes—
576C2×C4#56214yes—
577C2×C4C2×#13514yes—
578C2×C4C2×AGL(1,ℤ/8)14yes—
579C2×C4C23×Q811yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4C2×H(5,𝔽2)11yescharacteristic
582C2×C4C2×H(5,𝔽2)11yescharacteristic
583C2×C4C2×2−1+411yescharacteristic
584C2×C4C2×2−1+411yescharacteristic
585C8—24no—
586C8—42no—
587C8—42no—
588C8—42no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—816no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22#270414yes—
598C22#324714yes—
599C22#326214yes—
600C22#330014yes—
601C22#332514yes—
602C22#338014yes—
603C22#338614yes—
604C22C2×#53711yescharacteristic
605C22C2×#54611yescharacteristic
606C22C2×D4×Q811yescharacteristic
607C22C22×#13511yescharacteristic
608C22C22×AGL(1,ℤ/8)11yescharacteristic
609C22C2×#57611yescharacteristic
610C22C2×#57811yescharacteristic
611C4—24no—
612C4—24no—
613C4—44no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—44no—
618C4—42no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4#270414yes—
625C4C2×#54111yescharacteristic
626C4C2×#54411yescharacteristic
627C4C2×#55411yescharacteristic
628C4C2×#56211yescharacteristic
629C2—84no—
630C2#3187518yesdirect product factor
631C2C2×#270411yescharacteristic
632C2C2×#324711yescharacteristic
633C2C2×#326211yescharacteristic
634C2C2×#330011yescharacteristic
635C2C2×#332511yescharacteristic
636C2C2×#338011yescharacteristic
637C2C2×#338611yescharacteristic
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×#2508C2semidirect
3. #31875C2direct (N × H)
4. C2×#2087C2semidirect
5. C2×#2088C2semidirect
6. C2×#2085C2semidirect
9. C2×#2057C2semidirect
10. C2×#1987C2semidirect
11. C2×#1942C2semidirect
12. C2×#1939C2semidirect
33. #2508C22semidirect
54. #2088C22semidirect
55. #2087C22semidirect
56. #2085C22semidirect
59. #2057C22semidirect
60. #1987C22semidirect
61. #1942C22semidirect
62. #1939C22semidirect
136. C2×#102Q8semidirect
327. #102C2×Q8semidirect
630. C2#31875direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25082
C2×#24522
#318752
C2×#20872
C2×#20882
C2×#20852
C2×#20672
C2×#20652
C2×#20572
C2×#19872
C2×#19422
C2×#19392
C2×#19342
C2×#19252
C2×#18622
C2×#18042