Finite Group Databasebeta

← C2×#32585 #7510385

Subgroups — C2×#32585 #7510385

Order 512.

1944 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×#2538C211yesmaximal, characteristic
2C2×#2584C211yesmaximal, characteristic, complemented
3#32585C2116yesmaximal, direct product factor
4C2×#2099C211yesmaximal, characteristic, complemented
5C2×#2088C211yesmaximal, characteristic, complemented
6C2×#2084C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic
8C2×#2059C211yesmaximal, characteristic, complemented
9C2×#2025C211yesmaximal, characteristic
10C2×#1985C211yesmaximal, characteristic, complemented
11C2×#1944C211yesmaximal, characteristic, complemented
12C2×#1936C211yesmaximal, characteristic
13C2×#1926C211yesmaximal, characteristic
14C2×#1920C211yesmaximal, characteristic, complemented
15C2×#1864C211yesmaximal, characteristic
16C2×#1862C211yesmaximal, characteristic
17C22×#125C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#97C2211yescharacteristic
31#2584—24no—
32#2584C2218yescomplemented
33#2538—24no—
34#2538C2218yes—
35C2×#398—21no—
36C2×#396—21no—
37C2×#396—21no—
38C2×#395—21no—
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#389—21no—
42C2×#389—21no—
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C2×#380—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2099C2218yescomplemented
53#2088C2218yescomplemented
54#2084C2218yescomplemented
55#2065C2218yes—
56#2059C2218yescomplemented
57#2025C2218yes—
58#1985C2218yescomplemented
59#1944C2218yescomplemented
60#1936C2218yes—
61#1926C2218yes—
62#1920C2218yescomplemented
63#1864C2218yes—
64#1862C2218yes—
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8D411yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×#125—22no—
85C2×#125—42no—
86C2×#125—42no—
87C2×#125C2314yes—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—48no—
90C2×C4×Q8—42no—
91C2×C4×Q8—42no—
92C2×C4×Q8D414yes—
93C2×C4×Q8C2314yes—
94C2×C4×D4—22no—
95C2×C4×D4—42no—
96C2×C4×D4—42no—
97C2×C4×D4C2314yes—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C2×#107—42no—
128C2×#107C2314yes—
129C2×#106—24no—
130C2×#106—24no—
131C2×#106—42no—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8—42no—
136C2×C4⋊C8C2314yes—
137C2×#102—24no—
138C2×#102—42no—
139C2×#102—42no—
140C2×#102Q818yescomplemented
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#97—42no—
146C2×#97C2314yes—
147#398—28no—
148#396—28no—
149#396—28no—
150#395—28no—
151#390—22no—
152#390—22no—
153#390C2314yes—
154#390C2314yes—
155#389—28no—
156#389—28no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389C2314yes—
164#389C2314yes—
165#389C2314yes—
166#389C2314yes—
167#389C2314yes—
168#389C2314yes—
169#384—22no—
170#384C2314yes—
171#380—28no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4C2314yes—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2314yes—
193C25—21no—
194C22×Q8—24no—
195C22×Q8—42no—
196C22×Q8—42no—
197C22×Q8C2×D414yes—
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—44no—
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—
240#125—44no—
241#125—44no—
242#125—48no—
243C22⋊Q8—432no—
244C4×Q8—48no—
245C4×Q8—44no—
246C4×Q8—44no—
247C4×D4—44no—
248C4×D4—44no—
249C4×D4—48no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
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—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—82no—
290C2×C4⋊C4—82no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D414yes—
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2×Q814yes—
296C2×C4⋊C4#4114yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—82no—
313C2×C22⋊C4—82no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—82no—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2412yes—
326C2×C42C2412yes—
327C2×C42C2412yes—
328#107—44no—
329#106—28no—
330#106—28no—
331#106—44no—
332C4⋊C8—44no—
333#102—28no—
334#102—44no—
335#102—44no—
336#102C2×Q8116yescomplemented
337#97—44no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345C24—24no—
346C24—22no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24C22×D411yescharacteristic
354C2×Q8—28no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—416no—
358C2×D4—28no—
359C2×D4—24no—
360C2×D4—28no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—42no—
367C2×D4—44no—
368C2×D4—42no—
369C2×D4—44no—
370C2×D4C22×Q814yes—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
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—48no—
398C22×C4—48no—
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—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—48no—
419C22×C4—816no—
420C22×C4AGL(1,ℤ/8)14yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×Q812yes—
429C22×C4C2×#4112yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C42−1+412yes—
433C22×C42−1+412yes—
434C22×C4C2511yescharacteristic
435C2×C8—28no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—44no—
448C2×C8—44no—
449C2×C8—44no—
450C2×C8C22×Q814yes—
451C4⋊C4—24no—
452C4⋊C4—24no—
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—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4C22×Q814yes—
484C4⋊C4C2×#4114yes—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—84no—
489C22⋊C4—84no—
490C42—24no—
491C42—24no—
492C42—24no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—84no—
500C23—24no—
501C23—48no—
502C23—48no—
503C23—42no—
504C23—42no—
505C23—48no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—84no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23#53412yes—
515C23#54812yes—
516C23D4×Q812yes—
517C23C2×AGL(1,ℤ/8)12yes—
518C23#57512yes—
519C23#57712yes—
520C23#57712yes—
521C23C23×D411yescharacteristic
522Q8—416no—
523D4—24no—
524D4—24no—
525D4—42no—
526D4—48no—
527D4—42no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
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—22no—
547C2×C4—22no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
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—42no—
574C2×C4—816no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4#53412yes—
581C2×C4#54214yes—
582C2×C4#54414yes—
583C2×C4#54812yes—
584C2×C4D4×Q812yes—
585C2×C4#55414yes—
586C2×C4#56314yes—
587C2×C4C2×AGL(1,ℤ/8)14yes—
588C2×C4C23×Q811yescharacteristic
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—82no—
602C22—816no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22#272914yes—
607C22#319414yes—
608C22#320414yes—
609C22#336814yes—
610C22#337614yes—
611C22#338414yes—
612C22#338814yes—
613C22C2×#53411yescharacteristic
614C22C2×#54811yescharacteristic
615C22C2×D4×Q811yescharacteristic
616C22C22×AGL(1,ℤ/8)11yescharacteristic
617C22C2×#57511yescharacteristic
618C22C2×#57711yescharacteristic
619C22C2×#57711yescharacteristic
620C4—24no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—44no—
627C4—44no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4#272914yes—
634C4C2×#54211yescharacteristic
635C4C2×#54411yescharacteristic
636C4C2×#55411yescharacteristic
637C4C2×#56311yescharacteristic
638C2—84no—
639C2#3258518yesdirect product factor
640C2C2×#272911yescharacteristic
641C2C2×#319411yescharacteristic
642C2C2×#320411yescharacteristic
643C2C2×#336811yescharacteristic
644C2C2×#337611yescharacteristic
645C2C2×#338411yescharacteristic
646C2C2×#338811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2584C2semidirect
3. #32585C2direct (N × H)
4. C2×#2099C2semidirect
5. C2×#2088C2semidirect
6. C2×#2084C2semidirect
8. C2×#2059C2semidirect
10. C2×#1985C2semidirect
11. C2×#1944C2semidirect
14. C2×#1920C2semidirect
32. #2584C22semidirect
52. #2099C22semidirect
53. #2088C22semidirect
54. #2084C22semidirect
56. #2059C22semidirect
58. #1985C22semidirect
59. #1944C22semidirect
62. #1920C22semidirect
140. C2×#102Q8semidirect
336. #102C2×Q8semidirect
639. C2#32585direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#25842
#325852
C2×#20992
C2×#20882
C2×#20842
C2×#20652
C2×#20592
C2×#20252
C2×#19852
C2×#19442
C2×#19362
C2×#19262
C2×#19202
C2×#18642
C2×#18622