Finite Group Databasebeta

← C2×#32635 #7510554

Subgroups — C2×#32635 #7510554

Order 512.

2068 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2548C211yesmaximal, characteristic
2C2×#2584C211yesmaximal, characteristic, complemented
3#32635C2116yesmaximal, direct product factor
4C2×#2067C211yesmaximal, characteristic, complemented
5C2×#2065C211yesmaximal, characteristic, complemented
6C2×#2041C211yesmaximal, characteristic
7C2×#2041C211yesmaximal, characteristic
8C2×#2025C211yesmaximal, characteristic, complemented
9C2×#2025C211yesmaximal, characteristic, complemented
10C2×#1984C211yesmaximal, characteristic, complemented
11C2×#1934C211yesmaximal, characteristic, complemented
12C2×#1934C211yesmaximal, characteristic, complemented
13C2×#1926C211yesmaximal, characteristic, complemented
14C2×#1925C211yesmaximal, characteristic, complemented
15C2×#1921C211yesmaximal, characteristic, complemented
16C2×#1919C211yesmaximal, characteristic, complemented
17C22×#127C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#124—21no—
20C22×C4×Q8C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33#2584—24no—
34#2584C2218yescomplemented
35#2548—24no—
36#2548C2218yes—
37C2×#398—21no—
38C2×#395—21no—
39C2×#394—21no—
40C2×#394—21no—
41C2×#390—21no—
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#389—21no—
47C2×#389—21no—
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2067C2218yescomplemented
55#2065C2218yescomplemented
56#2041C2218yes—
57#2041C2218yes—
58#2025C2218yescomplemented
59#2025C2218yescomplemented
60#1984C2218yescomplemented
61#1934C2218yescomplemented
62#1934C2218yescomplemented
63#1926C2218yescomplemented
64#1925C2218yescomplemented
65#1921C2218yescomplemented
66#1919C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8D411yescharacteristic
78C23×D4—21no—
79C23×D4—21no—
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#127—22no—
87C2×#127—42no—
88C2×#127—42no—
89C2×#127C2314yes—
90C2×#125—22no—
91C2×#125—42no—
92C2×#125—42no—
93C2×#125C2314yes—
94C2×#124—28no—
95C2×#124—48no—
96C2×C4×Q8—42no—
97C2×C4×Q8—42no—
98C2×C4×Q8D414yes—
99C2×C4×Q8C2314yes—
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⋊C4D411yescharacteristic
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C42—21no—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C2×#107—24no—
129C2×#107C2314yes—
130C2×#106—24no—
131C2×#106C2314yes—
132C2×C4⋊C8—42no—
133C2×C4⋊C8C2314yes—
134C2×#102—24no—
135C2×#102—24no—
136C2×#102—24no—
137C2×#102—24no—
138C2×#102—42no—
139C2×#102—42no—
140C2×#102—42no—
141C2×#102—42no—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C4×C8—42no—
151C2×C4×C8C2314yes—
152#398—28no—
153#395—28no—
154#394—28no—
155#394—28no—
156#390—28no—
157#390—22no—
158#390—22no—
159#390—22no—
160#390—22no—
161#390C2314yes—
162#390C2314yes—
163#390C2314yes—
164#390C2314yes—
165#389—28no—
166#389—28no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389C2314yes—
172#389C2314yes—
173#389C2314yes—
174#389C2314yes—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2314yes—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C25—21no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—42no—
200C22×Q8C2×D414yes—
201C22×D4—24no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—44no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4—44no—
221C22×D4—82no—
222C22×D4—82no—
223C22×D4C2412yes—
224C22×D4C2412yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2411yescharacteristic
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254#127—44no—
255#127—44no—
256#127—48no—
257#125—44no—
258#125—44no—
259#125—48no—
260#124—432no—
261C4×Q8—48no—
262C4×Q8—44no—
263C4×Q8—44no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×D414yes—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4#4114yes—
299C2×C4⋊C4#4114yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—82no—
319C2×C22⋊C4—82no—
320C2×C22⋊C4—82no—
321C2×C22⋊C4—82no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—44no—
331C2×C42—82no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337C2×C42C2412yes—
338#107—28no—
339#106—28no—
340C4⋊C8—44no—
341#102—28no—
342#102—28no—
343#102—28no—
344#102—28no—
345#102—44no—
346#102—44no—
347#102—44no—
348#102—44no—
349C4×C8—44no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356C24—22no—
357C24—22no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24C22×D411yescharacteristic
373C2×Q8—28no—
374C2×Q8—28no—
375C2×Q8—44no—
376C2×Q8—416no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—48no—
407C2×D4—84no—
408C2×D4—84no—
409C2×D4—816no—
410C2×D4—816no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4#13514yes—
452C22×C4AGL(1,ℤ/8)14yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+412yes—
465C22×C42−1+412yes—
466C22×C4C2511yescharacteristic
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—44no—
480C2×C8—44no—
481C2×C8—44no—
482C2×C8—44no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4C2×#4114yes—
509C4⋊C4C2×#4114yes—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C42—24no—
519C42—24no—
520C42—24no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—48no—
528C42—84no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23#53412yes—
558C23#54812yes—
559C23#54812yes—
560C23C2×#13512yes—
561C23C2×AGL(1,ℤ/8)12yes—
562C23#57612yes—
563C23#57712yes—
564C23C23×D411yescharacteristic
565Q8—416no—
566D4—42no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—816no—
575D4—816no—
576D4—84no—
577D4—84no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4#53412yes—
625C2×C4#54214yes—
626C2×C4#54214yes—
627C2×C4#54812yes—
628C2×C4#54812yes—
629C2×C4#55514yes—
630C2×C4#56314yes—
631C2×C4C2×#13514yes—
632C2×C4C2×AGL(1,ℤ/8)14yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C2×C4C2×2−1+411yescharacteristic
639C8—42no—
640C8—42no—
641C8—42no—
642C8—42no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—168no—
658C22—168no—
659C22#273114yes—
660C22#318814yes—
661C22#319814yes—
662C22#336114yes—
663C22#336514yes—
664C22#336614yes—
665C22#336814yes—
666C22C2×#53411yescharacteristic
667C22C2×#54811yescharacteristic
668C22C2×#54811yescharacteristic
669C22C22×#13511yescharacteristic
670C22C22×AGL(1,ℤ/8)11yescharacteristic
671C22C2×#57611yescharacteristic
672C22C2×#57711yescharacteristic
673C4—24no—
674C4—24no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—44no—
680C4—44no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#273114yes—
686C4C2×#54211yescharacteristic
687C4C2×#54211yescharacteristic
688C4C2×#55511yescharacteristic
689C4C2×#56311yescharacteristic
690C2—162no—
691C2—162no—
692C2#3263518yesdirect product factor
693C2C2×#273111yescharacteristic
694C2C2×#318811yescharacteristic
695C2C2×#319811yescharacteristic
696C2C2×#336111yescharacteristic
697C2C2×#336511yescharacteristic
698C2C2×#336611yescharacteristic
699C2C2×#336811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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. #32635C2direct (N × H)
4. C2×#2067C2semidirect
5. C2×#2065C2semidirect
8. C2×#2025C2semidirect
9. C2×#2025C2semidirect
10. C2×#1984C2semidirect
11. C2×#1934C2semidirect
12. C2×#1934C2semidirect
13. C2×#1926C2semidirect
14. C2×#1925C2semidirect
15. C2×#1921C2semidirect
16. C2×#1919C2semidirect
34. #2584C22semidirect
54. #2067C22semidirect
55. #2065C22semidirect
58. #2025C22semidirect
59. #2025C22semidirect
60. #1984C22semidirect
61. #1934C22semidirect
62. #1934C22semidirect
63. #1926C22semidirect
64. #1925C22semidirect
65. #1921C22semidirect
66. #1919C22semidirect
692. C2#32635direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25482
C2×#25842
#326352
C2×#20672
C2×#20652
C2×#20412
C2×#20252
C2×#19842
C2×#19342
C2×#19262
C2×#19252
C2×#19212
C2×#19192