Finite Group Databasebeta

← C2×#33097 #6629621

Subgroups — C2×#33097 #6629621

Order 512.

2400 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic, complemented
3#33097C2116yesmaximal, direct product factor
4C2×#1926C211yesmaximal, characteristic, complemented
5C2×#1926C211yesmaximal, characteristic, complemented
6C2×#1925C211yesmaximal, characteristic, complemented
7C2×#1925C211yesmaximal, characteristic, complemented
8C2×#1921C211yesmaximal, characteristic, complemented
9C2×#1919C211yesmaximal, characteristic, complemented
10C2×#1920C211yesmaximal, characteristic, complemented
11C2×#1918C211yesmaximal, characteristic, complemented
12C2×#1875C211yesmaximal, characteristic
13C2×#1875C211yesmaximal, characteristic
14C2×#1854C211yesmaximal, characteristic, complemented
15C2×#1854C211yesmaximal, characteristic, complemented
16C2×#1817C211yesmaximal, characteristic, complemented
17C22×#123—21no—
18C22×#121C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32#2854—24no—
33#2854C2218yes—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#388—21no—
39C2×#388—21no—
40C2×#386—21no—
41C2×#380C2211yescharacteristic
42C4×#117—24no—
43C4×#117C2218yescomplemented
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4—21no—
49C2×C4×C22⋊C4—21no—
50C2×#376—21no—
51C2×C43—21no—
52#1926C2218yescomplemented
53#1926C2218yescomplemented
54#1925C2218yescomplemented
55#1925C2218yescomplemented
56#1921C2218yescomplemented
57#1920C2218yescomplemented
58#1919C2218yescomplemented
59#1918C2218yescomplemented
60#1875C2218yes—
61#1875C2218yes—
62#1854C2218yescomplemented
63#1854C2218yescomplemented
64#1817C2218yescomplemented
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×D4—21no—
78C23×D4C2311yescharacteristic
79C24×C4D411yescharacteristic
80C24×C4D411yescharacteristic
81C24×C4C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#123—28no—
87C2×#123—48no—
88C2×#121—22no—
89C2×#121—24no—
90C2×#121—24no—
91C2×#121C2314yes—
92C2×#117—22no—
93C2×#117—22no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117C2314yes—
99C2×#117C2314yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C42—21no—
118C22×C42—21no—
119C22×C42—24no—
120C22×C42—24no—
121C22×C42—21no—
122C22×C42—21no—
123C22×C42—21no—
124C22×C42—21no—
125C22×C42D414yes—
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42C2314yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C2×#107—24no—
132C2×#107—24no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#106—24no—
136C2×#106—24no—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×#102—24no—
140C2×#102—24no—
141C2×#102—24no—
142C2×#102—24no—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×C22⋊C8—24no—
148C2×C22⋊C8—24no—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151#390—22no—
152#390—22no—
153#390C2314yes—
154#390C2314yes—
155#389—22no—
156#389—22no—
157#389C2314yes—
158#389C2314yes—
159#388—28no—
160#388—28no—
161#386—28no—
162#380—22no—
163#380C2314yes—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4C2314yes—
169C4×C4⋊C4C2314yes—
170C4×C4⋊C4C2314yes—
171C4×C4⋊C4C2314yes—
172C4×C22⋊C4—28no—
173C4×C22⋊C4—28no—
174#376—28no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C43—28no—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C25C2×D411yescharacteristic
197C22×D4—24no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4C2412yes—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D412yes—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4#4114yes—
235C23×C4#4114yes—
236C23×C4C2412yes—
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#123—432no—
255#121—28no—
256#121—28no—
257#121—48no—
258#117—216no—
259#117—216no—
260#117—28no—
261#117—28no—
262#117—28no—
263#117—28no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
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—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4#4114yes—
286C2×C4⋊C4#4114yes—
287C2×C4⋊C4#4114yes—
288C2×C4⋊C4#4114yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C42—28no—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—216no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—28no—
324C2×C42—28no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—44no—
330C2×C42—44no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339#107—28no—
340#107—28no—
341#106—28no—
342#106—28no—
343#102—28no—
344#102—28no—
345#102—28no—
346#102—28no—
347C22⋊C8—28no—
348C22⋊C8—28no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24#12314yes—
371C24C22×D412yes—
372C24C22×D411yescharacteristic
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—816no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
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#12314yes—
449C22×C4#12318yes—
450C22×C4#13514yes—
451C22×C4AGL(1,ℤ/8)14yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4114yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
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—24no—
486C4⋊C4—24no—
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—22no—
496C4⋊C4—22no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—48no—
502C4⋊C4C2×#4114yes—
503C4⋊C4C2×#4114yes—
504C4⋊C4C2×#4114yes—
505C4⋊C4C2×#4114yes—
506C22⋊C4—216no—
507C22⋊C4—216no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C42—216no—
515C42—216no—
516C42—28no—
517C42—24no—
518C42—24no—
519C42—28no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—48no—
525C42—48no—
526C42—48no—
527C42—48no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23C2×#12314yes—
556C23C2×#12312yes—
557C23#52512yes—
558C23#52512yes—
559C23C2×#13512yes—
560C23C2×AGL(1,ℤ/8)12yes—
561C23#57512yes—
562C23#57512yes—
563C23C23×D411yescharacteristic
564D4—42no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—816no—
569D4—84no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—216no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—416no—
595C2×C4—416no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—42no—
603C2×C4—42no—
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—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4C2×#12314yes—
623C2×C4C2×#12312yes—
624C2×C4#52512yes—
625C2×C4#52512yes—
626C2×C4#52914yes—
627C2×C4#52914yes—
628C2×C4#52914yes—
629C2×C4#52914yes—
630C2×C4C2×#13514yes—
631C2×C4C2×AGL(1,ℤ/8)14yes—
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C8—42no—
639C8—42no—
640C8—42no—
641C8—42no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—416no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—168no—
658C22#285314yes—
659C22#309114yes—
660C22#309414yes—
661C22#310214yes—
662C22#310614yes—
663C22#311114yes—
664C22#311514yes—
665C22C22×#12311yescharacteristic
666C22C2×#52511yescharacteristic
667C22C2×#52511yescharacteristic
668C22C22×#13511yescharacteristic
669C22C22×AGL(1,ℤ/8)11yescharacteristic
670C22C2×#57511yescharacteristic
671C22C2×#57511yescharacteristic
672C4—24no—
673C4—24no—
674C4—24no—
675C4—42no—
676C4—42no—
677C4—44no—
678C4—44no—
679C4—44no—
680C4—44no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4#285314yes—
685C4C2×#52911yescharacteristic
686C4C2×#52911yescharacteristic
687C4C2×#52911yescharacteristic
688C4C2×#52911yescharacteristic
689C2—44no—
690C2—162no—
691C2#3309718yesdirect product factor
692C2C2×#285311yescharacteristic
693C2C2×#309111yescharacteristic
694C2C2×#309411yescharacteristic
695C2C2×#310211yescharacteristic
696C2C2×#310611yescharacteristic
697C2C2×#311511yescharacteristic
698C2C2×#311111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#117C2semidirect
3. #33097C2direct (N × H)
4. C2×#1926C2semidirect
5. C2×#1926C2semidirect
6. C2×#1925C2semidirect
7. C2×#1925C2semidirect
8. C2×#1921C2semidirect
9. C2×#1919C2semidirect
10. C2×#1920C2semidirect
11. C2×#1918C2semidirect
14. C2×#1854C2semidirect
15. C2×#1854C2semidirect
16. C2×#1817C2semidirect
43. C4×#117C22semidirect
52. #1926C22semidirect
53. #1926C22semidirect
54. #1925C22semidirect
55. #1925C22semidirect
56. #1921C22semidirect
57. #1920C22semidirect
58. #1919C22semidirect
59. #1918C22semidirect
62. #1854C22semidirect
63. #1854C22semidirect
64. #1817C22semidirect
691. C2#33097direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28542
C2×C4×#1172
#330972
C2×#19262
C2×#19252
C2×#19212
C2×#19192
C2×#19202
C2×#19182
C2×#18752
C2×#18542
C2×#18172