Finite Group Databasebeta

← C2×#9251 #198255

Subgroups — C2×#9251 #198255

Order 512.

1480 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic
2C2×D4×M16C211yesmaximal, characteristic
3C2×#1966C211yesmaximal, characteristic, complemented
4C2×#1546C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic
6#9251C218yesmaximal, direct product factor
7C2×#1292C211yesmaximal, characteristic, complemented
8C2×#1283C211yesmaximal, characteristic, complemented
9C23×M16—21no—
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#474—21no—
13C2×#474—21no—
14C2×#474—22no—
15C2×#474—21no—
16C2×#474—21no—
17C2×#474—22no—
18C2×#474C414yes—
19C2×#474C2214yes—
20D4×M16—22no—
21D4×M16—22no—
22D4×M16—22no—
23D4×M16C2214yes—
24C2×#436—21no—
25C2×#435—22no—
26C2×C8×D4—21no—
27C2×#431C2211yescharacteristic
28C22×#107C412yes—
29C2×#423C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#102C412yes—
33C22×#101—21no—
34C22×#101—21no—
35C2×#407—21no—
36C2×C4×M16C2211yescharacteristic
37#1966—22no—
38#1966C2214yescomplemented
39#1546—22no—
40#1546C2214yescomplemented
41C2×#332—22no—
42#1486—22no—
43#1486C2214yes—
44C2×D4⋊C8—21no—
45C2×D4⋊C8—21no—
46#1292C2214yescomplemented
47#1283C2214yescomplemented
48C23×Q8—21no—
49C23×D4C2×C411yescharacteristic
50C24×C4—21no—
51C22×M16—22no—
52C22×M16—24no—
53C22×M16—21no—
54C22×M16—24no—
55C22×M16—21no—
56C22×M16—44no—
57C22×M16—44no—
58C22×M16D411yescharacteristic
59C22×M16D411yescharacteristic
60C23×C8—22no—
61C23×C8C2×C412yes—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8C2×C412yes—
73C2×C4⋊Q8C2312yes—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4C2×C412yes—
81C2×C4×D4D411yescharacteristic
82C2×C4×D4D411yescharacteristic
83C2×C4×D4D412yes—
84C2×C4×D4D411yescharacteristic
85C2×C4×D4D411yescharacteristic
86C2×C4×D4D412yes—
87C2×C4×D4C2312yes—
88C22×C4⋊C4—21no—
89C22×C4⋊C4C2×C412yes—
90C22×C4⋊C4C2×C411yescharacteristic
91C22×C22⋊C4—21no—
92C22×C42C2311yescharacteristic
93#474—24no—
94#474—24no—
95#474—24no—
96#474—24no—
97#474—24no—
98#474—24no—
99#474—24no—
100#474—24no—
101#436—24no—
102#436—24no—
103#435—28no—
104#435—28no—
105C8×D4—24no—
106C8×D4—24no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431C2312yes—
111C2×#107—24no—
112C2×#107—22no—
113C2×#107—22no—
114C2×#107C2×C418yes—
115#423—21no—
116#423—21no—
117#423—21no—
118#423—21no—
119#423—21no—
120#423D412yes—
121#423C2312yes—
122#423C2312yes—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8—21no—
130C2×C4⋊C8—21no—
131C2×C4⋊C8C2×C412yes—
132C2×C4⋊C8D412yes—
133C2×C4⋊C8C2312yes—
134C2×#102—22no—
135C2×#102—24no—
136C2×#102—22no—
137C2×#102C2×C418yes—
138C2×#101—24no—
139C2×#101—24no—
140C2×#101—44no—
141C2×#101—44no—
142#407—24no—
143#407—24no—
144C2×C22⋊C8—22no—
145C2×C22⋊C8—22no—
146C2×C22⋊C8—21no—
147C4×M16—21no—
148C4×M16—21no—
149C4×M16D412yes—
150C4×M16C2312yes—
151C2×#97—21no—
152C2×C4×C8—21no—
153C2×C4×C8—21no—
154#332—28no—
155D4⋊C8—24no—
156D4⋊C8—24no—
157C25—21no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—41no—
161C22×Q8—41no—
162C22×Q8—41no—
163C22×Q8—41no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×D4—24no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—42no—
173C22×D4C22⋊C411yescharacteristic
174C22×D4C22⋊C412yes—
175C22×D4C22⋊C411yescharacteristic
176C22×D4C22×C412yes—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4#4111yescharacteristic
194C2×M16—21no—
195C2×M16—22no—
196C2×M16—28no—
197C2×M16—22no—
198C2×M16—21no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—48no—
204C2×M16—48no—
205C2×M16—416no—
206C2×M16—42no—
207C2×M16—42no—
208C2×M16—48no—
209C2×M16—41no—
210C2×M16—42no—
211C2×M16—42no—
212C2×M16D814yes—
213C2×M16C2×D412yes—
214C2×M16C2×D412yes—
215C22×C8—21no—
216C22×C8—28no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—21no—
221C22×C8—48no—
222C22×C8—44no—
223C22×C8—41no—
224C22×C8—41no—
225C22×C8C22×C414yes—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234C4⋊Q8—41no—
235C4⋊Q8—41no—
236C4⋊Q8—41no—
237C4⋊Q8—41no—
238C4⋊Q8—41no—
239C4⋊Q8—41no—
240C4⋊Q8—41no—
241C4⋊Q8—41no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—21no—
245C4×D4—22no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—22no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—22no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4C22⋊C412yes—
259C4×D4C22⋊C412yes—
260C4×D4C22⋊C412yes—
261C4×D4C22⋊C412yes—
262C4×D4C2×D412yes—
263C4×D4C2×D412yes—
264C4×D4C2×D412yes—
265C4×D4C2×D412yes—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
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—41no—
277C2×C4⋊C4—41no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4C22⋊C412yes—
281C2×C4⋊C4C22⋊C411yescharacteristic
282C2×C4⋊C4C22⋊C411yescharacteristic
283C2×C4⋊C4C22×C414yes—
284C2×C4⋊C4C22×C412yes—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—22no—
293C2×C42C22×C411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42#4111yescharacteristic
302C2×C42#4111yescharacteristic
303C2×C42#4111yescharacteristic
304C2×C42C2411yescharacteristic
305#107—28no—
306#107—28no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—22no—
314C4⋊C8—22no—
315C4⋊C8—22no—
316C4⋊C8—22no—
317C4⋊C8—41no—
318C4⋊C8—41no—
319C4⋊C8—41no—
320C4⋊C8C2×D414yes—
321#102—28no—
322#102—28no—
323#101—48no—
324#101—48no—
325C22⋊C8—28no—
326C22⋊C8—28no—
327C22⋊C8—22no—
328C22⋊C8—41no—
329#97—22no—
330#97—22no—
331C4×C8—22no—
332C4×C8—22no—
333C4×C8—22no—
334C4×C8—41no—
335C24—21no—
336C24—21no—
337C24—22no—
338C24—22no—
339C24—44no—
340C24—42no—
341C24—42no—
342C24—82no—
343C24C4×D411yescharacteristic
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—82no—
355C2×Q8—82no—
356C2×Q8—82no—
357C2×Q8—82no—
358C2×Q8—82no—
359C2×Q8—82no—
360C2×Q8—82no—
361C2×Q8—82no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4#10114yescomplemented
377C2×D4C2×C22⋊C412yes—
378C2×D4C2×C22⋊C412yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
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—24no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C4×D411yescharacteristic
422C22×C4C4×D411yescharacteristic
423C22×C4C4×D411yescharacteristic
424C22×C4C4×D411yescharacteristic
425C22×C4C4×D411yescharacteristic
426C22×C4#12011yescharacteristic
427C22×C4#12011yescharacteristic
428C22×C4#12111yescharacteristic
429C22×C4#12111yescharacteristic
430C22×C4#12111yescharacteristic
431C22×C4#12111yescharacteristic
432C22×C4#12311yescharacteristic
433C22×C4#12311yescharacteristic
434C22×C4C2×D812yes—
435C22×C4#13512yes—
436C22×C4AGL(1,ℤ/8)12yes—
437C22×C4AGL(1,ℤ/8)12yes—
438C22×C4C23×C411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4111yescharacteristic
441M16—22no—
442M16—24no—
443M16—22no—
444M16—22no—
445M16—416no—
446M16—42no—
447M16—42no—
448M16—44no—
449M16—42no—
450M16—42no—
451M16—42no—
452M16—82no—
453M16C2×D814yes—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—416no—
460C2×C8—44no—
461C2×C8—44no—
462C2×C8—44no—
463C2×C8—42no—
464C2×C8—44no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—82no—
468C2×C8—82no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4C2×C22⋊C412yes—
482C4⋊C4C2×C22⋊C412yes—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C42C2×C22⋊C411yescharacteristic
492C42C2×C22⋊C411yescharacteristic
493C42C4×D411yescharacteristic
494C42C4×D411yescharacteristic
495C42#12011yescharacteristic
496C42#12011yescharacteristic
497C42#12111yescharacteristic
498C42#12111yescharacteristic
499C42#12111yescharacteristic
500C42#12111yescharacteristic
501C42#12311yescharacteristic
502C42#12311yescharacteristic
503C42C22×D411yescharacteristic
504C42C22×D411yescharacteristic
505C42C22×D411yescharacteristic
506C42C2×#4111yescharacteristic
507C23—22no—
508C23—22no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—82no—
523C23—82no—
524C23—88no—
525C23—88no—
526C23#38611yescharacteristic
527C23#38611yescharacteristic
528C23C4×D812yes—
529C23#44212yes—
530C23C2×C4×D411yescharacteristic
531Q8—84no—
532Q8—84no—
533Q8—82no—
534Q8—82no—
535Q8—82no—
536Q8—82no—
537D4—22no—
538D4—22no—
539D4—22no—
540D4—24no—
541D4—44no—
542D4—44no—
543D4—42no—
544D4C2×#10114yescomplemented
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4#38611yescharacteristic
583C2×C4#38611yescharacteristic
584C2×C4#38611yescharacteristic
585C2×C4#38611yescharacteristic
586C2×C4#38611yescharacteristic
587C2×C4#38611yescharacteristic
588C2×C4C2×#10112yes—
589C2×C4#41312yes—
590C2×C4C4×D812yes—
591C2×C4#44212yes—
592C2×C4#44712yes—
593C2×C4#44712yes—
594C2×C4#44912yes—
595C2×C4#44912yes—
596C2×C4#45912yes—
597C2×C4#45912yes—
598C2×C4#46312yes—
599C2×C4#46312yes—
600C2×C4C22×C22⋊C411yescharacteristic
601C2×C4C2×C4×D411yescharacteristic
602C2×C4C2×#12011yescharacteristic
603C2×C4C2×#12111yescharacteristic
604C2×C4C2×#12111yescharacteristic
605C2×C4C2×#12311yescharacteristic
606C2×C4C22×D811yescharacteristic
607C2×C4C2×#13511yescharacteristic
608C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
609C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
610C8—24no—
611C8—44no—
612C8—44no—
613C8—42no—
614C8—82no—
615C8—82no—
616C22—21no—
617C22—21no—
618C22—24no—
619C22—24no—
620C22—21no—
621C22—21no—
622C22—24no—
623C22—42no—
624C22—48no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—88no—
630C22—84no—
631C22—84no—
632C22—84no—
633C22#187912yes—
634C22#187912yes—
635C22#190112yes—
636C22#190212yes—
637C22C2×#38611yescharacteristic
638C22C2×C4×D811yescharacteristic
639C22C2×#44211yescharacteristic
640C4—21no—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—42no—
647C4—42no—
648C4—84no—
649C4#187912yes—
650C4#187912yes—
651C4#190112yes—
652C4C22×#10111yescharacteristic
653C4C2×#41311yescharacteristic
654C4C2×#44711yescharacteristic
655C4C2×#44911yescharacteristic
656C4C2×#45911yescharacteristic
657C4C2×#46311yescharacteristic
658C2—22no—
659C2—21no—
660C2—21no—
661C2—44no—
662C2—82no—
663C2#925114yesdirect product factor
664C2C2×#187911yescharacteristic
665C2C2×#190111yescharacteristic
666C2C2×#190211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1966C2semidirect
4. C2×#1546C2semidirect
6. #9251C2direct (N × H)
7. C2×#1292C2semidirect
8. C2×#1283C2semidirect
38. #1966C22semidirect
40. #1546C22semidirect
46. #1292C22semidirect
47. #1283C22semidirect
376. C2×D4#101semidirect
544. D4C2×#101semidirect
663. C2#9251direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4742
C2×D4×M162
C2×#19662
C2×#15462
C2×#14862
#92512
C2×#12922
C2×#12832