Finite Group Databasebeta

← C2×#9096 #196171

Subgroups — C2×#9096 #196171

Order 512.

1420 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C8×#41C211yesmaximal, characteristic
2C2×C4×#137C211yesmaximal, characteristic
3C2×#1477C211yesmaximal, characteristic
4C22×Q8⋊C8C211yesmaximal, characteristic, complemented
5C2×#1461C211yesmaximal, characteristic, complemented
6#9096C218yesmaximal, direct product factor
7C2×#1283C211yesmaximal, characteristic, complemented
8C2×#1278C211yesmaximal, characteristic, complemented
9C22×#137C411yescharacteristic
10C2×C4×#41C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C8×#41—22no—
13C8×#41—22no—
14C8×#41—22no—
15C8×#41C2214yes—
16C2×C8×Q8—21no—
17C4×#137—22no—
18C4×#137—22no—
19C4×#137C414yes—
20C4×#137C2214yes—
21C2×#441—21no—
22C2×#440—21no—
23C2×C4×Q16—21no—
24C2×C4×SD16—21no—
25C2×C8×D4—22no—
26C2×C8×D4—21no—
27C2×#431—21no—
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#428C411yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C2×#417C411yescharacteristic
33C2×#419C411yescharacteristic
34C2×C4×M16C2211yescharacteristic
35C22×C4×C8—21no—
36C22×C4×C8C2211yescharacteristic
37#1477—22no—
38#1477C2214yes—
39C2×Q8⋊C8—21no—
40C2×Q8⋊C8—21no—
41C2×Q8⋊C8—22no—
42C2×Q8⋊C8—21no—
43C2×Q8⋊C8C2214yescomplemented
44C2×D4⋊C8—21no—
45#1461—22no—
46#1461C2214yescomplemented
47C2×#324—22no—
48#1283C2214yescomplemented
49#1278C2214yescomplemented
50C22×#41C2×C411yescharacteristic
51C23×Q8C2×C411yescharacteristic
52C2×#137—22no—
53C2×#137—24no—
54C2×#137C818yescomplemented
55C2×#137C2×C414yes—
56C22×Q16—21no—
57C22×SD16—21no—
58C22×M16C2×C411yescharacteristic
59C22×M16C2×C411yescharacteristic
60C23×C8—22no—
61C23×C8—21no—
62C23×C8—21no—
63C4×#41—21no—
64C4×#41—21no—
65C4×#41—21no—
66C4×#41—21no—
67C4×#41C2×C412yes—
68C4×#41D412yes—
69C4×#41D412yes—
70C4×#41C2312yes—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8—21no—
81C2×C4×Q8C2×C412yes—
82C2×C4×Q8D411yescharacteristic
83C2×C4×Q8D411yescharacteristic
84C2×C4×Q8C2312yes—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4D411yescharacteristic
88C2×C4×D4D411yescharacteristic
89C2×#117—21no—
90C2×#117C2×C411yescharacteristic
91C22×C4⋊C4—21no—
92C22×C4⋊C4C2×C411yescharacteristic
93C22×C42—21no—
94C22×C42—21no—
95C22×C42C2311yescharacteristic
96C8×Q8—24no—
97C8×Q8—24no—
98#441—24no—
99#441—24no—
100#440—24no—
101#440—24no—
102C4×Q16—24no—
103C4×Q16—24no—
104C4×SD16—24no—
105C4×SD16—24no—
106C8×D4—28no—
107C8×D4—24no—
108C8×D4—24no—
109C8×D4—28no—
110#431—24no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431—24no—
115#431—21no—
116#431—21no—
117#431D412yes—
118#431C2312yes—
119#431C2312yes—
120#428—22no—
121#428C2×C414yes—
122C2×#107—21no—
123C2×#106—21no—
124C2×C4⋊C8—22no—
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⋊C8—21no—
132C2×C4⋊C8C2312yes—
133#419—22no—
134#419C2×C414yes—
135#417—22no—
136#417C2×C414yes—
137C2×#103—21no—
138C2×#103—21no—
139C2×#103—21no—
140C2×#102—21no—
141C2×C22⋊C8—22no—
142C2×C22⋊C8—21no—
143C2×C22⋊C8D412yes—
144C4×M16—21no—
145C4×M16—21no—
146C4×M16C2×C412yes—
147C4×M16C2312yes—
148C2×#97—21no—
149C2×C4×C8—22no—
150C2×C4×C8—24no—
151C2×C4×C8—21no—
152C2×C4×C8—21no—
153C2×C4×C8—21no—
154C2×C4×C8—21no—
155C2×C4×C8—21no—
156C2×C4×C8—24no—
157C2×C4×C8—21no—
158C2×C4×C8D411yescharacteristic
159C2×C4×C8D411yescharacteristic
160C2×C4×C8D412yes—
161C2×C4×C8C2312yes—
162Q8⋊C8—24no—
163Q8⋊C8—24no—
164Q8⋊C8—24no—
165D4⋊C8—24no—
166#324—28no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—22no—
170C2×#41—22no—
171C2×#41C22⋊C412yes—
172C2×#41C2×C814yes—
173C2×#41M1614yes—
174C2×#41C22×C412yes—
175C22×Q8—21no—
176C22×Q8—22no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8C22⋊C411yescharacteristic
184C22×Q8C2×C814yescomplemented
185C22×Q8C22×C412yes—
186C22×D4—21no—
187C22×D4C22⋊C411yescharacteristic
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4#4111yescharacteristic
196#137—216no—
197#137—28no—
198#137C2×C8116yescomplemented
199C2×Q16—28no—
200C2×Q16—24no—
201C2×SD16—28no—
202C2×SD16—24no—
203C2×M16—21no—
204C2×M16—22no—
205C2×M16—21no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16C2×C814yescomplemented
209C2×M16C22×C412yes—
210C2×M16C22×C412yes—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—28no—
214C22×C8—21no—
215C22×C8—22no—
216C22×C8—21no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—44no—
221C22×C8—48no—
222C22×C8—41no—
223C22×C8—41no—
224C22×C8—41no—
225C22×C8—82no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—41no—
239C4×Q8—41no—
240C4×Q8—41no—
241C4×Q8—41no—
242C4×Q8—41no—
243C4×Q8—41no—
244C4×Q8—41no—
245C4×Q8—41no—
246C4×Q8C22⋊C412yes—
247C4×Q8C22⋊C412yes—
248C4×Q8C2×D412yes—
249C4×Q8C2×D412yes—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4C22⋊C412yes—
263C4×D4C22⋊C412yes—
264C4×D4C2×D412yes—
265C4×D4C2×D412yes—
266#117—21no—
267#117—21no—
268#117—22no—
269#117—22no—
270#117—22no—
271#117—22no—
272#117C22⋊C412yes—
273#117C22×C412yes—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—41no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—41no—
286C2×C4⋊C4—41no—
287C2×C4⋊C4C22⋊C411yescharacteristic
288C2×C4⋊C4C22⋊C411yescharacteristic
289C2×C4⋊C4C22×C412yes—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—41no—
303C2×C42—41no—
304C2×C42—41no—
305C2×C42—41no—
306C2×C42C22×C411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42#4111yescharacteristic
315C2×C42#4111yescharacteristic
316C2×C42#4111yescharacteristic
317C2×C42C2411yescharacteristic
318#107—24no—
319#106—24no—
320C4⋊C8—28no—
321C4⋊C8—22no—
322C4⋊C8—22no—
323C4⋊C8—22no—
324C4⋊C8—22no—
325C4⋊C8—22no—
326C4⋊C8—41no—
327C4⋊C8—41no—
328C4⋊C8—41no—
329#103—24no—
330#103—24no—
331#103—24no—
332#102—24no—
333C22⋊C8—28no—
334C22⋊C8—22no—
335C22⋊C8—22no—
336C22⋊C8—41no—
337C22⋊C8C2×D414yes—
338#97—22no—
339#97—22no—
340C4×C8—28no—
341C4×C8—22no—
342C4×C8—21no—
343C4×C8—22no—
344C4×C8—21no—
345C4×C8—22no—
346C4×C8—22no—
347C4×C8—22no—
348C4×C8—22no—
349C4×C8—22no—
350C4×C8—41no—
351C4×C8C2×D412yes—
352C4×C8C2×D412yes—
353C24—21no—
354C24C4×D411yescharacteristic
355#41—22no—
356#41—22no—
357#41—24no—
358#41—24no—
359#41—24no—
360#41—24no—
361#41—24no—
362#41C22⋊C818yes—
363#41C22×C814yes—
364#41C2×M1614yes—
365C2×Q8—22no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8C22⋊C814yes—
385C2×Q8C2×C22⋊C412yes—
386C2×Q8C22×C814yescomplemented
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—21no—
392C2×D4—22no—
393C2×D4C22⋊C814yes—
394C2×D4C2×C22⋊C412yes—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—82no—
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C4×D411yescharacteristic
424C22×C4C4×D411yescharacteristic
425C22×C4C4×D411yescharacteristic
426C22×C4C4×D411yescharacteristic
427C22×C4C4×D411yescharacteristic
428C22×C4#12011yescharacteristic
429C22×C4#12011yescharacteristic
430C22×C4#12111yescharacteristic
431C22×C4#12111yescharacteristic
432C22×C4#12111yescharacteristic
433C22×C4#12111yescharacteristic
434C22×C4#12311yescharacteristic
435C22×C4#12311yescharacteristic
436C22×C4C22×C812yes—
437C22×C4C2×M1612yes—
438C22×C4#13112yes—
439C22×C4#13112yes—
440C22×C4C23×C411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4111yescharacteristic
443Q16—216no—
444SD16—216no—
445M16—22no—
446M16—24no—
447M16—22no—
448M16—22no—
449M16—42no—
450M16C22×C814yescomplemented
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—44no—
459C2×C8—44no—
460C2×C8—416no—
461C2×C8—44no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—82no—
467C2×C8—82no—
468C2×C8—82no—
469C2×C8—82no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4C2×C22⋊C412yes—
485C4⋊C4C2×C22⋊C412yes—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—42no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42C2×C22⋊C411yescharacteristic
503C42C2×C22⋊C411yescharacteristic
504C42C4×D411yescharacteristic
505C42C4×D411yescharacteristic
506C42#12011yescharacteristic
507C42#12011yescharacteristic
508C42#12111yescharacteristic
509C42#12111yescharacteristic
510C42#12111yescharacteristic
511C42#12111yescharacteristic
512C42#12311yescharacteristic
513C42#12311yescharacteristic
514C42C22×D411yescharacteristic
515C42C22×D411yescharacteristic
516C42C22×D411yescharacteristic
517C42C2×#4111yescharacteristic
518C23—22no—
519C23—22no—
520C23—24no—
521C23—22no—
522C23—42no—
523C23—42no—
524C23#38611yescharacteristic
525C23#38611yescharacteristic
526C23C8×D412yes—
527C23#43512yes—
528C23C2×C4×D411yescharacteristic
529Q8—22no—
530Q8—24no—
531Q8—24no—
532Q8—24no—
533Q8—42no—
534Q8—44no—
535Q8—44no—
536Q8—42no—
537Q8C2×C22⋊C814yes—
538D4—28no—
539D4—24no—
540D4—24no—
541D4—22no—
542D4—24no—
543D4C2×C22⋊C814yes—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
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—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4#38611yescharacteristic
581C2×C4#38611yescharacteristic
582C2×C4#38611yescharacteristic
583C2×C4#38611yescharacteristic
584C2×C4#38611yescharacteristic
585C2×C4#38611yescharacteristic
586C2×C4C2×C22⋊C812yes—
587C2×C4C2×C22⋊C812yes—
588C2×C4#40812yes—
589C2×C4#40812yes—
590C2×C4C8×D412yes—
591C2×C4C8×D412yes—
592C2×C4C8×D412yes—
593C2×C4#43512yes—
594C2×C4#43512yes—
595C2×C4#43512yes—
596C2×C4#47112yes—
597C2×C4#47312yes—
598C2×C4C22×C22⋊C411yescharacteristic
599C2×C4C2×C4×D411yescharacteristic
600C2×C4C2×#12011yescharacteristic
601C2×C4C2×#12111yescharacteristic
602C2×C4C2×#12111yescharacteristic
603C2×C4C2×#12311yescharacteristic
604C2×C4C23×C811yescharacteristic
605C2×C4C22×M1611yescharacteristic
606C2×C4C2×#13111yescharacteristic
607C2×C4C2×#13111yescharacteristic
608C8—24no—
609C8—24no—
610C8—44no—
611C8—42no—
612C8—42no—
613C8—82no—
614C8—82no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—42no—
623C22—42no—
624C22—48no—
625C22#186912yes—
626C22#186912yes—
627C22#190612yes—
628C22#190812yes—
629C22C2×#38611yescharacteristic
630C22C2×C8×D411yescharacteristic
631C22C2×#43511yescharacteristic
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—24no—
639C4—42no—
640C4—42no—
641C4—44no—
642C4—82no—
643C4#186912yes—
644C4#186912yes—
645C4#190812yes—
646C4C22×C22⋊C811yescharacteristic
647C4C2×#40811yescharacteristic
648C4C2×C8×D411yescharacteristic
649C4C2×#43511yescharacteristic
650C4C2×#47111yescharacteristic
651C4C2×#47311yescharacteristic
652C2—22no—
653C2—21no—
654C2—21no—
655C2—44no—
656C2#909614yesdirect product factor
657C2C2×#186911yescharacteristic
658C2C2×#190611yescharacteristic
659C2C2×#190811yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23#386
prime radicalC22×#41C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C22×Q8⋊C8C2semidirect
5. C2×#1461C2semidirect
6. #9096C2direct (N × H)
7. C2×#1283C2semidirect
8. C2×#1278C2semidirect
43. C2×Q8⋊C8C22semidirect
46. #1461C22semidirect
48. #1283C22semidirect
49. #1278C22semidirect
54. C2×#137C8semidirect
184. C22×Q8C2×C8semidirect
198. #137C2×C8semidirect
208. C2×M16C2×C8semidirect
386. C2×Q8C22×C8semidirect
450. M16C22×C8semidirect
656. C2#9096direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C8×#412
C2×C4×#1372
C2×#14772
C22×Q8⋊C82
C2×#14612
#90962
C2×#12832
C2×#12782