Finite Group Databasebeta

← C2×#9233 #198013

Subgroups — C2×#9233 #198013

Order 512.

2114 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3100C211yesmaximal, characteristic
2C2×C42×D4C211yesmaximal, characteristic, complemented
3C2×#1962C211yesmaximal, characteristic, complemented
4C2×#1502C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic, complemented
6#9233C218yesmaximal, direct product factor
7C2×#1303C211yesmaximal, characteristic, complemented
8C2×#1276C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4×D4—21no—
11C22×C4×D4C2211yescharacteristic
12C23×C42—21no—
13#3100—22no—
14#3100—22no—
15#3100C414yescomplemented
16#3100C2214yes—
17C2×#480—21no—
18C2×#474C412yescomplemented
19C2×#459C412yescomplemented
20C2×#447—21no—
21C2×C4×D8—22no—
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C22×#104—21no—
26C22×#104—21no—
27C42×D4—22no—
28C42×D4—22no—
29C42×D4—22no—
30C42×D4C2214yescomplemented
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4—21no—
34C2×C4×C22⋊C4—22no—
35C2×C43C2211yescharacteristic
36#1962—22no—
37#1962C2214yescomplemented
38#1502—22no—
39#1502C2214yescomplemented
40C2×#327—22no—
41#1486—22no—
42#1486C2214yescomplemented
43C2×D4⋊C8—22no—
44#1303C2214yescomplemented
45#1276C2214yescomplemented
46C22×#41—21no—
47C23×D4D411yescharacteristic
48C24×C4—22no—
49C24×C4—21no—
50C22×D8—22no—
51C22×M16—21no—
52#532—21no—
53#532—21no—
54#532—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532C2×C412yescomplemented
59#532C2312yes—
60C2×C4⋊Q8C2×C411yescharacteristic, complemented
61C2×#127C2×C411yescharacteristic, complemented
62C2×#124—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×C4×D4—216no—
66C2×C4×D4—24no—
67C2×C4×D4—24no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2×C412yes—
77C2×C4×D4C2×C412yes—
78C2×C4×D4D412yes—
79C2×C4×D4D412yes—
80C2×C4×D4D412yes—
81C2×C4×D4C2312yes—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D411yescharacteristic
84C22×C22⋊C4—22no—
85C22×C22⋊C4—21no—
86C22×C42—21no—
87C22×C42—22no—
88C22×C42—24no—
89C22×C42—21no—
90C22×C42—21no—
91C22×C42—24no—
92C22×C42—44no—
93C22×C42—44no—
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42C2311yescharacteristic
97#480—24no—
98#480—24no—
99#474—24no—
100#474C2×C418yescomplemented
101#459—24no—
102#459C2×C418yescomplemented
103#447—24no—
104#447—24no—
105C4×D8—28no—
106C4×D8—28no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431C2×C412yescomplemented
113#431C2312yes—
114#431C2312yes—
115C2×#107—22no—
116#423—21no—
117#423—21no—
118#423—21no—
119#423C2312yes—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8C2×C412yescomplemented
124C2×#104—24no—
125C2×#104—24no—
126C2×#104—44no—
127C2×#104—44no—
128C2×#102—22no—
129C2×#102—22no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C4×C8—21no—
133C2×C4×C8—21no—
134C4×C4⋊C4—24no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4D412yes—
139C4×C4⋊C4C2312yes—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—28no—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—28no—
144C2×#95—22no—
145C2×#95D412yes—
146C43—21no—
147C43—21no—
148C43D412yes—
149C43C2312yes—
150#327—28no—
151D4⋊C8—28no—
152C25—21no—
153C2×#41—22no—
154C2×#41—22no—
155C2×#41—41no—
156C2×#41—41no—
157C2×#41—42no—
158C2×#41—42no—
159C2×#41—42no—
160C2×#41—42no—
161C22×Q8—21no—
162C22×D4—28no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—41no—
170C22×D4C22⋊C412yes—
171C22×D4C22⋊C412yes—
172C22×D4C2×D412yes—
173C23×C4—21no—
174C23×C4—216no—
175C23×C4—28no—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—48no—
186C23×C4—48no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—41no—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C2×D8—28no—
196C2×D8—48no—
197C2×M16—22no—
198C2×M16—41no—
199C2×M16—42no—
200C2×M16—42no—
201C22×C8—22no—
202C22×C8—41no—
203C22×C8—41no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8C22×C412yescomplemented
208#127—21no—
209#127—21no—
210#127—21no—
211#127C22×C412yescomplemented
212#124—22no—
213#124—22no—
214#124—41no—
215#124—41no—
216#121—22no—
217#121—22no—
218#121—22no—
219#121—22no—
220#121—41no—
221#121—41no—
222#121—41no—
223#121—41no—
224C4×D4—232no—
225C4×D4—232no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—41no—
243C4×D4—41no—
244C4×D4—41no—
245C4×D4—41no—
246C4×D4C22⋊C414yes—
247C4×D4C22⋊C414yes—
248C4×D4C22×C414yes—
249C4×D4C2×D414yes—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C22⋊C412yes—
259C2×C4⋊C4C22⋊C412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C22⋊C4—216no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C42—28no—
272C2×C42—22no—
273C2×C42—21no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—28no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—48no—
285C2×C42—416no—
286C2×C42—44no—
287C2×C42—48no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—48no—
291C2×C42D814yescomplemented
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
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×C42#4111yescharacteristic
305C2×C42C2411yescharacteristic
306#107—28no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—41no—
312C4⋊C8—41no—
313C4⋊C8—41no—
314C4⋊C8C22×C414yescomplemented
315#104—48no—
316#104—48no—
317#102—28no—
318#102—28no—
319C22⋊C8—22no—
320C22⋊C8—22no—
321C22⋊C8—22no—
322C22⋊C8—41no—
323C4×C8—22no—
324C4×C8—22no—
325C4×C8—22no—
326C4×C8—41no—
327#95—28no—
328#95—22no—
329#95C2×D414yes—
330C24—28no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—44no—
335C24—42no—
336C24—42no—
337C24—41no—
338C24#12011yescharacteristic
339#41—42no—
340#41—42no—
341#41—42no—
342#41—42no—
343#41—44no—
344#41—44no—
345#41—82no—
346#41—82no—
347#41—82no—
348#41—82no—
349C2×Q8—22no—
350C2×Q8—41no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×D4—28no—
354C2×D4—28no—
355C2×D4—28no—
356C2×D4—28no—
357C2×D4—28no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—24no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—44no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—82no—
374C2×D4—82no—
375C2×D4—82no—
376C2×D4#10418yescomplemented
377C2×D4C2×C22⋊C414yes—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—28no—
385C22×C4—232no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—42no—
396C22×C4—416no—
397C22×C4—416no—
398C22×C4—464no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C4×D411yescharacteristic
427C22×C4C4×D411yescharacteristic
428C22×C4C4×D411yescharacteristic
429C22×C4C4×D411yescharacteristic
430C22×C4#12011yescharacteristic
431C22×C4#12011yescharacteristic
432C22×C4#12011yescharacteristic
433C22×C4#12111yescharacteristic
434C22×C4#12111yescharacteristic
435C22×C4#12111yescharacteristic
436C22×C4#12111yescharacteristic
437C22×C4#12311yescharacteristic
438C22×C4#12311yescharacteristic
439C22×C4C2×D812yes—
440C22×C4#13512yes—
441C22×C4AGL(1,ℤ/8)12yes—
442C22×C4AGL(1,ℤ/8)12yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446D8—416no—
447M16—42no—
448M16—42no—
449M16—82no—
450C2×C8—24no—
451C2×C8—44no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—82no—
455C2×C8—82no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4C2×C22⋊C414yes—
466C22⋊C4—232no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C42—28no—
476C42—28no—
477C42—22no—
478C42—24no—
479C42—22no—
480C42—22no—
481C42—44no—
482C42—44no—
483C42—416no—
484C42—42no—
485C42—42no—
486C42—44no—
487C42—42no—
488C42C2×C22⋊C411yescharacteristic
489C42C2×C22⋊C411yescharacteristic
490C42C4×D411yescharacteristic
491C42C4×D411yescharacteristic
492C42C4×D411yescharacteristic
493C42C4×D411yescharacteristic
494C42#12111yescharacteristic
495C42#12111yescharacteristic
496C42#12111yescharacteristic
497C42#12111yescharacteristic
498C42#12311yescharacteristic
499C42#12311yescharacteristic
500C42C2×D814yescomplemented
501C42C23×C411yescharacteristic
502C42C22×D411yescharacteristic
503C42C2×#4111yescharacteristic
504C42C2×#4111yescharacteristic
505C23—22no—
506C23—22no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—24no—
511C23—44no—
512C23—416no—
513C23—44no—
514C23—416no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23#38611yescharacteristic
525C23#38611yescharacteristic
526C23#44712yes—
527C23#44912yes—
528C23C2×#12011yescharacteristic
529Q8—42no—
530Q8—42no—
531Q8—82no—
532D4—28no—
533D4—28no—
534D4—24no—
535D4—28no—
536D4—44no—
537D4—42no—
538D4—42no—
539D4—82no—
540D4—82no—
541D4—84no—
542D4—82no—
543D4—82no—
544D4—82no—
545D4—82no—
546D4C2×#10418yescomplemented
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—216no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—464no—
566C2×C4—42no—
567C2×C4—48no—
568C2×C4—44no—
569C2×C4—416no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—82no—
584C2×C4—84no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4#38611yescharacteristic
588C2×C4#38611yescharacteristic
589C2×C4#38611yescharacteristic
590C2×C4#38611yescharacteristic
591C2×C4#38611yescharacteristic
592C2×C4#38611yescharacteristic
593C2×C4C2×#10414yes—
594C2×C4C4×D812yes—
595C2×C4C4×D812yes—
596C2×C4#44212yes—
597C2×C4#44212yes—
598C2×C4#44712yes—
599C2×C4#44912yes—
600C2×C4#45912yes—
601C2×C4#45912yes—
602C2×C4#46312yes—
603C2×C4#46312yes—
604C2×C4C22×C22⋊C411yescharacteristic
605C2×C4C2×C4×D411yescharacteristic
606C2×C4C2×C4×D411yescharacteristic
607C2×C4C2×#12111yescharacteristic
608C2×C4C2×#12111yescharacteristic
609C2×C4C2×#12311yescharacteristic
610C2×C4C22×D811yescharacteristic
611C2×C4C2×#13511yescharacteristic
612C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
613C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
614C8—44no—
615C8—82no—
616C8—82no—
617C22—21no—
618C22—21no—
619C22—24no—
620C22—24no—
621C22—21no—
622C22—21no—
623C22—28no—
624C22—48no—
625C22—48no—
626C22—416no—
627C22—44no—
628C22—416no—
629C22—44no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—164no—
634C22#187912yes—
635C22#187912yes—
636C22#189612yes—
637C22#190612yes—
638C22C2×#38611yescharacteristic
639C22C2×#44711yescharacteristic
640C22C2×#44911yescharacteristic
641C4—28no—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—44no—
649C4—48no—
650C4—44no—
651C4—82no—
652C4#187912yes—
653C4#187912yes—
654C4#189612yes—
655C4C22×#10412yes—
656C4C2×C4×D811yescharacteristic
657C4C2×#44211yescharacteristic
658C4C2×#45911yescharacteristic
659C4C2×#46311yescharacteristic
660C2—21no—
661C2—21no—
662C2—22no—
663C2—48no—
664C2—162no—
665C2#923314yesdirect product factor
666C2C2×#187911yescharacteristic
667C2C2×#189611yescharacteristic
668C2C2×#190611yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC23#386
prime radicalC2×#3100C2
LCSG ▷ C42 ▷ C22 ▷ 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
2. C2×C42×D4C2semidirect
3. C2×#1962C2semidirect
4. C2×#1502C2semidirect
5. C2×#1486C2semidirect
6. #9233C2direct (N × H)
7. C2×#1303C2semidirect
8. C2×#1276C2semidirect
15. #3100C4semidirect
18. C2×#474C4semidirect
19. C2×#459C4semidirect
30. C42×D4C22semidirect
37. #1962C22semidirect
39. #1502C22semidirect
42. #1486C22semidirect
44. #1303C22semidirect
45. #1276C22semidirect
58. #532C2×C4semidirect
60. C2×C4⋊Q8C2×C4semidirect
61. C2×#127C2×C4semidirect
100. #474C2×C4semidirect
102. #459C2×C4semidirect
112. #431C2×C4semidirect
123. C2×C4⋊C8C2×C4semidirect
207. C4⋊Q8C22×C4semidirect
211. #127C22×C4semidirect
291. C2×C42D8semidirect
314. C4⋊C8C22×C4semidirect
376. C2×D4#104semidirect
500. C42C2×D8semidirect
546. D4C2×#104semidirect
665. C2#9233direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31002
C2×C42×D42
C2×#19622
C2×#15022
C2×#14862
#92332
C2×#13032
C2×#12762