Finite Group Databasebeta

← C2×#9249 #198253

Subgroups — C2×#9249 #198253

Order 512.

1436 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2C2×D4×M16C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic, complemented
4C2×#1549C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic
6#9249C218yesmaximal, direct product factor
7C2×#1292C211yesmaximal, characteristic, complemented
8C2×#1283C211yesmaximal, characteristic, complemented
9C23×M16—21no—
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#476—21no—
13C2×#476—21no—
14C2×#476—22no—
15C2×#476—21no—
16C2×#476—21no—
17C2×#476—22no—
18C2×#476C414yes—
19C2×#476C2214yes—
20D4×M16—22no—
21D4×M16—22no—
22D4×M16—22no—
23D4×M16C2214yes—
24C2×#436—21no—
25C2×#435—22no—
26C2×C8×D4—21no—
27C2×#431C2211yescharacteristic
28C22×#106C412yes—
29C2×#423C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#102C412yes—
33C2×#407—21no—
34C2×C4×M16C2211yescharacteristic
35#1997—22no—
36#1997C2214yescomplemented
37C2×#343—21no—
38C2×#343—21no—
39#1549—22no—
40#1549C2214yescomplemented
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#476—24no—
94#476—24no—
95#476—24no—
96#476—24no—
97#476—24no—
98#476—24no—
99#476—24no—
100#476—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×#106—24no—
112C2×#106—22no—
113C2×#106—22no—
114C2×#106C2×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—
138#407—24no—
139#407—24no—
140C2×C22⋊C8—22no—
141C2×C22⋊C8—22no—
142C2×C22⋊C8—21no—
143C4×M16—21no—
144C4×M16—21no—
145C4×M16D412yes—
146C4×M16C2312yes—
147C2×#97—21no—
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150#343—24no—
151#343—24no—
152#332—28no—
153D4⋊C8—24no—
154D4⋊C8—24no—
155C25—21no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—41no—
161C22×Q8—41no—
162C22×Q8—84no—
163C22×D4—24no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—42no—
168C22×D4C22⋊C411yescharacteristic
169C22×D4C22⋊C412yes—
170C22×D4C22⋊C411yescharacteristic
171C22×D4C22×C412yes—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4C22×C411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4#4111yescharacteristic
189C2×M16—21no—
190C2×M16—22no—
191C2×M16—28no—
192C2×M16—22no—
193C2×M16—21no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—48no—
199C2×M16—48no—
200C2×M16—416no—
201C2×M16—42no—
202C2×M16—42no—
203C2×M16—48no—
204C2×M16—41no—
205C2×M16—42no—
206C2×M16—42no—
207C2×M16SD1614yes—
208C2×M16C2×D412yes—
209C2×M16C2×D412yes—
210C22×C8—21no—
211C22×C8—28no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—21no—
216C22×C8—48no—
217C22×C8—44no—
218C22×C8—41no—
219C22×C8—41no—
220C22×C8C22×C414yes—
221C4⋊Q8—22no—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—22no—
225C4⋊Q8—22no—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—41no—
230C4⋊Q8—41no—
231C4⋊Q8—41no—
232C4⋊Q8—41no—
233C4⋊Q8—41no—
234C4⋊Q8—41no—
235C4⋊Q8—41no—
236C4⋊Q8—41no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—21no—
240C4×D4—22no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—22no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—22no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4C22⋊C412yes—
254C4×D4C22⋊C412yes—
255C4×D4C22⋊C412yes—
256C4×D4C22⋊C412yes—
257C4×D4C2×D412yes—
258C4×D4C2×D412yes—
259C4×D4C2×D412yes—
260C4×D4C2×D412yes—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—41no—
273C2×C4⋊C4—41no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4C22⋊C412yes—
276C2×C4⋊C4C22⋊C411yescharacteristic
277C2×C4⋊C4C22⋊C411yescharacteristic
278C2×C4⋊C4C22×C414yes—
279C2×C4⋊C4C22×C412yes—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—22no—
288C2×C42C22×C411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42#4111yescharacteristic
297C2×C42#4111yescharacteristic
298C2×C42#4111yescharacteristic
299C2×C42C2411yescharacteristic
300#106—28no—
301#106—28no—
302C4⋊C8—22no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—41no—
313C4⋊C8—41no—
314C4⋊C8—41no—
315C4⋊C8C2×D414yes—
316#102—28no—
317#102—28no—
318C22⋊C8—28no—
319C22⋊C8—28no—
320C22⋊C8—22no—
321C22⋊C8—41no—
322#97—22no—
323#97—22no—
324C4×C8—22no—
325C4×C8—22no—
326C4×C8—22no—
327C4×C8—41no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—22no—
332C24—44no—
333C24—42no—
334C24—42no—
335C24—82no—
336C24C4×D411yescharacteristic
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—84no—
342C2×Q8—84no—
343C2×Q8—84no—
344C2×Q8—84no—
345C2×Q8—84no—
346C2×Q8—84no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—22no—
350C2×D4—21no—
351C2×D4—22no—
352C2×D4—21no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4#10114yes—
362C2×D4C2×C22⋊C412yes—
363C2×D4C2×C22⋊C412yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—84no—
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C4×D411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4C4×D411yescharacteristic
408C22×C4C4×D411yescharacteristic
409C22×C4C4×D411yescharacteristic
410C22×C4#12011yescharacteristic
411C22×C4#12011yescharacteristic
412C22×C4#12111yescharacteristic
413C22×C4#12111yescharacteristic
414C22×C4#12111yescharacteristic
415C22×C4#12111yescharacteristic
416C22×C4#12311yescharacteristic
417C22×C4#12311yescharacteristic
418C22×C4C2×SD1612yes—
419C22×C4#13512yes—
420C22×C4AGL(1,ℤ/8)12yes—
421C22×C4#13712yes—
422C22×C4C23×C411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2×#4111yescharacteristic
425M16—22no—
426M16—24no—
427M16—22no—
428M16—22no—
429M16—416no—
430M16—42no—
431M16—42no—
432M16—44no—
433M16—42no—
434M16—42no—
435M16—42no—
436M16—82no—
437M16C2×SD1614yes—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—416no—
444C2×C8—44no—
445C2×C8—44no—
446C2×C8—44no—
447C2×C8—42no—
448C2×C8—44no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—82no—
452C2×C8—82no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4C2×C22⋊C412yes—
466C4⋊C4C2×C22⋊C412yes—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C42C2×C22⋊C411yescharacteristic
476C42C2×C22⋊C411yescharacteristic
477C42C4×D411yescharacteristic
478C42C4×D411yescharacteristic
479C42#12011yescharacteristic
480C42#12011yescharacteristic
481C42#12111yescharacteristic
482C42#12111yescharacteristic
483C42#12111yescharacteristic
484C42#12111yescharacteristic
485C42#12311yescharacteristic
486C42#12311yescharacteristic
487C42C22×D411yescharacteristic
488C42C22×D411yescharacteristic
489C42C22×D411yescharacteristic
490C42C2×#4111yescharacteristic
491C23—22no—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—82no—
507C23—82no—
508C23—88no—
509C23—88no—
510C23#38611yescharacteristic
511C23#38611yescharacteristic
512C23C4×SD1612yes—
513C23#44012yes—
514C23C2×C4×D411yescharacteristic
515Q8—84no—
516Q8—84no—
517Q8—84no—
518Q8—84no—
519D4—22no—
520D4—22no—
521D4—22no—
522D4—24no—
523D4—44no—
524D4—44no—
525D4—42no—
526D4C2×#10114yes—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4#38611yescharacteristic
564C2×C4#38611yescharacteristic
565C2×C4#38611yescharacteristic
566C2×C4#38611yescharacteristic
567C2×C4#38611yescharacteristic
568C2×C4#38611yescharacteristic
569C2×C4C2×#10112yes—
570C2×C4#41312yes—
571C2×C4C4×SD1612yes—
572C2×C4#44012yes—
573C2×C4#45012yes—
574C2×C4#45012yes—
575C2×C4#45212yes—
576C2×C4#45212yes—
577C2×C4#46112yes—
578C2×C4#46112yes—
579C2×C4#46312yes—
580C2×C4#46312yes—
581C2×C4C22×C22⋊C411yescharacteristic
582C2×C4C2×C4×D411yescharacteristic
583C2×C4C2×#12011yescharacteristic
584C2×C4C2×#12111yescharacteristic
585C2×C4C2×#12111yescharacteristic
586C2×C4C2×#12311yescharacteristic
587C2×C4C22×SD1611yescharacteristic
588C2×C4C2×#13511yescharacteristic
589C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
590C2×C4C2×#13711yescharacteristic
591C8—24no—
592C8—44no—
593C8—44no—
594C8—42no—
595C8—82no—
596C8—82no—
597C22—21no—
598C22—21no—
599C22—24no—
600C22—24no—
601C22—21no—
602C22—21no—
603C22—24no—
604C22—42no—
605C22—48no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—88no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22#187712yes—
615C22#187712yes—
616C22#190112yes—
617C22#190312yes—
618C22C2×#38611yescharacteristic
619C22C2×C4×SD1611yescharacteristic
620C22C2×#44011yescharacteristic
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—42no—
628C4—42no—
629C4—84no—
630C4#187712yes—
631C4#187712yes—
632C4#190112yes—
633C4C22×#10111yescharacteristic
634C4C2×#41311yescharacteristic
635C4C2×#45011yescharacteristic
636C4C2×#45211yescharacteristic
637C4C2×#46111yescharacteristic
638C4C2×#46311yescharacteristic
639C2—22no—
640C2—21no—
641C2—21no—
642C2—44no—
643C2—82no—
644C2#924914yesdirect product factor
645C2C2×#187711yescharacteristic
646C2C2×#190111yescharacteristic
647C2C2×#190311yescharacteristic
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×#1997C2semidirect
4. C2×#1549C2semidirect
6. #9249C2direct (N × H)
7. C2×#1292C2semidirect
8. C2×#1283C2semidirect
36. #1997C22semidirect
40. #1549C22semidirect
46. #1292C22semidirect
47. #1283C22semidirect
644. C2#9249direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4762
C2×D4×M162
C2×#19972
C2×#15492
C2×#14862
#92492
C2×#12922
C2×#12832