Finite Group Databasebeta

← C2×#8312 #184841

Subgroups — C2×#8312 #184841

Order 512.

3850 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#41C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
4C22×Q8⋊C8C212yesmaximal, complemented
5C22×D4⋊C8C212yesmaximal, complemented
6#8312C218yesmaximal, direct product factor
7C23×#41C411yescharacteristic
8C2×C4×#41—22no—
9C2×C4×#41—22no—
10C2×C4×#41—28no—
11C2×C4×#41—28no—
12C2×C4×#41C414yes—
13C2×C4×#41C2214yes—
14C22×C4×Q8C411yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C22×C4×D4—21no—
17C22×C4×D4—21no—
18C22×C4×D4C411yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117—21no—
21C22×#117C411yescharacteristic
22C23×C42—21no—
23C23×C42C2211yescharacteristic
24C22×C4⋊C8C2212yes—
25C22×C22⋊C8—21no—
26C22×C22⋊C8—21no—
27C22×C4×C8C2212yes—
28#1797—22no—
29#1797C2214yescomplemented
30C4×C22⋊C8—22no—
31C4×C22⋊C8C2214yescomplemented
32C2×#336—21no—
33C2×#336—21no—
34C2×Q8⋊C8—28no—
35C2×Q8⋊C8C2218yescomplemented
36C2×D4⋊C8—28no—
37C2×D4⋊C8C2218yescomplemented
38C22×#41—24no—
39C22×#41—24no—
40C22×#41—24no—
41C22×#41—24no—
42C22×#41—22no—
43C22×#41—24no—
44C22×#41C818yescomplemented
45C22×#41C2×C414yes—
46C23×Q8C2×C411yescharacteristic
47C23×D4—21no—
48C23×D4C2×C411yescharacteristic
49C24×C4—21no—
50C24×C4—21no—
51C24×C4—21no—
52C24×C4C2×C411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C23×C8—22no—
56C23×C8—22no—
57C4×#41—232no—
58C4×#41—232no—
59C4×#41—216no—
60C4×#41—216no—
61C4×#41—216no—
62C4×#41—216no—
63C2×C4×Q8—24no—
64C2×C4×Q8—24no—
65C2×C4×Q8—22no—
66C2×C4×Q8—21no—
67C2×C4×Q8—24no—
68C2×C4×Q8—21no—
69C2×C4×Q8C2×C414yes—
70C2×C4×Q8C2×C412yes—
71C2×C4×Q8D418yes—
72C2×C4×Q8C2312yes—
73C2×C4×D4—216no—
74C2×C4×D4—216no—
75C2×C4×D4—24no—
76C2×C4×D4—24no—
77C2×C4×D4—24no—
78C2×C4×D4—24no—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C2×C4×D4—22no—
82C2×C4×D4—24no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4C2×C414yes—
86C2×C4×D4C2×C412yes—
87C2×C4×D4D418yes—
88C2×C4×D4C2312yes—
89C2×#117—216no—
90C2×#117—22no—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117—24no—
95C2×#117C2×C414yes—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4C2×C411yescharacteristic
99C22×C4⋊C4C2×C411yescharacteristic
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C42—216no—
104C22×C42—21no—
105C22×C42—21no—
106C22×C42—21no—
107C22×C42—24no—
108C22×C42—24no—
109C22×C42—24no—
110C22×C42—24no—
111C22×C42C2×C411yescharacteristic
112C22×C42C2×C412yes—
113C22×C42D412yes—
114C22×C42D411yescharacteristic
115C22×C42D412yes—
116C22×C42D411yescharacteristic
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C2×C4⋊C8—24no—
120C2×C4⋊C8—24no—
121C2×C4⋊C8—22no—
122C2×C4⋊C8C2314yes—
123C2×C22⋊C8—28no—
124C2×C22⋊C8—24no—
125C2×C22⋊C8—24no—
126C2×C22⋊C8—44no—
127C2×C4×C8—24no—
128C2×C4×C8—24no—
129C2×C4×C8—22no—
130C2×C4×C8C2314yes—
131#336—24no—
132#336—24no—
133Q8⋊C8—216no—
134D4⋊C8—216no—
135C25—21no—
136C25C22⋊C411yescharacteristic
137C2×#41—216no—
138C2×#41—216no—
139C2×#41—232no—
140C2×#41—232no—
141C2×#41—232no—
142C2×#41—28no—
143C2×#41—28no—
144C2×#41—232no—
145C2×#41—28no—
146C2×#41—28no—
147C2×#41—216no—
148C2×#41—28no—
149C2×#41C2×C8116yescomplemented
150C22×Q8—21no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8C22⋊C414yes—
155C22×Q8C2×C814yes—
156C22×Q8M1614yes—
157C22×Q8C22×C412yes—
158C22×D4—28no—
159C22×D4—28no—
160C22×D4—28no—
161C22×D4—24no—
162C22×D4—22no—
163C22×D4—21no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4C22⋊C414yes—
167C22×D4C2×C814yes—
168C22×D4M1614yes—
169C22×D4C22×C412yes—
170C23×C4—28no—
171C23×C4—28no—
172C23×C4—21no—
173C23×C4—28no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—28no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—28no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—42no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22⋊C412yes—
200C23×C4C22⋊C412yes—
201C23×C4C22⋊C412yes—
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22⋊C411yescharacteristic
204C23×C4C2×C814yes—
205C23×C4M1614yes—
206C23×C4C22×C412yes—
207C23×C4C22×C411yescharacteristic
208C23×C4C2×D412yes—
209C23×C4C2×D412yes—
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C22×C8—28no—
213C22×C8—24no—
214C22×C8—24no—
215C22×C8—44no—
216C22×C8—44no—
217C22×C8—44no—
218C22×C8—44no—
219C22×C8—44no—
220C4×Q8—216no—
221C4×Q8—28no—
222C4×Q8—28no—
223C4×Q8—24no—
224C4×Q8—28no—
225C4×Q8—28no—
226C4×Q8—24no—
227C4×Q8C22⋊C418yes—
228C4×Q8C2×D418yes—
229C4×D4—232no—
230C4×D4—232no—
231C4×D4—232no—
232C4×D4—232no—
233C4×D4—216no—
234C4×D4—28no—
235C4×D4—28no—
236C4×D4—24no—
237C4×D4—28no—
238C4×D4—28no—
239C4×D4—24no—
240C4×D4C22⋊C418yes—
241C4×D4C2×D418yes—
242#117—216no—
243#117—232no—
244#117—232no—
245#117—28no—
246#117—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4C22⋊C414yes—
257C2×C4⋊C4C22⋊C414yes—
258C2×C4⋊C4C22×C412yes—
259C2×C4⋊C4C22×C412yes—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—22no—
267C2×C42—28no—
268C2×C42—28no—
269C2×C42—232no—
270C2×C42—232no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—28no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—28no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42C22⋊C412yes—
287C2×C42C22⋊C412yes—
288C2×C42C22⋊C412yes—
289C2×C42C22×C412yes—
290C2×C42C22×C411yescharacteristic
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2411yescharacteristic
298C4⋊C8—24no—
299C4⋊C8—24no—
300C4⋊C8—44no—
301C22⋊C8—216no—
302C22⋊C8—48no—
303C4×C8—24no—
304C4×C8—24no—
305C4×C8—44no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—24no—
310C24—22no—
311C24—24no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24C22⋊C814yes—
321C24C2×C22⋊C412yes—
322C24C2×C22⋊C411yescharacteristic
323#41—232no—
324#41—232no—
325#41—2128no—
326#41—232no—
327#41—232no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—28no—
334C2×Q8—28no—
335C2×Q8—24no—
336C2×Q8—28no—
337C2×Q8C22⋊C8116yescomplemented
338C2×Q8C2×C22⋊C414yes—
339C2×Q8C22×C814yes—
340C2×Q8C2×M1614yes—
341C2×D4—216no—
342C2×D4—216no—
343C2×D4—264no—
344C2×D4—216no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—24no—
349C2×D4—28no—
350C2×D4—24no—
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—24no—
354C2×D4C22⋊C8116yescomplemented
355C2×D4C2×C22⋊C414yes—
356C2×D4C22×C814yes—
357C2×D4C2×M1614yes—
358C22×C4—216no—
359C22×C4—216no—
360C22×C4—264no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—216no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—216no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—416no—
406C22×C4—48no—
407C22×C4—432no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—416no—
420C22×C4—48no—
421C22×C4—416no—
422C22×C4C22⋊C814yes—
423C22×C4C22⋊C818yes—
424C22×C4#10414yes—
425C22×C4C2×C22⋊C412yes—
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C412yes—
428C22×C4C2×C22⋊C412yes—
429C22×C4C2×C22⋊C412yes—
430C22×C4C2×C22⋊C412yes—
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4#12014yes—
434C22×C4#12014yes—
435C22×C4C22×C814yes—
436C22×C4C22×C812yes—
437C22×C4C2×M1614yes—
438C22×C4C2×M1612yes—
439C22×C4C2×M1612yes—
440C22×C4C2×M1612yes—
441C22×C4#13514yes—
442C22×C4AGL(1,ℤ/8)12yes—
443C22×C4#13712yes—
444C22×C4C23×C411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C2×C8—28no—
448C2×C8—44no—
449C2×C8—416no—
450C2×C8—44no—
451C2×C8—44no—
452C2×C8—44no—
453C2×C8—44no—
454C2×C8—88no—
455C4⋊C4—216no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4C2×C22⋊C414yes—
463C4⋊C4C2×C22⋊C414yes—
464C22⋊C4—216no—
465C22⋊C4—216no—
466C22⋊C4—28no—
467C22⋊C4—28no—
468C42—216no—
469C42—216no—
470C42—22no—
471C42—22no—
472C42—24no—
473C42—28no—
474C42—24no—
475C42—24no—
476C42—28no—
477C42—24no—
478C42—24no—
479C42C2×C22⋊C412yes—
480C42C2×C22⋊C412yes—
481C42#12014yes—
482C42#12014yes—
483C42C22×D412yes—
484C42C22×D412yes—
485C23—24no—
486C23—22no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—28no—
493C23—28no—
494C23—22no—
495C23—24no—
496C23—28no—
497C23—216no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—48no—
503C23—48no—
504C23—44no—
505C23—48no—
506C23—44no—
507C23—48no—
508C23—44no—
509C23—48no—
510C23—44no—
511C23—48no—
512C23#37912yes—
513C23C2×C22⋊C814yes—
514C23C2×C22⋊C812yes—
515C23#40712yes—
516C23#41612yes—
517C23#41712yes—
518C23C2×#10412yes—
519C23#42112yes—
520C23C22×C22⋊C411yescharacteristic
521Q8—24no—
522Q8—24no—
523Q8—24no—
524Q8—216no—
525Q8C2×C22⋊C818yescomplemented
526D4—264no—
527D4—24no—
528D4—24no—
529D4—24no—
530D4—216no—
531D4C2×C22⋊C818yescomplemented
532C2×C4—264no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—216no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—216no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—432no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—416no—
574C2×C4#37912yes—
575C2×C4#37912yes—
576C2×C4#37912yes—
577C2×C4C2×C22⋊C814yes—
578C2×C4C2×C22⋊C814yes—
579C2×C4C2×C22⋊C814yes—
580C2×C4C2×C22⋊C812yes—
581C2×C4#40714yes—
582C2×C4#40714yes—
583C2×C4#40712yes—
584C2×C4#41612yes—
585C2×C4#41712yes—
586C2×C4C2×#10414yes—
587C2×C4#44918yes—
588C2×C4#45218yes—
589C2×C4C22×C22⋊C411yescharacteristic
590C2×C4C22×C22⋊C411yescharacteristic
591C2×C4C2×#12012yes—
592C2×C4C2×#12012yes—
593C2×C4C23×C811yescharacteristic
594C2×C4C22×M1611yescharacteristic
595C2×C4C22×M1611yescharacteristic
596C2×C4C22×M1611yescharacteristic
597C2×C4C2×#13512yes—
598C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
599C2×C4C2×#13711yescharacteristic
600C8—48no—
601C8—84no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—28no—
606C22—44no—
607C22—44no—
608C22—42no—
609C22—44no—
610C22—42no—
611C22—44no—
612C22—42no—
613C22—44no—
614C22—416no—
615C22—432no—
616C22—42no—
617C22—44no—
618C22#147614yes—
619C22#147714yes—
620C22#148814yes—
621C22#148914yes—
622C22#177914yes—
623C22#178814yes—
624C22#178914yes—
625C22C2×#37911yescharacteristic
626C22C22×C22⋊C811yescharacteristic
627C22C2×#40711yescharacteristic
628C22C2×#41611yescharacteristic
629C22C2×#41711yescharacteristic
630C22C22×#10411yescharacteristic
631C22C2×#42111yescharacteristic
632C4—24no—
633C4—28no—
634C4—22no—
635C4—24no—
636C4—22no—
637C4—42no—
638C4—44no—
639C4—44no—
640C4#177914yes—
641C4#178814yes—
642C4C22×C22⋊C811yescharacteristic
643C4C22×C22⋊C811yescharacteristic
644C4C2×#40711yescharacteristic
645C4C2×#40711yescharacteristic
646C4C2×#44912yes—
647C4C2×#45212yes—
648C2—44no—
649C2—48no—
650C2#831218yesdirect product factor
651C2C2×#147611yescharacteristic
652C2C2×#147711yescharacteristic
653C2C2×#148811yescharacteristic
654C2C2×#148911yescharacteristic
655C2C2×#177911yescharacteristic
656C2C2×#178811yescharacteristic
657C2C2×#178911yescharacteristic
subgroupG/H
centerC23×C4C2×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC24C2×C22⋊C4
prime radicalC23×#41C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1797C2semidirect
3. C2×C4×C22⋊C8C2semidirect
4. C22×Q8⋊C8C2semidirect
5. C22×D4⋊C8C2semidirect
6. #8312C2direct (N × H)
29. #1797C22semidirect
31. C4×C22⋊C8C22semidirect
35. C2×Q8⋊C8C22semidirect
37. C2×D4⋊C8C22semidirect
44. C22×#41C8semidirect
149. C2×#41C2×C8semidirect
337. C2×Q8C22⋊C8semidirect
354. C2×D4C22⋊C8semidirect
525. Q8C2×C22⋊C8semidirect
531. D4C2×C22⋊C8semidirect
650. C2#8312direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×#412
C2×#17972
C2×C4×C22⋊C82
C22×Q8⋊C82
C22×D4⋊C82
#83122