Finite Group Databasebeta

← C2×#23759 #6432457

Subgroups — C2×#23759 #6432457

Order 512.

2118 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic
3#23759C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2097C211yesmaximal, characteristic, complemented
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2077C211yesmaximal, characteristic
8C2×#2075C211yesmaximal, characteristic
9C2×#2074C211yesmaximal, characteristic, complemented
10C2×#2036C211yesmaximal, characteristic
11C2×#2035C211yesmaximal, characteristic
12C2×C4×#107C211yesmaximal, characteristic, complemented
13C2×C4×#106C211yesmaximal, characteristic, complemented
14C2×C4×#103C211yesmaximal, characteristic, complemented
15C2×C4×#103C211yesmaximal, characteristic, complemented
16C2×C4×C22⋊C8C211yesmaximal, characteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32C4×C22⋊Q8—24no—
33C4×C22⋊Q8C2218yes—
34C2×#391C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#387—21no—
37C2×#385—21no—
38#2283—24no—
39#2283C2218yes—
40C2×#383—21no—
41C2×#382C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×#381—21no—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4—21no—
52C2×C4×C22⋊C4—21no—
53C2×#376—21no—
54#2101C2218yescomplemented
55#2097C2218yescomplemented
56#2085C2218yescomplemented
57#2077C2218yes—
58#2075C2218yes—
59#2074C2218yescomplemented
60#2036C2218yes—
61#2035C2218yes—
62C4×#107C2218yescomplemented
63C4×#106C2218yescomplemented
64C4×#103C2218yescomplemented
65C4×#103C2218yescomplemented
66C4×C22⋊C8C2218yes—
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C24×C4D411yescharacteristic
79C24×C4D411yescharacteristic
80C24×C4C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8C2314yes—
89C2×C4×Q8—22no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8C2314yes—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C42—24no—
108C22×C42—24no—
109C22×C42—21no—
110C22×C42—21no—
111C22×C42—21no—
112C22×C42—21no—
113C22×C42D411yescharacteristic
114C22×C42D414yes—
115C22×C42D411yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2314yes—
118C22×C42C2311yescharacteristic
119C2×#107—24no—
120C2×#107—24no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106—24no—
124C2×#106—24no—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×#103—24no—
128C2×#103—24no—
129C2×#103—24no—
130C2×#103—24no—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×C22⋊C8—24no—
136C2×C22⋊C8—24no—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×C4×C8—24no—
140C2×C4×C8—24no—
141C2×C4×C8C2314yes—
142C2×C4×C8C2314yes—
143#391—22no—
144#391C2314yes—
145#389—22no—
146#389C2314yes—
147#387—28no—
148#385—28no—
149#383—28no—
150#382—22no—
151#382—22no—
152#382C2314yes—
153#382C2314yes—
154#381—28no—
155#380—22no—
156#380—22no—
157#380—22no—
158#380C2314yes—
159#380C2314yes—
160#380C2314yes—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4C2314yes—
166C4×C4⋊C4C2314yes—
167C4×C4⋊C4C2314yes—
168C4×C22⋊C4—28no—
169C4×C22⋊C4—28no—
170#376—28no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25C2×D411yescharacteristic
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8#4114yes—
202C22×Q8C2412yes—
203C23×C4—24no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—42no—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D412yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8#4114yes—
244C22×C8#4114yes—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22⋊Q8—216no—
250C22⋊Q8—28no—
251C22⋊Q8—28no—
252C4×Q8—28no—
253C4×Q8—28no—
254C4×Q8—48no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
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—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4#4114yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—216no—
302C2×C42—28no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42C2×D414yes—
313C2×C42C2×D412yes—
314C2×C42C2×D414yes—
315C2×C42C2×D412yes—
316C2×C42#4114yes—
317C2×C42#4114yes—
318C2×C42C2412yes—
319C2×C42C2412yes—
320#107—28no—
321#107—28no—
322#106—28no—
323#106—28no—
324#103—28no—
325#103—28no—
326#103—28no—
327#103—28no—
328C22⋊C8—28no—
329C22⋊C8—28no—
330C4×C8—28no—
331C4×C8—28no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24#13514yes—
352C24C22×D412yes—
353C24C22×D411yescharacteristic
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—28no—
357C2×Q8—28no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—42no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—42no—
366C2×Q8C2×#4114yes—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—44no—
403C22×C4—416no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—416no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4#12314yes—
432C22×C4#12314yes—
433C22×C4#13514yes—
434C22×C4#13714yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2511yescharacteristic
448C2×C8—28no—
449C2×C8—28no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—44no—
461C2×C8—44no—
462C2×C8C2×#4114yes—
463C2×C8C2×#4114yes—
464C4⋊C4—216no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4C2×#4114yes—
488C22⋊C4—216no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C42—28no—
493C42—28no—
494C42—24no—
495C42—22no—
496C42—22no—
497C42—48no—
498C42—48no—
499C42—48no—
500C42—48no—
501C42#12318yes—
502C42C2×#4114yes—
503C42C2×#4114yes—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23C2×#12312yes—
518C23#52512yes—
519C23#53212yes—
520C23C2×#13514yes—
521C23C2×#13512yes—
522C23C2×#13512yes—
523C23C2×#13712yes—
524C23#57512yes—
525C23C23×D411yescharacteristic
526Q8—24no—
527Q8—24no—
528Q8—48no—
529Q8—42no—
530Q8—42no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—416no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—416no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
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—42no—
578C2×C4—42no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4#485116yes—
586C2×C4C2×#12314yes—
587C2×C4C2×#12314yes—
588C2×C4C2×#12312yes—
589C2×C4#52512yes—
590C2×C4#52914yes—
591C2×C4#52914yes—
592C2×C4#52914yes—
593C2×C4#52914yes—
594C2×C4#53212yes—
595C2×C4C2×#13514yes—
596C2×C4C2×#13714yes—
597C2×C4C22×#4111yescharacteristic
598C2×C4C22×#4111yescharacteristic
599C2×C4C22×#4111yescharacteristic
600C2×C4C22×#4111yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C8—24no—
604C8—24no—
605C8—42no—
606C8—42no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—416no—
614C22—42no—
615C22#239514yes—
616C22C2×#48514yes—
617C22#309114yes—
618C22#310214yes—
619C22#310814yes—
620C22#316814yes—
621C22#317214yes—
622C22C22×#12311yescharacteristic
623C22C2×#52511yescharacteristic
624C22C2×#53211yescharacteristic
625C22C22×#13511yescharacteristic
626C22C22×#13511yescharacteristic
627C22C22×#13711yescharacteristic
628C22C2×#57511yescharacteristic
629C4—24no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—42no—
634C4—44no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4#239514yes—
642C4C2×#48518yes—
643C4C2×#52911yescharacteristic
644C4C2×#52911yescharacteristic
645C4C2×#52911yescharacteristic
646C4C2×#52911yescharacteristic
647C2—44no—
648C2#2375918yesdirect product factor
649C2C2×#239511yescharacteristic
650C2C22×#48511yescharacteristic
651C2C2×#309111yescharacteristic
652C2C2×#310211yescharacteristic
653C2C2×#310811yescharacteristic
654C2C2×#316811yescharacteristic
655C2C2×#317211yescharacteristic
subgroupG/H
centerC23×C4C2×D4
commutator, Frattini, cosocleC22×C4C25
socleC24C22×D4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #23759C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2097C2semidirect
6. C2×#2085C2semidirect
9. C2×#2074C2semidirect
12. C2×C4×#107C2semidirect
13. C2×C4×#106C2semidirect
14. C2×C4×#103C2semidirect
15. C2×C4×#103C2semidirect
54. #2101C22semidirect
55. #2097C22semidirect
56. #2085C22semidirect
59. #2074C22semidirect
62. C4×#107C22semidirect
63. C4×#106C22semidirect
64. C4×#103C22semidirect
65. C4×#103C22semidirect
648. C2#23759direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C2×#22832
#237592
C2×#21012
C2×#20972
C2×#20852
C2×#20772
C2×#20752
C2×#20742
C2×#20362
C2×#20352
C2×C4×#1072
C2×C4×#1062
C2×C4×#1032
C2×C4×C22⋊C82