Finite Group Databasebeta

← C2×#29261 #6463594

Subgroups — C2×#29261 #6463594

Order 512.

2106 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#125C211yesmaximal, characteristic, complemented
2C2×C4×#124C211yesmaximal, characteristic
3#29261C2116yesmaximal, direct product factor
4C2×#2097C211yesmaximal, characteristic, complemented
5C2×#2092C211yesmaximal, characteristic
6C2×#2026C211yesmaximal, characteristic, complemented
7C2×#2025C211yesmaximal, characteristic
8C2×#1983C211yesmaximal, characteristic
9C2×#1983C211yesmaximal, characteristic
10C2×#1949C211yesmaximal, characteristic, complemented
11C2×#1949C211yesmaximal, characteristic, complemented
12C2×#1865C211yesmaximal, characteristic, complemented
13C2×#1863C211yesmaximal, characteristic, complemented
14C2×#1864C211yesmaximal, characteristic
15C2×#1862C211yesmaximal, characteristic
16C2×#1831C211yesmaximal, characteristic, complemented
17C22×#125C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C4×#125—24no—
36C4×#125C2218yescomplemented
37C4×#124—24no—
38C4×#124C2218yes—
39C2×#391C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44C2×#388—21no—
45C2×#387—21no—
46C2×#387—21no—
47C2×#382C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4—21no—
54C2×C4×C22⋊C4—21no—
55C2×C43C2211yescharacteristic
56#2097C2218yescomplemented
57#2092C2218yes—
58#2026C2218yescomplemented
59#2025C2218yes—
60#1983C2218yes—
61#1983C2218yes—
62#1949C2218yescomplemented
63#1949C2218yescomplemented
64#1865C2218yescomplemented
65#1864C2218yes—
66#1863C2218yescomplemented
67#1862C2218yes—
68#1831C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×#125—24no—
85C2×#125—24no—
86C2×#125—22no—
87C2×#125C2314yes—
88C2×#124—22no—
89C2×#124—24no—
90C2×#124Q818yes—
91C2×#124C2314yes—
92C2×C4×Q8—22no—
93C2×C4×Q8—42no—
94C2×C4×Q8—42no—
95C2×C4×Q8C2314yes—
96C2×C4×D4—22no—
97C2×C4×D4—42no—
98C2×C4×D4—42no—
99C2×C4×D4C2314yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C42—21no—
117C22×C42—21no—
118C22×C42—21no—
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C2×#107—42no—
127C2×#107—42no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106—42no—
131C2×#106—42no—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8—24no—
135C2×C4⋊C8—24no—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2314yes—
138C2×#103—24no—
139C2×#103—24no—
140C2×#103—42no—
141C2×#103—42no—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102—24no—
147C2×#102—24no—
148C2×#102—42no—
149C2×#102—42no—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154#391—22no—
155#391C2314yes—
156#390—22no—
157#390C2314yes—
158#389—22no—
159#389—22no—
160#389C2314yes—
161#389C2314yes—
162#388—28no—
163#388—28no—
164#387—28no—
165#387—28no—
166#382—22no—
167#382—22no—
168#382C2314yes—
169#382C2314yes—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—28no—
173C4×C4⋊C4—28no—
174C4×C4⋊C4C2314yes—
175C4×C4⋊C4C2314yes—
176C4×C22⋊C4—28no—
177C4×C22⋊C4—28no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C43D414yes—
189C43C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#336C2314yes—
195#336C2314yes—
196#336C2314yes—
197#336C2314yes—
198C25—21no—
199C22×Q8—21no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8C2×Q814yes—
206C22×Q8C2412yes—
207C22×D4—21no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4C2×Q814yes—
214C22×D4C2412yes—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—24no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
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—22no—
244C22×C8—22no—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249#125—216no—
250#125—28no—
251#125—28no—
252#124—216no—
253#124—28no—
254#124C2×Q8116yes—
255C4×Q8—44no—
256C4×Q8—44no—
257C4×Q8—48no—
258C4×D4—44no—
259C4×D4—44no—
260C4×D4—48no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4#4114yes—
288C2×C4⋊C4#4114yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
324C2×C42C2×D414yes—
325C2×C42C2×D412yes—
326C2×C42C2×Q814yes—
327C2×C42#4114yes—
328C2×C42C2412yes—
329C2×C42C2412yes—
330C2×C42C2412yes—
331#107—44no—
332#107—44no—
333#106—44no—
334#106—44no—
335C4⋊C8—28no—
336C4⋊C8—28no—
337#103—28no—
338#103—28no—
339#103—44no—
340#103—44no—
341#102—28no—
342#102—28no—
343#102—44no—
344#102—44no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355C24—24no—
356C24—22no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24C22×D411yescharacteristic
364C2×Q8—24no—
365C2×Q8—22no—
366C2×Q8—28no—
367C2×Q8—28no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8C22×Q814yes—
377C2×D4—24no—
378C2×D4—28no—
379C2×D4—28no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4C22×Q814yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—48no—
416C22×C4—48no—
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—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—48no—
436C22×C4—816no—
437C22×C4C22⋊Q814yes—
438C22×C4C22⋊Q814yes—
439C22×C4#13514yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×Q812yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C4C2511yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—44no—
467C2×C8—44no—
468C2×C8—44no—
469C2×C8—44no—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—48no—
491C4⋊C4C2×#4114yes—
492C4⋊C4C2×#4114yes—
493C22⋊C4—216no—
494C22⋊C4—216no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C42—24no—
499C42—24no—
500C42—28no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—44no—
506C42—44no—
507C42—44no—
508C42—48no—
509C42—48no—
510C42—48no—
511C42C22⋊Q818yes—
512C42C22×Q814yes—
513C42C2×#4114yes—
514C23—24no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—84no—
525C23—84no—
526C23—84no—
527C23—84no—
528C23C2×C22⋊Q812yes—
529C23#52512yes—
530C23#53512yes—
531C23C2×#13512yes—
532C23#57512yes—
533C23#57512yes—
534C23#57512yes—
535C23C23×D411yescharacteristic
536Q8—24no—
537Q8—24no—
538Q8—48no—
539Q8—42no—
540Q8—42no—
541D4—24no—
542D4—24no—
543D4—48no—
544D4—42no—
545D4—42no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—48no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—816no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4C2×C22⋊Q814yes—
597C2×C4C2×C22⋊Q814yes—
598C2×C4C2×C22⋊Q812yes—
599C2×C4#52512yes—
600C2×C4#53512yes—
601C2×C4#55414yes—
602C2×C4#55414yes—
603C2×C4#55914yes—
604C2×C4#55914yes—
605C2×C4C2×#13514yes—
606C2×C4C23×Q811yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×H(5,𝔽2)11yescharacteristic
612C8—42no—
613C8—42no—
614C8—42no—
615C8—42no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—816no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22#258014yes—
625C22#307614yes—
626C22#308014yes—
627C22#310214yes—
628C22#311114yes—
629C22#321114yes—
630C22#321914yes—
631C22C22×C22⋊Q811yescharacteristic
632C22C2×#52511yescharacteristic
633C22C2×#53511yescharacteristic
634C22C22×#13511yescharacteristic
635C22C2×#57511yescharacteristic
636C22C2×#57511yescharacteristic
637C22C2×#57511yescharacteristic
638C4—24no—
639C4—24no—
640C4—24no—
641C4—44no—
642C4—42no—
643C4—44no—
644C4—42no—
645C4—42no—
646C4—44no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4#258014yes—
652C4C2×#55411yescharacteristic
653C4C2×#55411yescharacteristic
654C4C2×#55911yescharacteristic
655C4C2×#55911yescharacteristic
656C2—84no—
657C2#2926118yesdirect product factor
658C2C2×#258011yescharacteristic
659C2C2×#307611yescharacteristic
660C2C2×#308011yescharacteristic
661C2C2×#310211yescharacteristic
662C2C2×#311111yescharacteristic
663C2C2×#321111yescharacteristic
664C2C2×#321911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×#125C2semidirect
3. #29261C2direct (N × H)
4. C2×#2097C2semidirect
6. C2×#2026C2semidirect
10. C2×#1949C2semidirect
11. C2×#1949C2semidirect
12. C2×#1865C2semidirect
13. C2×#1863C2semidirect
16. C2×#1831C2semidirect
36. C4×#125C22semidirect
56. #2097C22semidirect
58. #2026C22semidirect
62. #1949C22semidirect
63. #1949C22semidirect
64. #1865C22semidirect
66. #1863C22semidirect
68. #1831C22semidirect
657. C2#29261direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#1252
C2×C4×#1242
#292612
C2×#20972
C2×#20922
C2×#20262
C2×#20252
C2×#19832
C2×#19492
C2×#18652
C2×#18632
C2×#18642
C2×#18622
C2×#18312