Finite Group Databasebeta

← C2×#29436 #6461609

Subgroups — C2×#29436 #6461609

Order 512.

1960 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2644C211yesmaximal, characteristic, complemented
2C2×#2319C211yesmaximal, characteristic, complemented
3#29436C2116yesmaximal, direct product factor
4C2×#2087C211yesmaximal, characteristic, complemented
5C2×#2067C211yesmaximal, characteristic, complemented
6C2×#2067C211yesmaximal, characteristic, complemented
7C2×#2054C211yesmaximal, characteristic
8C2×#1970C211yesmaximal, characteristic
9C2×#1942C211yesmaximal, characteristic, complemented
10C2×#1936C211yesmaximal, characteristic, complemented
11C2×#1936C211yesmaximal, characteristic, complemented
12C2×#1925C211yesmaximal, characteristic, complemented
13C2×#1921C211yesmaximal, characteristic, complemented
14C2×#1879C211yesmaximal, characteristic
15C2×#1862C211yesmaximal, characteristic, complemented
16C2×#1768C211yesmaximal, characteristic, complemented
17C23×D8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#97C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2644—24no—
33#2644C2218yescomplemented
34C2×#397—21no—
35C2×#397—21no—
36C2×#397—21no—
37C2×#396—21no—
38#2319—24no—
39#2319C2218yescomplemented
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#388—21no—
51C2×#388—21no—
52C2×#381—21no—
53C2×#380—21no—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×#376C2211yescharacteristic
56#2087C2218yescomplemented
57#2067C2218yescomplemented
58#2067C2218yescomplemented
59#2054C2218yes—
60#1970C2218yes—
61#1942C2218yescomplemented
62#1936C2218yescomplemented
63#1936C2218yescomplemented
64#1925C2218yescomplemented
65#1921C2218yescomplemented
66#1879C2218yes—
67#1862C2218yescomplemented
68#1768C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C22×D8—24no—
81C22×D8—24no—
82C22×D8—42no—
83C22×D8C2314yes—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4—22no—
91C2×C4×D4C2314yes—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
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×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C42—21no—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C2×#107—24no—
119C2×#107—24no—
120C2×#107—42no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×C4⋊C8—24no—
125C2×C4⋊C8—24no—
126C2×C4⋊C8C2314yes—
127C2×C4⋊C8C2314yes—
128C2×#102—24no—
129C2×#102—24no—
130C2×#102—24no—
131C2×#102—24no—
132C2×#102—42no—
133C2×#102—42no—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#97—42no—
141C2×#97C2314yes—
142C2×C4×C8—42no—
143C2×C4×C8C2314yes—
144#397—28no—
145#397—28no—
146#397—28no—
147#396—28no—
148#390—22no—
149#390—22no—
150#390—22no—
151#390—22no—
152#390—22no—
153#390C2314yes—
154#390C2314yes—
155#390C2314yes—
156#390C2314yes—
157#390C2314yes—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389C2314yes—
164#389C2314yes—
165#389C2314yes—
166#389C2314yes—
167#389C2314yes—
168#388—28no—
169#388—28no—
170#381—28no—
171#380—28no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4C2314yes—
174#376D414yes—
175#376C2314yes—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95D411yescharacteristic
183C2×#95D411yescharacteristic
184C2×#95D411yescharacteristic
185C2×#95D411yescharacteristic
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25—21no—
195C25—21no—
196C22×D4—21no—
197C22×D4—21no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4#4114yes—
210C22×D4C2412yes—
211C22×D4C2412yes—
212C23×C4—21no—
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—82no—
221C23×C4—82no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2411yescharacteristic
229C2×D8—28no—
230C2×D8—28no—
231C2×D8—44no—
232C2×D8—416no—
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—
249C4×D4—216no—
250C4×D4—28no—
251C4×D4—28no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
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—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—82no—
275C2×C4⋊C4—82no—
276C2×C4⋊C4—82no—
277C2×C4⋊C4—82no—
278C2×C4⋊C4—82no—
279C2×C4⋊C4#4114yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—82no—
299C2×C22⋊C4—82no—
300C2×C22⋊C4—82no—
301C2×C22⋊C4—82no—
302C2×C22⋊C4—82no—
303C2×C22⋊C4—82no—
304C2×C22⋊C4—82no—
305C2×C22⋊C4—82no—
306C2×C22⋊C4—82no—
307C2×C22⋊C4—82no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—82no—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2412yes—
320C2×C42C2412yes—
321C2×C42C2412yes—
322#107—28no—
323#107—28no—
324#107—44no—
325C4⋊C8—28no—
326C4⋊C8—28no—
327#102—28no—
328#102—28no—
329#102—28no—
330#102—28no—
331#102—44no—
332#102—44no—
333#97—44no—
334C4×C8—44no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95C2×D414yes—
342#95C2×D414yes—
343#95C2×D414yes—
344#95C2×D414yes—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24C22×D411yescharacteristic
362C2×D4—24no—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4C2×#4114yes—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
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—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—216no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
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—84no—
432C22×C4—84no—
433C22×C4—816no—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C42−1+412yes—
447C22×C4C2511yescharacteristic
448D8—416no—
449C2×C8—28no—
450C2×C8—28no—
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—22no—
461C2×C8—44no—
462C2×C8—44no—
463C2×C8C2×#4114yes—
464C2×C8C2×#4114yes—
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—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4C2×#4114yes—
488C22⋊C4—216no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C22⋊C4—84no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C42—24no—
500C42—24no—
501C42—24no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—84no—
509C23—24no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—84no—
527C23—84no—
528C23—84no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23#52512yes—
538C23#53212yes—
539C23#53712yes—
540C23#57612yes—
541C23#57612yes—
542C23#57612yes—
543C23#57812yes—
544C23C23×D411yescharacteristic
545D4—24no—
546D4—24no—
547D4—48no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—84no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
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—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—816no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4#52512yes—
605C2×C4#53212yes—
606C2×C4#53712yes—
607C2×C4#55114yes—
608C2×C4#55214yes—
609C2×C4#55314yes—
610C2×C4#55614yes—
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×2−1+411yescharacteristic
617C8—24no—
618C8—24no—
619C8—42no—
620C8—42no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—816no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—168no—
637C22#260114yes—
638C22#312214yes—
639C22#312414yes—
640C22#317514yes—
641C22#317614yes—
642C22#325914yes—
643C22#326214yes—
644C22C2×#52511yescharacteristic
645C22C2×#53211yescharacteristic
646C22C2×#53711yescharacteristic
647C22C2×#57611yescharacteristic
648C22C2×#57611yescharacteristic
649C22C2×#57611yescharacteristic
650C22C2×#57811yescharacteristic
651C4—42no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4#260114yes—
664C4C2×#55111yescharacteristic
665C4C2×#55211yescharacteristic
666C4C2×#55311yescharacteristic
667C4C2×#55611yescharacteristic
668C2—84no—
669C2—162no—
670C2#2943618yesdirect product factor
671C2C2×#260111yescharacteristic
672C2C2×#312211yescharacteristic
673C2C2×#312411yescharacteristic
674C2C2×#317511yescharacteristic
675C2C2×#317611yescharacteristic
676C2C2×#325911yescharacteristic
677C2C2×#326211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2644C2semidirect
2. C2×#2319C2semidirect
3. #29436C2direct (N × H)
4. C2×#2087C2semidirect
5. C2×#2067C2semidirect
6. C2×#2067C2semidirect
9. C2×#1942C2semidirect
10. C2×#1936C2semidirect
11. C2×#1936C2semidirect
12. C2×#1925C2semidirect
13. C2×#1921C2semidirect
15. C2×#1862C2semidirect
16. C2×#1768C2semidirect
33. #2644C22semidirect
39. #2319C22semidirect
56. #2087C22semidirect
57. #2067C22semidirect
58. #2067C22semidirect
61. #1942C22semidirect
62. #1936C22semidirect
63. #1936C22semidirect
64. #1925C22semidirect
65. #1921C22semidirect
67. #1862C22semidirect
68. #1768C22semidirect
670. C2#29436direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26442
C2×#23192
#294362
C2×#20872
C2×#20672
C2×#20542
C2×#19702
C2×#19422
C2×#19362
C2×#19252
C2×#19212
C2×#18792
C2×#18622
C2×#17682