Finite Group Databasebeta

← C2×#26431 #6441920

Subgroups — C2×#26431 #6441920

Order 512.

2256 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×#2440C211yesmaximal, characteristic
2C2×#2362C211yesmaximal, characteristic
3#26431C2116yesmaximal, direct product factor
4C2×#2099C211yesmaximal, characteristic, complemented
5C2×#2086C211yesmaximal, characteristic, complemented
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2024C211yesmaximal, characteristic, complemented
8C2×#1987C211yesmaximal, characteristic, complemented
9C2×#1920C211yesmaximal, characteristic, complemented
10C2×#1894C211yesmaximal, characteristic
11C2×#1872C211yesmaximal, characteristic
12C2×#1865C211yesmaximal, characteristic, complemented
13C2×#1854C211yesmaximal, characteristic
14C2×#1851C211yesmaximal, characteristic
15C2×#1803C211yesmaximal, characteristic, complemented
16C2×#1736C211yesmaximal, characteristic
17C23×M16C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33#2440—24no—
34#2440C2218yes—
35C2×#398—21no—
36C2×#395—21no—
37#2362—24no—
38#2362C2218yes—
39C2×#391C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44C2×#388—21no—
45C2×#387—21no—
46C2×#385—21no—
47C2×#384C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#2099C2218yescomplemented
53#2086C2218yescomplemented
54#2068C2218yescomplemented
55#2024C2218yescomplemented
56#1987C2218yescomplemented
57#1920C2218yescomplemented
58#1894C2218yes—
59#1872C2218yes—
60#1865C2218yescomplemented
61#1854C2218yes—
62#1851C2218yes—
63#1803C2218yescomplemented
64#1736C2218yes—
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×Q8C2311yescharacteristic
79C24×C4—21no—
80C24×C4C2311yescharacteristic
81C22×M16—24no—
82C22×M16—42no—
83C22×M16—42no—
84C22×M16C2314yes—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—42no—
92C2×C4⋊Q8C2314yes—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—48no—
97C2×C22⋊Q8Q818yescomplemented
98C2×C22⋊Q8C2314yes—
99C2×C4×Q8—22no—
100C2×C4×Q8—24no—
101C2×C4×Q8—42no—
102C2×C4×Q8C2314yes—
103C2×#117—22no—
104C2×#117—24no—
105C2×#117—42no—
106C2×#117C2314yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—42no—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4D411yescharacteristic
131C22×C22⋊C4D411yescharacteristic
132C22×C42—21no—
133C22×C42—21no—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C2×#107—42no—
138C2×#107C2314yes—
139C2×#106—24no—
140C2×#106C2314yes—
141C2×C4⋊C8—42no—
142C2×C4⋊C8—42no—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#103—24no—
146C2×#103—42no—
147C2×#103—42no—
148C2×#103—42no—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×C22⋊C8—24no—
154C2×C22⋊C8C2314yes—
155#398—28no—
156#395—28no—
157#391—22no—
158#391C2314yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389C2314yes—
163#389C2314yes—
164#389C2314yes—
165#388—28no—
166#388—28no—
167#387—28no—
168#385—28no—
169#384—22no—
170#384C2314yes—
171#380—22no—
172#380C2314yes—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—28no—
175C4×C4⋊C4C2314yes—
176C4×C22⋊C4D414yes—
177C4×C22⋊C4C2314yes—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95D411yescharacteristic
186C2×#95D411yescharacteristic
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199C25C2×D411yescharacteristic
200C22×Q8—21no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—44no—
208C22×Q8—44no—
209C22×Q8—82no—
210C22×Q8C2×Q814yes—
211C22×Q8C2412yes—
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—21no—
223C23×C4—21no—
224C23×C4—24no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—82no—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D412yes—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×Q814yes—
235C23×C4#4114yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C2×M16—28no—
239C2×M16—44no—
240C2×M16—44no—
241C2×M16—416no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4⋊Q8—28no—
259C4⋊Q8—44no—
260C4⋊Q8—48no—
261C22⋊Q8—216no—
262C22⋊Q8—28no—
263C22⋊Q8—432no—
264C22⋊Q8C2×Q8116yescomplemented
265C4×Q8—28no—
266C4×Q8—44no—
267C4×Q8—48no—
268#117—28no—
269#117—44no—
270#117—48no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—48no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4C2×Q814yes—
311C2×C4⋊C4#4114yes—
312C2×C4⋊C4#4114yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4C2×D414yes—
329C2×C22⋊C4C2×D414yes—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—44no—
338C2×C42—44no—
339C2×C42—44no—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344#107—44no—
345#106—28no—
346C4⋊C8—44no—
347C4⋊C8—44no—
348#103—28no—
349#103—44no—
350#103—44no—
351#103—44no—
352C22⋊C8—28no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95C2×D414yes—
361#95C2×D414yes—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24C22⋊Q814yes—
369C24C22×D412yes—
370C24C22×D411yescharacteristic
371C2×Q8—24no—
372C2×Q8—22no—
373C2×Q8—28no—
374C2×Q8—28no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—84no—
386C2×Q8—816no—
387C2×Q8C22×Q814yes—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—48no—
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—48no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—84no—
445C22×C4—816no—
446C22×C4C22⋊Q814yes—
447C22×C4C22⋊Q818yes—
448C22×C4AGL(1,ℤ/8)14yes—
449C22×C4#13714yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×Q814yes—
457C22×C4C22×Q812yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4C2511yescharacteristic
465M16—416no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—44no—
479C2×C8—44no—
480C2×C8—44no—
481C2×C8—44no—
482C4⋊C4—216no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—84no—
509C4⋊C4C22×Q814yes—
510C4⋊C4C2×#4114yes—
511C4⋊C4C2×#4114yes—
512C22⋊C4—216no—
513C22⋊C4—28no—
514C22⋊C4—28no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C42—24no—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—48no—
524C42—48no—
525C42—48no—
526C23—28no—
527C23—28no—
528C23—22no—
529C23—22no—
530C23—28no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23C2×C22⋊Q814yes—
540C23C2×C22⋊Q812yes—
541C23#54612yes—
542C23#54612yes—
543C23C2×AGL(1,ℤ/8)12yes—
544C23C2×#13712yes—
545C23#57712yes—
546C23#57812yes—
547C23C23×D411yescharacteristic
548Q8—24no—
549Q8—24no—
550Q8—42no—
551Q8—42no—
552Q8—48no—
553Q8—816no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—216no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—48no—
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—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—816no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4C2×C22⋊Q814yes—
610C2×C4C2×C22⋊Q812yes—
611C2×C4#52914yes—
612C2×C4#53814yes—
613C2×C4#54612yes—
614C2×C4#54612yes—
615C2×C4#55414yes—
616C2×C4#55414yes—
617C2×C4C2×AGL(1,ℤ/8)14yes—
618C2×C4C2×#13714yes—
619C2×C4C23×Q811yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C8—42no—
626C8—42no—
627C8—42no—
628C8—42no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—416no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22#245014yes—
638C22#307914yes—
639C22#308214yes—
640C22#330814yes—
641C22#331214yes—
642C22#332414yes—
643C22#333114yes—
644C22C22×C22⋊Q811yescharacteristic
645C22C2×#54611yescharacteristic
646C22C2×#54611yescharacteristic
647C22C22×AGL(1,ℤ/8)11yescharacteristic
648C22C22×#13711yescharacteristic
649C22C2×#57711yescharacteristic
650C22C2×#57811yescharacteristic
651C4—24no—
652C4—24no—
653C4—24no—
654C4—44no—
655C4—44no—
656C4—42no—
657C4—44no—
658C4—42no—
659C4—44no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4#245014yes—
665C4C2×#52911yescharacteristic
666C4C2×#53811yescharacteristic
667C4C2×#55411yescharacteristic
668C4C2×#55411yescharacteristic
669C2—44no—
670C2#2643118yesdirect product factor
671C2C2×#245011yescharacteristic
672C2C2×#307911yescharacteristic
673C2C2×#308211yescharacteristic
674C2C2×#330811yescharacteristic
675C2C2×#331211yescharacteristic
676C2C2×#332411yescharacteristic
677C2C2×#333111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #26431C2direct (N × H)
4. C2×#2099C2semidirect
5. C2×#2086C2semidirect
6. C2×#2068C2semidirect
7. C2×#2024C2semidirect
8. C2×#1987C2semidirect
9. C2×#1920C2semidirect
12. C2×#1865C2semidirect
15. C2×#1803C2semidirect
52. #2099C22semidirect
53. #2086C22semidirect
54. #2068C22semidirect
55. #2024C22semidirect
56. #1987C22semidirect
57. #1920C22semidirect
60. #1865C22semidirect
63. #1803C22semidirect
97. C2×C22⋊Q8Q8semidirect
264. C22⋊Q8C2×Q8semidirect
670. C2#26431direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24402
C2×#23622
#264312
C2×#20992
C2×#20862
C2×#20682
C2×#20242
C2×#19872
C2×#19202
C2×#18942
C2×#18722
C2×#18652
C2×#18542
C2×#18512
C2×#18032
C2×#17362