Finite Group Databasebeta

← C2×#25998 #6452282

Subgroups — C2×#25998 #6452282

Order 512.

2174 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic
3#25998C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2087C211yesmaximal, characteristic, complemented
6C2×#2088C211yesmaximal, characteristic, complemented
7C2×#2059C211yesmaximal, characteristic, complemented
8C2×#1989C211yesmaximal, characteristic, complemented
9C2×#1949C211yesmaximal, characteristic, complemented
10C2×#1921C211yesmaximal, characteristic, complemented
11C2×#1919C211yesmaximal, characteristic, complemented
12C2×#1852C211yesmaximal, characteristic
13C2×#1850C211yesmaximal, characteristic
14C2×#1823C211yesmaximal, characteristic
15C2×#1807C211yesmaximal, characteristic
16C22×#340C211yesmaximal, characteristic
17C24×C8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#117—21no—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29#2442—24no—
30#2442C2218yes—
31C2×#400—21no—
32C2×#398—21no—
33#2344—24no—
34#2344C2218yes—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#387—21no—
40C2×#387—21no—
41C2×#385—21no—
42C2×#382C2211yescharacteristic
43C2×#381—21no—
44C2×#380—21no—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2101C2218yescomplemented
51#2088C2218yescomplemented
52#2087C2218yescomplemented
53#2059C2218yescomplemented
54#1989C2218yescomplemented
55#1949C2218yescomplemented
56#1921C2218yescomplemented
57#1919C2218yescomplemented
58#1852C2218yes—
59#1850C2218yes—
60#1823C2218yes—
61#1807C2218yes—
62C2×#340C2211yescharacteristic
63C2×#340C2211yescharacteristic
64C2×#340C2211yescharacteristic
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2218yes—
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C24×C4—21no—
78C24×C4—21no—
79C24×C4C2311yescharacteristic
80C23×C8—24no—
81C23×C8—24no—
82C23×C8—42no—
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2314yes—
87C23×C8C2311yescharacteristic
88C2×#125—24no—
89C2×#125—22no—
90C2×#125—42no—
91C2×#125C2314yes—
92C2×#117—28no—
93C2×#117—48no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—42no—
107C22×C4⋊C4Q818yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4D411yescharacteristic
121C22×C22⋊C4D411yescharacteristic
122C22×C42—21no—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C2×#107—24no—
127C2×#107—42no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106—24no—
131C2×#106—42no—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8—24no—
135C2×C4⋊C8—42no—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2314yes—
138C2×C22⋊C8—24no—
139C2×C22⋊C8C2314yes—
140#400—28no—
141#398—28no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389C2314yes—
147#389C2314yes—
148#389C2314yes—
149#389C2314yes—
150#387—28no—
151#387—28no—
152#385—28no—
153#382—22no—
154#382C2314yes—
155#381—28no—
156#380—22no—
157#380—22no—
158#380—28no—
159#380C2314yes—
160#380C2314yes—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4C2314yes—
164C4×C4⋊C4C2314yes—
165C4×C22⋊C4D414yes—
166C4×C22⋊C4C2314yes—
167C2×#95—21no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95D411yescharacteristic
178C2×#95D411yescharacteristic
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192#336C2314yes—
193C25C2×D411yescharacteristic
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—24no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—82no—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D412yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×Q814yes—
221C23×C4#4114yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—28no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—28no—
237C22×C8—44no—
238C22×C8—416no—
239C22×C8#4114yes—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244#125—28no—
245#125—44no—
246#125—48no—
247#117—432no—
248C2×C4⋊C4—216no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—28no—
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—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—48no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—82no—
289C2×C4⋊C4—82no—
290C2×C4⋊C4C2×Q8116yes—
291C2×C4⋊C4C2×Q814yes—
292C2×C4⋊C4C2×Q814yes—
293C2×C4⋊C4#4114yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4C2×D414yes—
309C2×C22⋊C4C2×D414yes—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321C2×C42C2412yes—
322#107—28no—
323#107—44no—
324#106—28no—
325#106—44no—
326C4⋊C8—28no—
327C4⋊C8—44no—
328C22⋊C8—28no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95C2×D414yes—
340#95C2×D414yes—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24C22⋊Q814yes—
346C24#13514yes—
347C24AGL(1,ℤ/8)14yes—
348C24C22×D412yes—
349C24C22×D411yescharacteristic
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—216no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
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—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—48no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—432no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—48no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—84no—
413C22×C4—816no—
414C22×C4C22⋊Q814yes—
415C22×C4C22⋊Q818yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×Q814yes—
423C22×C4C22×Q812yes—
424C22×C4C2×#4114yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C42−1+412yes—
430C22×C4C2511yescharacteristic
431C2×C8—28no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—416no—
444C2×C8—44no—
445C2×C8—44no—
446C2×C8—44no—
447C2×C8C2×#4114yes—
448C4⋊C4—216no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4C22×Q814yes—
478C4⋊C4C22×Q814yes—
479C4⋊C4C2×#4114yes—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C42—24no—
485C42—24no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—48no—
491C42—48no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—22no—
497C23—22no—
498C23—28no—
499C23—28no—
500C23—22no—
501C23—22no—
502C23#478116yes—
503C23C2×C22⋊Q814yes—
504C23C2×C22⋊Q812yes—
505C23#54612yes—
506C23#54712yes—
507C23C2×#13514yes—
508C23C2×#13512yes—
509C23C2×AGL(1,ℤ/8)14yes—
510C23C2×AGL(1,ℤ/8)12yes—
511C23#57712yes—
512C23#57812yes—
513C23C23×D411yescharacteristic
514C2×C4—28no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—216no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—432no—
538C2×C4—48no—
539C2×C4—48no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
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—816no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4C2×C22⋊Q814yes—
573C2×C4C2×C22⋊Q812yes—
574C2×C4#53314yes—
575C2×C4#54414yes—
576C2×C4#54612yes—
577C2×C4#54712yes—
578C2×C4#55114yes—
579C2×C4#55214yes—
580C2×C4C23×Q811yescharacteristic
581C2×C4C22×#4111yescharacteristic
582C2×C4C22×#4111yescharacteristic
583C2×C4C22×#4111yescharacteristic
584C2×C4C2×H(5,𝔽2)11yescharacteristic
585C2×C4C2×2−1+411yescharacteristic
586C8—24no—
587C8—42no—
588C8—42no—
589C8—42no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—232no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22#243814yes—
598C22C2×#47818yes—
599C22C2×#47814yes—
600C22#308214yes—
601C22#330014yes—
602C22#330614yes—
603C22#333714yes—
604C22#334314yes—
605C22C22×C22⋊Q811yescharacteristic
606C22C2×#54611yescharacteristic
607C22C2×#54711yescharacteristic
608C22C22×#13511yescharacteristic
609C22C22×AGL(1,ℤ/8)11yescharacteristic
610C22C2×#57711yescharacteristic
611C22C2×#57811yescharacteristic
612C4—24no—
613C4—42no—
614C4—42no—
615C4—44no—
616C4—44no—
617C4—42no—
618C4—42no—
619C4—44no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4#243814yes—
626C4C2×#53311yescharacteristic
627C4C2×#54411yescharacteristic
628C4C2×#55111yescharacteristic
629C4C2×#55211yescharacteristic
630C2—28no—
631C2#2599818yesdirect product factor
632C2C2×#243811yescharacteristic
633C2C22×#47811yescharacteristic
634C2C2×#308211yescharacteristic
635C2C2×#330011yescharacteristic
636C2C2×#330611yescharacteristic
637C2C2×#333711yescharacteristic
638C2C2×#334311yescharacteristic
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. #25998C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2087C2semidirect
6. C2×#2088C2semidirect
7. C2×#2059C2semidirect
8. C2×#1989C2semidirect
9. C2×#1949C2semidirect
10. C2×#1921C2semidirect
11. C2×#1919C2semidirect
50. #2101C22semidirect
51. #2088C22semidirect
52. #2087C22semidirect
53. #2059C22semidirect
54. #1989C22semidirect
55. #1949C22semidirect
56. #1921C22semidirect
57. #1919C22semidirect
631. C2#25998direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24422
C2×#23442
#259982
C2×#21012
C2×#20872
C2×#20882
C2×#20592
C2×#19892
C2×#19492
C2×#19212
C2×#19192
C2×#18522
C2×#18502
C2×#18232
C2×#18072
C22×#3402