Finite Group Databasebeta

← C2×#25486 #6446154

Subgroups — C2×#25486 #6446154

Order 512.

2240 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C22×#446C211yesmaximal, characteristic, complemented
3C2×#2538C211yesmaximal, characteristic
4C2×#2584C211yesmaximal, characteristic, complemented
5#25486C2116yesmaximal, direct product factor
6C2×#2088C211yesmaximal, characteristic, complemented
7C2×#2086C211yesmaximal, characteristic, complemented
8C2×#2065C211yesmaximal, characteristic
9C2×#2065C211yesmaximal, characteristic
10C2×#1992C211yesmaximal, characteristic, complemented
11C2×#1948C211yesmaximal, characteristic, complemented
12C2×#1944C211yesmaximal, characteristic, complemented
13C2×#1941C211yesmaximal, characteristic, complemented
14C2×#1936C211yesmaximal, characteristic
15C2×#1934C211yesmaximal, characteristic
16C2×#1762C211yesmaximal, characteristic
17C22×#125C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#478—28no—
22C2×#478C2218yes—
23C2×#446—28no—
24C2×#446C2218yescomplemented
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#97C2211yescharacteristic
37C22×#97C2211yescharacteristic
38C22×C4×C8C2211yescharacteristic
39#2584—24no—
40#2584C2218yescomplemented
41#2538—24no—
42#2538C2218yes—
43C2×#398—21no—
44C2×#396—21no—
45C2×#396—21no—
46C2×#395—21no—
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#389—21no—
50C2×#389—21no—
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#384C2211yescharacteristic
58C2×#380—21no—
59C2×C4×C4⋊C4C2211yescharacteristic
60#2088C2218yescomplemented
61#2086C2218yescomplemented
62#2065C2218yes—
63#2065C2218yes—
64#1992C2218yescomplemented
65#1948C2218yescomplemented
66#1944C2218yescomplemented
67#1941C2218yescomplemented
68#1936C2218yes—
69#1934C2218yes—
70#1762C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8D411yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#125—22no—
87C2×#125—24no—
88C2×#125—24no—
89C2×#125C2314yes—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—48no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8D414yes—
95C2×C4×Q8C2314yes—
96C2×C4×D4—22no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
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⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#478—232no—
130#446—232no—
131C2×#107—24no—
132C2×#107C2314yes—
133C2×#106—24no—
134C2×#106—24no—
135C2×#106—24no—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8—24no—
140C2×C4⋊C8—24no—
141C2×C4⋊C8D418yes—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#102—24no—
146C2×#102—24no—
147C2×#102—24no—
148C2×#102Q818yescomplemented
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#97—24no—
154C2×#97—24no—
155C2×#97C2314yes—
156C2×#97C2314yes—
157C2×C4×C8—24no—
158C2×C4×C8C2314yes—
159#398—28no—
160#396—28no—
161#396—28no—
162#395—28no—
163#390—22no—
164#390—22no—
165#390C2314yes—
166#390C2314yes—
167#389—28no—
168#389—28no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389C2314yes—
176#389C2314yes—
177#389C2314yes—
178#389C2314yes—
179#389C2314yes—
180#389C2314yes—
181#384—22no—
182#384C2314yes—
183#380—28no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4C2314yes—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201C25—21no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—42no—
205C22×Q8C2×D414yes—
206C22×D4—22no—
207C22×D4—21no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4C2×Q814yes—
213C22×D4C2412yes—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—24no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2411yescharacteristic
232C22×C8—22no—
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×C8C2×D414yes—
242C22×C8C2×D414yes—
243C22×C8C2×Q814yes—
244C22×C8C2412yes—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248#125—28no—
249#125—28no—
250#125—48no—
251C22⋊Q8—432no—
252C4×Q8—216no—
253C4×Q8—28no—
254C4×Q8—28no—
255C4×D4—28no—
256C4×D4—28no—
257C4×D4—48no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
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—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4C2×D414yes—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×Q814yes—
304C2×C4⋊C4#4114yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—82no—
321C2×C22⋊C4—82no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—82no—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D414yes—
332C2×C42#4114yes—
333C2×C42C2412yes—
334C2×C42C2412yes—
335C2×C42C2412yes—
336#107—28no—
337#106—28no—
338#106—28no—
339#106—28no—
340C4⋊C8—28no—
341C4⋊C8—28no—
342C4⋊C8C2×D4116yes—
343#102—28no—
344#102—28no—
345#102—28no—
346#102C2×Q8116yescomplemented
347#97—28no—
348#97—28no—
349C4×C8—28no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357C24—24no—
358C24—22no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24C22×D411yescharacteristic
366C2×Q8—28no—
367C2×Q8—28no—
368C2×Q8—44no—
369C2×Q8—416no—
370C2×D4—28no—
371C2×D4—28no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—42no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—42no—
382C2×D4C22×Q814yes—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
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—48no—
408C22×C4—48no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
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—48no—
429C22×C4—816no—
430C22×C4#12114yes—
431C22×C4#12114yes—
432C22×C4AGL(1,ℤ/8)14yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×Q812yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C42−1+412yes—
445C22×C42−1+412yes—
446C22×C4C2511yescharacteristic
447C2×C8—28no—
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—44no—
460C2×C8C22×D414yes—
461C2×C8C22×D414yes—
462C2×C8C22×Q814yes—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—24no—
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—22no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4C22×Q814yes—
496C4⋊C4C2×#4114yes—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C42—24no—
503C42—24no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—84no—
509C42#12118yes—
510C42C22×D414yes—
511C42C2×#4114yes—
512C23—24no—
513C23—48no—
514C23—48no—
515C23—42no—
516C23—48no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—84no—
523C23—84no—
524C23—84no—
525C23—84no—
526C23C2×#12112yes—
527C23#54812yes—
528C23D4×Q812yes—
529C23C2×AGL(1,ℤ/8)12yes—
530C23#57512yes—
531C23#57712yes—
532C23#57712yes—
533C23C23×D411yescharacteristic
534Q8—416no—
535D4—24no—
536D4—24no—
537D4—48no—
538D4—42no—
539D4—42no—
540C2×C4—22no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
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—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—816no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4C2×#12114yes—
591C2×C4C2×#12114yes—
592C2×C4C2×#12112yes—
593C2×C4#53314yes—
594C2×C4#54114yes—
595C2×C4#54714yes—
596C2×C4#54812yes—
597C2×C4D4×Q812yes—
598C2×C4D4×Q814yes—
599C2×C4C2×AGL(1,ℤ/8)14yes—
600C2×C4C23×D411yescharacteristic
601C2×C4C23×Q811yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C2×2−1+411yescharacteristic
605C2×C4C2×2−1+411yescharacteristic
606C8—24no—
607C8—24no—
608C8—24no—
609C8—42no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—816no—
618C22#242914yes—
619C22#305514yes—
620C22#306014yes—
621C22#336814yes—
622C22#337614yes—
623C22#338414yes—
624C22#338814yes—
625C22C22×#12111yescharacteristic
626C22C2×#54811yescharacteristic
627C22C2×D4×Q811yescharacteristic
628C22C22×AGL(1,ℤ/8)11yescharacteristic
629C22C2×#57511yescharacteristic
630C22C2×#57711yescharacteristic
631C22C2×#57711yescharacteristic
632C4—24no—
633C4—24no—
634C4—24no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—44no—
639C4—44no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4#242914yes—
646C4C2×#53311yescharacteristic
647C4C2×#54111yescharacteristic
648C4C2×#54711yescharacteristic
649C4C2×D4×Q811yescharacteristic
650C2—84no—
651C2#2548618yesdirect product factor
652C2C2×#242911yescharacteristic
653C2C2×#305511yescharacteristic
654C2C2×#306011yescharacteristic
655C2C2×#336811yescharacteristic
656C2C2×#337611yescharacteristic
657C2C2×#338411yescharacteristic
658C2C2×#338811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#446C2semidirect
4. C2×#2584C2semidirect
5. #25486C2direct (N × H)
6. C2×#2088C2semidirect
7. C2×#2086C2semidirect
10. C2×#1992C2semidirect
11. C2×#1948C2semidirect
12. C2×#1944C2semidirect
13. C2×#1941C2semidirect
24. C2×#446C22semidirect
40. #2584C22semidirect
60. #2088C22semidirect
61. #2086C22semidirect
64. #1992C22semidirect
65. #1948C22semidirect
66. #1944C22semidirect
67. #1941C22semidirect
148. C2×#102Q8semidirect
346. #102C2×Q8semidirect
651. C2#25486direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4782
C22×#4462
C2×#25382
C2×#25842
#254862
C2×#20882
C2×#20862
C2×#20652
C2×#19922
C2×#19482
C2×#19442
C2×#19412
C2×#19362
C2×#19342
C2×#17622