Finite Group Databasebeta

← C2×#32654 #7511115

Subgroups — C2×#32654 #7511115

Order 512.

2068 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2581C211yesmaximal, characteristic, complemented
3#32654C2116yesmaximal, direct product factor
4C2×#2095C211yesmaximal, characteristic, complemented
5C2×#2092C211yesmaximal, characteristic
6C2×#2084C211yesmaximal, characteristic, complemented
7C2×#2085C211yesmaximal, characteristic, complemented
8C2×#2066C211yesmaximal, characteristic, complemented
9C2×#2067C211yesmaximal, characteristic
10C2×#1984C211yesmaximal, characteristic, complemented
11C2×#1968C211yesmaximal, characteristic
12C2×#1942C211yesmaximal, characteristic, complemented
13C2×#1880C211yesmaximal, characteristic
14C2×#1877C211yesmaximal, characteristic
15C2×#1863C211yesmaximal, characteristic, complemented
16C2×#1862C211yesmaximal, characteristic
17C23×SD16C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35#2581—24no—
36#2581C2218yescomplemented
37#2538—24no—
38#2538C2218yes—
39C2×#400—21no—
40C2×#396—21no—
41C2×#396—21no—
42C2×#395—21no—
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#387—21no—
52C2×#387—21no—
53C2×#384C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C2×#380—21no—
56#2095C2218yescomplemented
57#2092C2218yes—
58#2085C2218yescomplemented
59#2084C2218yescomplemented
60#2067C2218yes—
61#2066C2218yescomplemented
62#1984C2218yescomplemented
63#1968C2218yes—
64#1942C2218yescomplemented
65#1880C2218yes—
66#1877C2218yes—
67#1863C2218yescomplemented
68#1862C2218yes—
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8D411yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C22×SD16—42no—
82C22×SD16—42no—
83C22×SD16Q818yescomplemented
84C22×SD16C2314yes—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—48no—
91C2×C4×Q8—22no—
92C2×C4×Q8—42no—
93C2×C4×Q8—42no—
94C2×C4×Q8—42no—
95C2×C4×Q8—42no—
96C2×C4×Q8D414yes—
97C2×C4×Q8C2314yes—
98C2×C4×Q8C2314yes—
99C2×C4×D4—22no—
100C2×C4×D4—42no—
101C2×C4×D4—42no—
102C2×C4×D4C2314yes—
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×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C2×#107—24no—
129C2×#107—42no—
130C2×#107—42no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106—42no—
135C2×#106—42no—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8—42no—
139C2×C4⋊C8C2314yes—
140C2×#103—24no—
141C2×#103—42no—
142C2×#103—42no—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102—24no—
147C2×#102—42no—
148C2×#102—42no—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C4×C8—42no—
153C2×C4×C8C2314yes—
154#400—28no—
155#396—28no—
156#396—28no—
157#395—28no—
158#391—22no—
159#391—22no—
160#391C2314yes—
161#391C2314yes—
162#390—22no—
163#390—22no—
164#390C2314yes—
165#390C2314yes—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389C2314yes—
171#389C2314yes—
172#389C2314yes—
173#389C2314yes—
174#387—28no—
175#387—28no—
176#384—22no—
177#384C2314yes—
178#382—22no—
179#382C2314yes—
180#380—28no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#336C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199C25—21no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—42no—
209C22×Q8C2×D414yes—
210C22×Q8C2×Q814yes—
211C22×Q8C2412yes—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4C2×Q814yes—
219C22×D4C2412yes—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—24no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2411yescharacteristic
238C2×SD16—416no—
239C2×SD16—44no—
240C2×SD16—44no—
241C2×SD16C2×Q8116yescomplemented
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×C8C2×Q814yes—
253C22×C8#4114yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22⋊Q8—432no—
259C4×Q8—48no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×Q8—44no—
264C4×Q8—48no—
265C4×D4—44no—
266C4×D4—44no—
267C4×D4—48no—
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—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2×D414yes—
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—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—44no—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2412yes—
334C2×C42C2412yes—
335C2×C42C2412yes—
336#107—28no—
337#107—44no—
338#107—44no—
339#106—44no—
340#106—44no—
341C4⋊C8—44no—
342#103—28no—
343#103—44no—
344#103—44no—
345#102—28no—
346#102—44no—
347#102—44no—
348C4×C8—44no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359C24—24no—
360C24—22no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24C22×D411yescharacteristic
368C2×Q8—24no—
369C2×Q8—28no—
370C2×Q8—28no—
371C2×Q8—22no—
372C2×Q8—28no—
373C2×Q8—28no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—44no—
379C2×Q8—416no—
380C2×Q8—42no—
381C2×Q8—44no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8C22×Q814yes—
385C2×D4—28no—
386C2×D4—24no—
387C2×D4—28no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—42no—
394C2×D4—44no—
395C2×D4—42no—
396C2×D4—44no—
397C2×D4C22×Q814yes—
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—24no—
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—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—48no—
423C22×C4—48no—
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—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—48no—
445C22×C4—816no—
446C22×C4#13514yes—
447C22×C4AGL(1,ℤ/8)14yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×Q812yes—
456C22×C4C2×#4112yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C42−1+412yes—
461C22×C4C2511yescharacteristic
462SD16—416no—
463C2×C8—28no—
464C2×C8—28no—
465C2×C8—22no—
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—44no—
476C2×C8—44no—
477C2×C8C22×Q814yes—
478C2×C8C2×#4114yes—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—84no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C42—24no—
513C42—24no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—48no—
522C23—24no—
523C23—48no—
524C23—48no—
525C23—42no—
526C23—42no—
527C23—48no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23#53412yes—
537C23D4×Q812yes—
538C23#55012yes—
539C23C2×#13512yes—
540C23C2×AGL(1,ℤ/8)12yes—
541C23#57712yes—
542C23#57812yes—
543C23C23×D411yescharacteristic
544Q8—24no—
545Q8—24no—
546Q8—416no—
547Q8—42no—
548Q8—48no—
549Q8—42no—
550D4—24no—
551D4—24no—
552D4—42no—
553D4—48no—
554D4—42no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
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—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—48no—
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—42no—
599C2×C4—816no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4#53412yes—
607C2×C4#53914yes—
608C2×C4#54314yes—
609C2×C4D4×Q812yes—
610C2×C4#55012yes—
611C2×C4#55414yes—
612C2×C4#56214yes—
613C2×C4C2×#13514yes—
614C2×C4C2×AGL(1,ℤ/8)14yes—
615C2×C4C23×Q811yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C2×C4C2×H(5,𝔽2)11yescharacteristic
620C2×C4C2×2−1+411yescharacteristic
621C8—24no—
622C8—24no—
623C8—42no—
624C8—42no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—82no—
629C22—82no—
630C22—816no—
631C22—82no—
632C22—82no—
633C22#273214yes—
634C22#318814yes—
635C22#320114yes—
636C22#338114yes—
637C22#338614yes—
638C22#339314yes—
639C22#339714yes—
640C22C2×#53411yescharacteristic
641C22C2×D4×Q811yescharacteristic
642C22C2×#55011yescharacteristic
643C22C22×#13511yescharacteristic
644C22C22×AGL(1,ℤ/8)11yescharacteristic
645C22C2×#57711yescharacteristic
646C22C2×#57811yescharacteristic
647C4—24no—
648C4—24no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4#273214yes—
661C4C2×#53911yescharacteristic
662C4C2×#54311yescharacteristic
663C4C2×#55411yescharacteristic
664C4C2×#56211yescharacteristic
665C2—84no—
666C2#3265418yesdirect product factor
667C2C2×#273211yescharacteristic
668C2C2×#318811yescharacteristic
669C2C2×#320111yescharacteristic
670C2C2×#338111yescharacteristic
671C2C2×#338611yescharacteristic
672C2C2×#339311yescharacteristic
673C2C2×#339711yescharacteristic
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. C2×#2581C2semidirect
3. #32654C2direct (N × H)
4. C2×#2095C2semidirect
6. C2×#2084C2semidirect
7. C2×#2085C2semidirect
8. C2×#2066C2semidirect
10. C2×#1984C2semidirect
12. C2×#1942C2semidirect
15. C2×#1863C2semidirect
36. #2581C22semidirect
56. #2095C22semidirect
58. #2085C22semidirect
59. #2084C22semidirect
61. #2066C22semidirect
62. #1984C22semidirect
64. #1942C22semidirect
67. #1863C22semidirect
83. C22×SD16Q8semidirect
241. C2×SD16C2×Q8semidirect
666. C2#32654direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#25812
#326542
C2×#20952
C2×#20922
C2×#20842
C2×#20852
C2×#20662
C2×#20672
C2×#19842
C2×#19682
C2×#19422
C2×#18802
C2×#18772
C2×#18632
C2×#18622