Finite Group Databasebeta

← C2×#28784 #6459820

Subgroups — C2×#28784 #6459820

Order 512.

2174 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2439C211yesmaximal, characteristic
2C2×#2440C211yesmaximal, characteristic
3#28784C2116yesmaximal, direct product factor
4C2×#2085C211yesmaximal, characteristic, complemented
5C2×#2083C211yesmaximal, characteristic, complemented
6C2×#2058C211yesmaximal, characteristic, complemented
7C2×#2057C211yesmaximal, characteristic, complemented
8C2×#1993C211yesmaximal, characteristic, complemented
9C2×#1990C211yesmaximal, characteristic, complemented
10C2×#1944C211yesmaximal, characteristic, complemented
11C2×#1942C211yesmaximal, characteristic, complemented
12C2×#1854C211yesmaximal, characteristic
13C2×#1852C211yesmaximal, characteristic
14C2×#1809C211yesmaximal, characteristic
15C2×#1753C211yesmaximal, characteristic
16C2×#1736C211yesmaximal, characteristic
17C23×M16C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C22⋊Q8—21no—
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×#97C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2440—24no—
33#2440C2218yes—
34#2439—24no—
35#2439C2218yes—
36C2×#398—21no—
37C2×#396—21no—
38C2×#395—21no—
39C2×#393—21no—
40C2×#391—21no—
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#387—21no—
46C2×#385—21no—
47C2×#384C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#2085C2218yescomplemented
53#2083C2218yescomplemented
54#2058C2218yescomplemented
55#2057C2218yescomplemented
56#1993C2218yescomplemented
57#1990C2218yescomplemented
58#1944C2218yescomplemented
59#1942C2218yescomplemented
60#1854C2218yes—
61#1852C2218yes—
62#1809C2218yes—
63#1753C2218yes—
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×Q8—21no—
79C24×C4—21no—
80C24×C4C2311yescharacteristic
81C22×M16—42no—
82C22×M16—42no—
83C22×M16Q818yescomplemented
84C22×M16C2314yes—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—42no—
94C2×C4⋊Q8—42no—
95C2×C4⋊Q8C2314yes—
96C2×C4⋊Q8C2314yes—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—48no—
99C2×#117—22no—
100C2×#117—24no—
101C2×#117—42no—
102C2×#117C2314yes—
103C22×C4⋊C4—24no—
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⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—42no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4D411yescharacteristic
128C22×C22⋊C4D411yescharacteristic
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C2×#107—24no—
133C2×#107—42no—
134C2×#107—42no—
135C2×#107—42no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106—24no—
141C2×#106—42no—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C22⋊C8Q818yescomplemented
145C2×C22⋊C8C2314yes—
146C2×#97—42no—
147C2×#97C2314yes—
148C2×C4×C8—42no—
149C2×C4×C8C2314yes—
150#398—28no—
151#396—28no—
152#395—28no—
153#393—28no—
154#391—28no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389C2314yes—
160#389C2314yes—
161#389C2314yes—
162#389C2314yes—
163#387—28no—
164#385—28no—
165#384—22no—
166#384C2314yes—
167#382—22no—
168#382C2314yes—
169#380—22no—
170#380—22no—
171#380C2314yes—
172#380C2314yes—
173C4×C22⋊C4D414yes—
174C4×C22⋊C4C2314yes—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95D411yescharacteristic
184C2×#95D411yescharacteristic
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#336C2314yes—
196#336C2314yes—
197C25C2×D411yescharacteristic
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×Q8—44no—
203C22×Q8—44no—
204C22×Q8—82no—
205C22×Q8—82no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—24no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—82no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D412yes—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×Q814yes—
229C23×C4C2×Q814yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C2×M16—416no—
233C2×M16—44no—
234C2×M16—44no—
235C2×M16C2×Q8116yescomplemented
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8C2×Q814yes—
245C22×C8C2×Q814yes—
246C22×C8C2×Q814yes—
247C22×C8C2×Q814yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—28no—
253C4⋊Q8—28no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256C4⋊Q8—48no—
257C4⋊Q8—48no—
258C22⋊Q8—432no—
259#117—28no—
260#117—44no—
261#117—48no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
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—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—21no—
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—24no—
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—24no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—48no—
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—44no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4#4114yes—
306C2×C4⋊C4#4114yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—82no—
323C2×C22⋊C4C2×D414yes—
324C2×C22⋊C4C2×D414yes—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—44no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2412yes—
334C2×C42C2412yes—
335#107—28no—
336#107—44no—
337#107—44no—
338#107—44no—
339#106—28no—
340#106—44no—
341C22⋊C8C2×Q8116yescomplemented
342#97—44no—
343C4×C8—44no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95C2×D414yes—
353#95C2×D414yes—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24C4⋊Q814yes—
361C24C22×D412yes—
362C24C22×D411yescharacteristic
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×Q8—84no—
368C2×Q8—84no—
369C2×Q8—816no—
370C2×Q8—816no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
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—48no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
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—84no—
428C22×C4—816no—
429C22×C4C4⋊Q814yes—
430C22×C4C4⋊Q818yes—
431C22×C4AGL(1,ℤ/8)14yes—
432C22×C4#13714yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×Q814yes—
440C22×C4C22×Q814yes—
441C22×C4C22×Q812yes—
442C22×C4C22×Q812yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C4C2511yescharacteristic
448M16—416no—
449C2×C8—28no—
450C2×C8—28no—
451C2×C8—28no—
452C2×C8—28no—
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×C8C22×Q814yes—
462C2×C8C22×Q814yes—
463C2×C8C22×Q814yes—
464C2×C8C22×Q814yes—
465C4⋊C4—24no—
466C4⋊C4—24no—
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—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C4⋊C4C2×#4114yes—
497C4⋊C4C2×#4114yes—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—84no—
503C42—24no—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—48no—
510C23—28no—
511C23—28no—
512C23—22no—
513C23—22no—
514C23—28no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23C2×C4⋊Q814yes—
524C23C2×C4⋊Q812yes—
525C23#54612yes—
526C23#54612yes—
527C23C2×AGL(1,ℤ/8)12yes—
528C23C2×#13712yes—
529C23#57612yes—
530C23#57812yes—
531C23C23×D411yescharacteristic
532Q8—816no—
533Q8—816no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—216no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
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—816no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4C2×C4⋊Q814yes—
590C2×C4C2×C4⋊Q812yes—
591C2×C4#53314yes—
592C2×C4#54314yes—
593C2×C4#54612yes—
594C2×C4#54612yes—
595C2×C4#55414yes—
596C2×C4#55414yes—
597C2×C4C2×AGL(1,ℤ/8)14yes—
598C2×C4C2×#13714yes—
599C2×C4C23×Q811yescharacteristic
600C2×C4C23×Q811yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4C2×H(5,𝔽2)11yescharacteristic
605C8—24no—
606C8—24no—
607C8—24no—
608C8—24no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—416no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22#254914yes—
618C22#316414yes—
619C22#316514yes—
620C22#330814yes—
621C22#331414yes—
622C22#332514yes—
623C22#332814yes—
624C22C22×C4⋊Q811yescharacteristic
625C22C2×#54611yescharacteristic
626C22C2×#54611yescharacteristic
627C22C22×AGL(1,ℤ/8)11yescharacteristic
628C22C22×#13711yescharacteristic
629C22C2×#57611yescharacteristic
630C22C2×#57811yescharacteristic
631C4—24no—
632C4—24no—
633C4—24no—
634C4—44no—
635C4—42no—
636C4—44no—
637C4—42no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4#254914yes—
645C4C2×#53311yescharacteristic
646C4C2×#54311yescharacteristic
647C4C2×#55411yescharacteristic
648C4C2×#55411yescharacteristic
649C2—44no—
650C2#2878418yesdirect product factor
651C2C2×#254911yescharacteristic
652C2C2×#316411yescharacteristic
653C2C2×#316511yescharacteristic
654C2C2×#331411yescharacteristic
655C2C2×#330811yescharacteristic
656C2C2×#332511yescharacteristic
657C2C2×#332811yescharacteristic
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. #28784C2direct (N × H)
4. C2×#2085C2semidirect
5. C2×#2083C2semidirect
6. C2×#2058C2semidirect
7. C2×#2057C2semidirect
8. C2×#1993C2semidirect
9. C2×#1990C2semidirect
10. C2×#1944C2semidirect
11. C2×#1942C2semidirect
52. #2085C22semidirect
53. #2083C22semidirect
54. #2058C22semidirect
55. #2057C22semidirect
56. #1993C22semidirect
57. #1990C22semidirect
58. #1944C22semidirect
59. #1942C22semidirect
83. C22×M16Q8semidirect
144. C2×C22⋊C8Q8semidirect
235. C2×M16C2×Q8semidirect
341. C22⋊C8C2×Q8semidirect
650. C2#28784direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24392
C2×#24402
#287842
C2×#20852
C2×#20832
C2×#20582
C2×#20572
C2×#19932
C2×#19902
C2×#19442
C2×#19422
C2×#18542
C2×#18522
C2×#18092
C2×#17532
C2×#17362