Finite Group Databasebeta

← C2×#28974 #6460202

Subgroups — C2×#28974 #6460202

Order 512.

2066 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2506C211yesmaximal, characteristic, complemented
2C2×#2452C211yesmaximal, characteristic
3#28974C2116yesmaximal, direct product factor
4C2×#2093C211yesmaximal, characteristic
5C2×#2087C211yesmaximal, characteristic, complemented
6C2×#2065C211yesmaximal, characteristic
7C2×#2058C211yesmaximal, characteristic, complemented
8C2×#1990C211yesmaximal, characteristic, complemented
9C2×#1944C211yesmaximal, characteristic, complemented
10C2×#1943C211yesmaximal, characteristic, complemented
11C2×#1936C211yesmaximal, characteristic
12C2×#1934C211yesmaximal, characteristic
13C2×#1921C211yesmaximal, characteristic, complemented
14C2×#1864C211yesmaximal, characteristic
15C2×#1771C211yesmaximal, characteristic, complemented
16C2×#1762C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125—21no—
19C22×C22⋊Q8—21no—
20C22×C4×D4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#97C2211yescharacteristic
31C22×#97C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33#2506—24no—
34#2506C2218yescomplemented
35#2452—24no—
36#2452C2218yes—
37C2×#400—21no—
38C2×#398—21no—
39C2×#396—21no—
40C2×#393—21no—
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#387—21no—
50C2×#385—21no—
51C2×#384C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2093C2218yes—
55#2087C2218yescomplemented
56#2065C2218yes—
57#2058C2218yescomplemented
58#1990C2218yescomplemented
59#1944C2218yescomplemented
60#1943C2218yescomplemented
61#1936C2218yes—
62#1934C2218yes—
63#1921C2218yescomplemented
64#1864C2218yes—
65#1771C2218yescomplemented
66#1762C2218yes—
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8C2314yes—
88C2×#125—28no—
89C2×#125—48no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—48no—
92C2×C4×D4—24no—
93C2×C4×D4—22no—
94C2×C4×D4—42no—
95C2×C4×D4C2314yes—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C2×#107—24no—
124C2×#107—24no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106—24no—
128C2×#106—42no—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8—24no—
132C2×C4⋊C8C2314yes—
133C2×#102—24no—
134C2×#102—24no—
135C2×#102—42no—
136C2×#102Q818yescomplemented
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#97—24no—
142C2×#97—42no—
143C2×#97C2314yes—
144C2×#97C2314yes—
145C2×C4×C8—42no—
146C2×C4×C8C2314yes—
147#400—28no—
148#398—28no—
149#396—28no—
150#393—28no—
151#390—22no—
152#390—22no—
153#390C2314yes—
154#390C2314yes—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389C2314yes—
162#389C2314yes—
163#389C2314yes—
164#389C2314yes—
165#389C2314yes—
166#389C2314yes—
167#387—28no—
168#385—28no—
169#384—22no—
170#384C2314yes—
171C4×C4⋊C4—22no—
172C4×C4⋊C4D414yes—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2314yes—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95D411yescharacteristic
183C2×#95D411yescharacteristic
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25—21no—
195C22×Q8—24no—
196C22×Q8—44no—
197C22×Q8—44no—
198C22×Q8—82no—
199C22×D4—21no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4C2×Q814yes—
206C22×D4C2412yes—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—24no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—82no—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2411yescharacteristic
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8C2×Q814yes—
235C22×C8#4114yes—
236C22×C8#4114yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C4⋊Q8—28no—
242C4⋊Q8—44no—
243C4⋊Q8—48no—
244#125—432no—
245C22⋊Q8—432no—
246C4×D4—28no—
247C4×D4—44no—
248C4×D4—48no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
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—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×Q814yes—
291C2×C4⋊C4#4114yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—82no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—44no—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321C2×C42C2412yes—
322C2×C42C2412yes—
323#107—28no—
324#107—28no—
325#106—28no—
326#106—44no—
327C4⋊C8—28no—
328#102—28no—
329#102—28no—
330#102—44no—
331#102C2×Q8116yescomplemented
332#97—28no—
333#97—44no—
334C4×C8—44no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95C2×D414yes—
343#95C2×D414yes—
344C24—24no—
345C24—22no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24C22×D411yescharacteristic
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—84no—
356C2×Q8—816no—
357C2×D4—28no—
358C2×D4—24no—
359C2×D4—28no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—42no—
366C2×D4—44no—
367C2×D4—42no—
368C2×D4—44no—
369C2×D4C22×Q814yes—
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—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
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—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—48no—
396C22×C4—48no—
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—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—84no—
417C22×C4—816no—
418C22×C4AGL(1,ℤ/8)14yes—
419C22×C4AGL(1,ℤ/8)14yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×Q812yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C42−1+412yes—
433C22×C4C2511yescharacteristic
434C2×C8—28no—
435C2×C8—28no—
436C2×C8—28no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—44no—
447C2×C8C22×Q814yes—
448C2×C8C2×#4114yes—
449C2×C8C2×#4114yes—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—28no—
453C4⋊C4—28no—
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—22no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4C22×Q814yes—
481C4⋊C4C2×#4114yes—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—84no—
487C42—24no—
488C42—24no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—48no—
497C23—24no—
498C23—48no—
499C23—48no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—48no—
505C23—42no—
506C23—42no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23#53212yes—
512C23#54612yes—
513C23D4×Q812yes—
514C23C2×AGL(1,ℤ/8)12yes—
515C23C2×AGL(1,ℤ/8)12yes—
516C23#57712yes—
517C23#57812yes—
518C23C23×D411yescharacteristic
519Q8—816no—
520D4—24no—
521D4—24no—
522D4—42no—
523D4—42no—
524D4—48no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—48no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—48no—
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—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—816no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4#53212yes—
577C2×C4#53314yes—
578C2×C4#53914yes—
579C2×C4#54612yes—
580C2×C4D4×Q812yes—
581C2×C4#55214yes—
582C2×C4#55414yes—
583C2×C4C2×AGL(1,ℤ/8)14yes—
584C2×C4C2×AGL(1,ℤ/8)14yes—
585C2×C4C23×Q811yescharacteristic
586C2×C4C22×#4111yescharacteristic
587C2×C4C22×#4111yescharacteristic
588C2×C4C22×#4111yescharacteristic
589C2×C4C2×H(5,𝔽2)11yescharacteristic
590C2×C4C2×2−1+411yescharacteristic
591C8—24no—
592C8—24no—
593C8—24no—
594C8—42no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—82no—
599C22—82no—
600C22—816no—
601C22—82no—
602C22—82no—
603C22#255614yes—
604C22#317814yes—
605C22#318414yes—
606C22#330814yes—
607C22#331014yes—
608C22#338414yes—
609C22#338614yes—
610C22C2×#53211yescharacteristic
611C22C2×#54611yescharacteristic
612C22C2×D4×Q811yescharacteristic
613C22C22×AGL(1,ℤ/8)11yescharacteristic
614C22C22×AGL(1,ℤ/8)11yescharacteristic
615C22C2×#57711yescharacteristic
616C22C2×#57811yescharacteristic
617C4—24no—
618C4—24no—
619C4—42no—
620C4—42no—
621C4—44no—
622C4—42no—
623C4—44no—
624C4—42no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4#255614yes—
631C4C2×#53311yescharacteristic
632C4C2×#53911yescharacteristic
633C4C2×#55211yescharacteristic
634C4C2×#55411yescharacteristic
635C2—84no—
636C2#2897418yesdirect product factor
637C2C2×#255611yescharacteristic
638C2C2×#317811yescharacteristic
639C2C2×#318411yescharacteristic
640C2C2×#331011yescharacteristic
641C2C2×#330811yescharacteristic
642C2C2×#338411yescharacteristic
643C2C2×#338611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2506C2semidirect
3. #28974C2direct (N × H)
5. C2×#2087C2semidirect
7. C2×#2058C2semidirect
8. C2×#1990C2semidirect
9. C2×#1944C2semidirect
10. C2×#1943C2semidirect
13. C2×#1921C2semidirect
15. C2×#1771C2semidirect
34. #2506C22semidirect
55. #2087C22semidirect
57. #2058C22semidirect
58. #1990C22semidirect
59. #1944C22semidirect
60. #1943C22semidirect
63. #1921C22semidirect
65. #1771C22semidirect
136. C2×#102Q8semidirect
331. #102C2×Q8semidirect
636. C2#28974direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25062
C2×#24522
#289742
C2×#20932
C2×#20872
C2×#20652
C2×#20582
C2×#19902
C2×#19442
C2×#19432
C2×#19362
C2×#19342
C2×#19212
C2×#18642
C2×#17712
C2×#17622