Finite Group Databasebeta

← C2×#25716 #6450123

Subgroups — C2×#25716 #6450123

Order 512.

3408 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C22×#501C211yesmaximal, characteristic, complemented
3C22×#477C211yesmaximal, characteristic, complemented
4C2×#2547C211yesmaximal, characteristic
5#25716C2116yesmaximal, direct product factor
6C2×#2036C212yesmaximal, complemented
7C2×#2035C212yesmaximal, complemented
8C2×#1989C211yesmaximal, characteristic, complemented
9C2×#1955C211yesmaximal, characteristic
10C2×#1854C212yesmaximal, complemented
11C2×#1852C212yesmaximal, complemented
12C2×#1763C211yesmaximal, characteristic, complemented
13C2×#552—28no—
14C2×#552—28no—
15C2×#552—24no—
16C2×#552C2218yes—
17C22×C4⋊Q8C2211yescharacteristic
18C22×#126—21no—
19C22×#125—21no—
20C22×#125C2211yescharacteristic
21C22×#124—21no—
22C22×#123—21no—
23C22×C22⋊Q8C2211yescharacteristic
24C22×C22⋊Q8C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C22×#117—21no—
28C22×#117C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30C2×#501—28no—
31C2×#501C2218yescomplemented
32C2×#477—28no—
33C2×#477C2218yescomplemented
34C22×#107C2212yes—
35C22×#107C2212yes—
36C22×#106C2212yes—
37C22×C4⋊C8C2211yescharacteristic
38C22×#103C2212yes—
39C22×#103C2212yes—
40C22×C22⋊C8C2212yes—
41C22×C22⋊C8C2212yes—
42C22×#97C2211yescharacteristic
43#2547—24no—
44#2547C2218yes—
45C2×#397—21no—
46C2×#397—21no—
47C2×#391—21no—
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2036C22116yescomplemented
52#2035C22116yescomplemented
53#1989C2218yescomplemented
54#1955C2218yes—
55#1854C22116yescomplemented
56#1852C22116yescomplemented
57#1763C2218yescomplemented
58C2×#340C2212yes—
59C2×#340C2212yes—
60C2×#340C2212yes—
61C23×Q8—21no—
62C23×Q8C2311yescharacteristic
63C23×D4D411yescharacteristic
64C24×C4—21no—
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C23×C8C2312yes—
68C23×C8C2312yes—
69#552—264no—
70#552—232no—
71#552—232no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8C2314yes—
76C2×#126—216no—
77C2×#126—28no—
78C2×#125—216no—
79C2×#125—28no—
80C2×#125—22no—
81C2×#125—24no—
82C2×#125D418yes—
83C2×#125C2314yes—
84C2×#124—28no—
85C2×#124—48no—
86C2×#123—216no—
87C2×#123—28no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8C2314yes—
95C2×C22⋊Q8C2314yes—
96C2×C4×Q8—24no—
97C2×C4×Q8—22no—
98C2×C4×Q8D418yes—
99C2×C4×Q8C2314yes—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4D414yes—
103C2×C4×D4C2314yes—
104C2×#117—216no—
105C2×#117—22no—
106C2×#117—24no—
107C2×#117—24no—
108C2×#117—28no—
109C2×#117C2314yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—24no—
118C22×C4⋊C4—24no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—22no—
131C22×C22⋊C4—22no—
132C22×C22⋊C4D411yescharacteristic
133C22×C22⋊C4D411yescharacteristic
134C22×C42—21no—
135C22×C42—21no—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138#501—232no—
139#477—232no—
140C2×#107—28no—
141C2×#107—28no—
142C2×#107C2318yes—
143C2×#107C2318yes—
144C2×#106—28no—
145C2×#106C2318yes—
146C2×C4⋊C8—24no—
147C2×C4⋊C8C2314yes—
148C2×#103—28no—
149C2×#103—28no—
150C2×#103C2318yes—
151C2×#103C2318yes—
152C2×C22⋊C8—28no—
153C2×C22⋊C8—28no—
154C2×C22⋊C8C2318yes—
155C2×C22⋊C8C2318yes—
156C2×#97—24no—
157C2×#97C2314yes—
158#397—28no—
159#397—28no—
160#391—28no—
161#380—24no—
162#380—24no—
163#380C2318yes—
164#380C2318yes—
165C4×C4⋊C4—22no—
166C4×C4⋊C4C2314yes—
167C2×#95—21no—
168C2×#95—22no—
169C2×#95—22no—
170#340C2318yes—
171#340C2318yes—
172#340C2318yes—
173C25C2×D411yescharacteristic
174C25C2×D411yescharacteristic
175C22×Q8—22no—
176C22×Q8—21no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8C2×D414yes—
184C22×Q8C2412yes—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—42no—
188C22×D4C2×D414yes—
189C23×C4—28no—
190C23×C4—24no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4C2×D412yes—
208C23×C4C2×D412yes—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C22×C8—24no—
217C22×C8—24no—
218C22×C8—24no—
219C22×C8—24no—
220C22×C8—24no—
221C22×C8#4118yes—
222C22×C8C2414yes—
223C22×C8C2414yes—
224C4⋊Q8—28no—
225C4⋊Q8—28no—
226C4⋊Q8—48no—
227#126—264no—
228#125—264no—
229#125—216no—
230#125—28no—
231#125C2×D4116yes—
232#124—432no—
233#123—264no—
234C22⋊Q8—216no—
235C22⋊Q8—216no—
236C22⋊Q8—28no—
237C22⋊Q8—28no—
238C22⋊Q8—28no—
239C22⋊Q8—28no—
240C4×Q8—216no—
241C4×Q8—28no—
242C4×Q8C2×D4116yes—
243C4×D4—216no—
244C4×D4—28no—
245C4×D4—28no—
246#117—216no—
247#117—264no—
248#117—28no—
249#117—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—216no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2414yes—
285C2×C4⋊C4C2414yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—48no—
300C2×C22⋊C4—48no—
301C2×C22⋊C4C2×D414yes—
302C2×C22⋊C4C2×D414yes—
303C2×C42—28no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—44no—
309C2×C42C2×D412yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42C2412yes—
313#107—216no—
314#107—216no—
315#106—216no—
316C4⋊C8—28no—
317#103—216no—
318#103—216no—
319C22⋊C8—216no—
320C22⋊C8—216no—
321#97—28no—
322#95—24no—
323#95—28no—
324#95—28no—
325C24—22no—
326C24—24no—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—24no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24AGL(1,ℤ/8)14yes—
335C24C22×D412yes—
336C24C22×D412yes—
337C24C22×D411yescharacteristic
338C2×Q8—24no—
339C2×Q8—22no—
340C2×Q8—28no—
341C2×Q8—28no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—44no—
348C2×Q8—48no—
349C2×Q8—816no—
350C2×Q8C22×D414yes—
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—44no—
354C2×D4—416no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—216no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—432no—
386C22×C4—48no—
387C22×C4—44no—
388C22×C4—48no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4#12014yes—
407C22×C4#12014yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C2×#4114yes—
416C22×C42−1+414yes—
417C22×C4C2511yescharacteristic
418C2×C8—216no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—48no—
425C2×C8C2×#4118yes—
426C4⋊C4—216no—
427C4⋊C4—216no—
428C4⋊C4—216no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—216no—
432C4⋊C4—216no—
433C4⋊C4—216no—
434C4⋊C4—24no—
435C4⋊C4—28no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—44no—
446C4⋊C4C22×D414yes—
447C4⋊C4C22×D414yes—
448C4⋊C4C22×D414yes—
449C22⋊C4—216no—
450C22⋊C4—216no—
451C22⋊C4—216no—
452C22⋊C4—28no—
453C22⋊C4—28no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C42—216no—
457C42—216no—
458C42—48no—
459C42#12018yes—
460C42C22×D414yes—
461C42C22×D414yes—
462C23—216no—
463C23—22no—
464C23—24no—
465C23—28no—
466C23—22no—
467C23—24no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—48no—
474C23—44no—
475C23—44no—
476C23—48no—
477C23—48no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23C2×#12012yes—
482C23#54714yes—
483C23C2×AGL(1,ℤ/8)14yes—
484C23C2×AGL(1,ℤ/8)12yes—
485C23#57512yes—
486C23#57814yes—
487C23C23×D411yescharacteristic
488Q8—24no—
489Q8—24no—
490Q8—48no—
491Q8—44no—
492Q8—816no—
493D4—416no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—432no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—48no—
511C2×C4—44no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—48no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—88no—
535C2×C4—88no—
536C2×C4C2×#12014yes—
537C2×C4C2×#12014yes—
538C2×C4C2×#12012yes—
539C2×C4#53214yes—
540C2×C4#53614yes—
541C2×C4#54714yes—
542C2×C4#54818yes—
543C2×C4C23×D411yescharacteristic
544C2×C4C23×D411yescharacteristic
545C2×C4C22×#4112yes—
546C2×C4C2×2−1+412yes—
547C8—28no—
548C8—44no—
549C22—28no—
550C22—24no—
551C22—44no—
552C22—416no—
553C22—42no—
554C22—42no—
555C22—44no—
556C22—44no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—88no—
561C22#243214yes—
562C22#300614yes—
563C22#301114yes—
564C22#334318yes—
565C22#334918yes—
566C22C22×#12011yescharacteristic
567C22C2×#54712yes—
568C22C22×AGL(1,ℤ/8)11yescharacteristic
569C22C2×#57511yescharacteristic
570C22C2×#57812yes—
571C4—24no—
572C4—24no—
573C4—42no—
574C4—42no—
575C4—44no—
576C4—44no—
577C4—44no—
578C4—44no—
579C4—84no—
580C4—84no—
581C4#243214yes—
582C4C2×#53211yescharacteristic
583C4C2×#53611yescharacteristic
584C4C2×#54812yes—
585C2—44no—
586C2—82no—
587C2#2571618yesdirect product factor
588C2C2×#243211yescharacteristic
589C2C2×#300611yescharacteristic
590C2C2×#301111yescharacteristic
591C2C2×#334312yes—
592C2C2×#334912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#501C2semidirect
3. C22×#477C2semidirect
5. #25716C2direct (N × H)
6. C2×#2036C2semidirect
7. C2×#2035C2semidirect
8. C2×#1989C2semidirect
10. C2×#1854C2semidirect
11. C2×#1852C2semidirect
12. C2×#1763C2semidirect
31. C2×#501C22semidirect
33. C2×#477C22semidirect
51. #2036C22semidirect
52. #2035C22semidirect
53. #1989C22semidirect
55. #1854C22semidirect
56. #1852C22semidirect
57. #1763C22semidirect
587. C2#25716direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5522
C22×#5012
C22×#4772
C2×#25472
#257162
C2×#20362
C2×#20352
C2×#19892
C2×#19552
C2×#18542
C2×#18522
C2×#17632