Finite Group Databasebeta

← C2×#15910 #609690

Subgroups — C2×#15910 #609690

Order 512.

5422 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#126C211yesmaximal, characteristic
2C22×#395C211yesmaximal, characteristic, complemented
3C22×#393C211yesmaximal, characteristic
4C22×#391C211yesmaximal, characteristic, complemented
5#15910C2116yesmaximal, direct product factor
6C2×#1445C211yesmaximal, characteristic, complemented
7C2×#1442C212yesmaximal, complemented
8C2×#1440C212yesmaximal
9C2×#1439C212yesmaximal
10C2×#1437C211yesmaximal, characteristic
11C2×#1436C212yesmaximal, complemented
12C2×#1436C211yesmaximal, characteristic, complemented
13C24×Q8C2211yescharacteristic
14C22×#126—28no—
15C22×#126—28no—
16C22×#126—28no—
17C22×#126C2218yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2212yes—
25C23×C22⋊C4C2212yes—
26C23×C22⋊C4C2211yescharacteristic
27C23×C42C2211yescharacteristic
28C23×C42C2211yescharacteristic
29C2×#395—28no—
30C2×#395C2218yescomplemented
31C2×#393—28no—
32C2×#393C2218yes—
33C2×#391—28no—
34C2×#391C2218yescomplemented
35C22×#95C2212yes—
36C22×#95C2212yes—
37C22×#95C2212yes—
38C22×#95C2212yes—
39C22×#95C2212yes—
40C22×#95C2212yes—
41C22×#95C2212yes—
42C22×#95C2212yes—
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2212yes—
46#1445C2218yescomplemented
47#1442C22116yescomplemented
48#1440C22116yes—
49#1439C22116yes—
50#1437C2218yes—
51#1436C22116yescomplemented
52#1436C2218yescomplemented
53C26C2311yescharacteristic
54C23×Q8—24no—
55C23×Q8—24no—
56C23×Q8—24no—
57C23×Q8—28no—
58C23×Q8—44no—
59C23×Q8C2314yes—
60C24×C4C2312yes—
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C24×C4C2312yes—
67C24×C4C2312yes—
68C24×C4C2312yes—
69C24×C4C2311yescharacteristic
70C2×#126—2128no—
71C2×#126—232no—
72C2×#126—232no—
73C2×#126—232no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—28no—
82C22×C4⋊C4—28no—
83C22×C4⋊C4—44no—
84C22×C4⋊C4D418yes—
85C22×C4⋊C4D418yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2318yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C22⋊C4—24no—
91C22×C22⋊C4—24no—
92C22×C22⋊C4—24no—
93C22×C22⋊C4—24no—
94C22×C22⋊C4—24no—
95C22×C22⋊C4—24no—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—44no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4D418yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2318yes—
108C22×C22⋊C4C2318yes—
109C22×C22⋊C4C2314yes—
110C22×C42—24no—
111C22×C42—24no—
112C22×C42—24no—
113C22×C42—24no—
114C22×C42—44no—
115C22×C42C2314yes—
116C22×C42C2314yes—
117#395—232no—
118#393—232no—
119#391—232no—
120C2×#95—28no—
121C2×#95—24no—
122C2×#95—28no—
123C2×#95—24no—
124C2×#95—28no—
125C2×#95—28no—
126C2×#95—28no—
127C2×#95—28no—
128C2×#95—28no—
129C2×#95—28no—
130C2×#95—44no—
131C2×#95C2318yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2318yes—
135C2×#95C2318yes—
136C2×#95C2318yes—
137C2×#95C2318yes—
138C2×#95C2318yes—
139C2×#95C2318yes—
140C2×#95C2318yes—
141C2×#95C2318yes—
142C25—22no—
143C25—22no—
144C25—24no—
145C25—24no—
146C25—24no—
147C25—24no—
148C25—24no—
149C25—24no—
150C25C2×D414yes—
151C25C2412yes—
152C25C2411yescharacteristic
153C22×Q8—232no—
154C22×Q8—28no—
155C22×Q8—28no—
156C22×Q8—28no—
157C22×Q8—216no—
158C22×Q8—48no—
159C22×Q8—432no—
160C22×Q8—432no—
161C22×Q8—416no—
162C22×Q8—416no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4#4118yes—
219C23×C4#4118yes—
220C23×C4#4118yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231#126—2256no—
232C2×C4⋊C4—232no—
233C2×C4⋊C4—232no—
234C2×C4⋊C4—232no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—216no—
243C2×C4⋊C4—216no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—432no—
246C2×C4⋊C4C2×D4116yes—
247C2×C4⋊C4C2×D4116yes—
248C2×C22⋊C4—232no—
249C2×C22⋊C4—232no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—232no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—216no—
260C2×C22⋊C4—216no—
261C2×C22⋊C4—216no—
262C2×C22⋊C4—216no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4—48no—
265C2×C22⋊C4—432no—
266C2×C22⋊C4—432no—
267C2×C22⋊C4C2×D4116yes—
268C2×C42—232no—
269C2×C42—28no—
270C2×C42—28no—
271C2×C42—28no—
272C2×C42—28no—
273C2×C42—48no—
274C2×C42—416no—
275#95—216no—
276#95—216no—
277#95—216no—
278#95—28no—
279#95—28no—
280#95—216no—
281#95—216no—
282#95—216no—
283#95—216no—
284#95—216no—
285#95—48no—
286C24—28no—
287C24—28no—
288C24—28no—
289C24—216no—
290C24—216no—
291C24—22no—
292C24—22no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—44no—
300C24—48no—
301C24—44no—
302C24—44no—
303C24—44no—
304C24—48no—
305C24—48no—
306C24—48no—
307C24—48no—
308C24—48no—
309C24—48no—
310C24—48no—
311C24—48no—
312C24—48no—
313C24—48no—
314C24—48no—
315C24C22×D414yes—
316C24C22×D412yes—
317C24C22×D412yes—
318C24C22×D412yes—
319C24C2×#4114yes—
320C24C2×#4114yes—
321C24C2×#4114yes—
322C24H(5,𝔽2)14yes—
323C242−1+414yes—
324C242−1+414yes—
325C24C2511yescharacteristic
326C2×Q8—232no—
327C2×Q8—432no—
328C2×Q8—432no—
329C2×Q8—4128no—
330C2×Q8—416no—
331C2×Q8—416no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—216no—
335C22×C4—216no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—216no—
344C22×C4—216no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—216no—
348C22×C4—216no—
349C22×C4—216no—
350C22×C4—216no—
351C22×C4—216no—
352C22×C4—216no—
353C22×C4—216no—
354C22×C4—216no—
355C22×C4—216no—
356C22×C4—216no—
357C22×C4—216no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—48no—
404C22×C4—44no—
405C22×C4—48no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—48no—
410C22×C4—44no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4#120116yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D414yes—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C2×#4118yes—
446C22×C4C2×#4118yes—
447C22×C4C2×#4118yes—
448C4⋊C4—232no—
449C4⋊C4—232no—
450C4⋊C4—232no—
451C4⋊C4—432no—
452C22⋊C4—232no—
453C22⋊C4—232no—
454C22⋊C4—232no—
455C22⋊C4—432no—
456C22⋊C4—432no—
457C42—232no—
458C42—416no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—28no—
463C23—28no—
464C23—416no—
465C23—42no—
466C23—44no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—416no—
474C23—416no—
475C23—432no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—432no—
485C23—816no—
486C23—816no—
487C23—816no—
488C23—816no—
489C23C2×#12014yes—
490C23#52514yes—
491C23#52918yes—
492C23#53214yes—
493C23#53214yes—
494C23#53514yes—
495C23#53614yes—
496C23#53614yes—
497C23#53918yes—
498C23#54118yes—
499C23#54218yes—
500C23#54618yes—
501C23#54618yes—
502C23#54718yes—
503C23#54718yes—
504C23#54818yes—
505C23#54818yes—
506C23#55118yes—
507C23#55618yes—
508C23#55618yes—
509C23#56318yes—
510C23C23×D411yescharacteristic
511C23C23×D411yescharacteristic
512C23C23×D411yescharacteristic
513C23C22×#4112yes—
514C23C22×#4112yes—
515C23C22×#4112yes—
516C23C2×H(5,𝔽2)12yes—
517C23C2×2−1+412yes—
518C23C2×2−1+412yes—
519Q8—464no—
520C2×C4—232no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—432no—
546C2×C4—416no—
547C2×C4—432no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—416no—
563C2×C4—416no—
564C2×C4—416no—
565C2×C4—416no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—432no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—816no—
587C2×C4C2×#12018yes—
588C22—48no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—48no—
593C22—84no—
594C22—832no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22#240418yes—
599C22#243218yes—
600C22#243218yes—
601C22#2469116yes—
602C22#2521116yes—
603C22#2560116yes—
604C22#2689116yes—
605C22#2716116yes—
606C22#2723116yes—
607C22C22×#12011yescharacteristic
608C22C2×#52511yescharacteristic
609C22C2×#52912yes—
610C22C2×#53211yescharacteristic
611C22C2×#53211yescharacteristic
612C22C2×#53511yescharacteristic
613C22C2×#53611yescharacteristic
614C22C2×#53611yescharacteristic
615C22C2×#53912yes—
616C22C2×#54112yes—
617C22C2×#54212yes—
618C22C2×#54612yes—
619C22C2×#54612yes—
620C22C2×#54712yes—
621C22C2×#54712yes—
622C22C2×#54812yes—
623C22C2×#54812yes—
624C22C2×#55112yes—
625C22C2×#55612yes—
626C22C2×#55612yes—
627C22C2×#56312yes—
628C4—28no—
629C4—48no—
630C4—44no—
631C4—44no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—48no—
636C4—48no—
637C4—84no—
638C2—84no—
639C2#15910116yesdirect product factor
640C2C2×#246912yes—
641C2C2×#243211yescharacteristic
642C2C2×#243211yescharacteristic
643C2C2×#240411yescharacteristic
644C2C2×#252112yes—
645C2C2×#256012yes—
646C2C2×#268912yes—
647C2C2×#271612yes—
648C2C2×#272312yes—
subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#395C2semidirect
4. C22×#391C2semidirect
5. #15910C2direct (N × H)
6. C2×#1445C2semidirect
7. C2×#1442C2semidirect
11. C2×#1436C2semidirect
12. C2×#1436C2semidirect
30. C2×#395C22semidirect
34. C2×#391C22semidirect
46. #1445C22semidirect
47. #1442C22semidirect
51. #1436C22semidirect
52. #1436C22semidirect
639. C2#15910direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#1262
C22×#3952
C22×#3932
C22×#3912
#159102
C2×#14452
C2×#14422
C2×#14402
C2×#14392
C2×#14372
C2×#14362