Finite Group Databasebeta

← C2×#15927 #609731

Subgroups — C2×#15927 #609731

Order 512.

5102 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#126C211yesmaximal, characteristic
2C22×#399C211yesmaximal, characteristic
3C22×#398C211yesmaximal, characteristic, complemented
4C22×#382C211yesmaximal, characteristic, complemented
5#15927C2116yesmaximal, direct product factor
6C2×#1445C211yesmaximal, characteristic, complemented
7C2×#1444C212yesmaximal, complemented
8C2×#1440C212yesmaximal
9C2×#1439C212yesmaximal
10C2×#1437C211yesmaximal, characteristic
11C2×#1436C212yesmaximal, complemented
12C2×#1436C211yesmaximal, characteristic, complemented
13C22×#126—28no—
14C22×#126—28no—
15C22×#126—28no—
16C22×#126C2218yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25C23×C22⋊C4C2212yes—
26C23×C22⋊C4C2212yes—
27C23×C22⋊C4C2211yescharacteristic
28C23×C42C2211yescharacteristic
29C23×C42C2211yescharacteristic
30C2×#399—28no—
31C2×#399C2218yes—
32C2×#398—28no—
33C2×#398C2218yescomplemented
34C2×#382—28no—
35C2×#382C2218yescomplemented
36C22×#95C2212yes—
37C22×#95C2212yes—
38C22×#95C2212yes—
39C22×#95C2212yes—
40C22×#95C2212yes—
41C22×#95C2212yes—
42C22×#95C2212yes—
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2212yes—
46#1445C2218yescomplemented
47#1444C22116yescomplemented
48#1440C22116yes—
49#1439C22116yes—
50#1437C2218yes—
51#1436C22116yescomplemented
52#1436C2218yescomplemented
53C26C2311yescharacteristic
54C24×C4C2312yes—
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C24×C4C2312yes—
61C24×C4C2312yes—
62C24×C4C2312yes—
63C24×C4C2311yescharacteristic
64C2×#126—2128no—
65C2×#126—232no—
66C2×#126—232no—
67C2×#126—232no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—28no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—28no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—28no—
81C22×C4⋊C4—44no—
82C22×C4⋊C4—44no—
83C22×C4⋊C4—44no—
84C22×C4⋊C4D418yes—
85C22×C4⋊C4D418yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2318yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2318yes—
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—24no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4—44no—
106C22×C22⋊C4D418yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2318yes—
110C22×C22⋊C4C2318yes—
111C22×C22⋊C4C2314yes—
112C22×C42—24no—
113C22×C42—24no—
114C22×C42—24no—
115C22×C42—24no—
116C22×C42—28no—
117C22×C42C2314yes—
118C22×C42C2314yes—
119#399—232no—
120#398—232no—
121#382—232no—
122C2×#95—24no—
123C2×#95—28no—
124C2×#95—24no—
125C2×#95—28no—
126C2×#95—28no—
127C2×#95—28no—
128C2×#95—28no—
129C2×#95—28no—
130C2×#95—28no—
131C2×#95—44no—
132C2×#95C2318yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
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
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
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×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2×D414yes—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4#4118yes—
209C23×C4#4118yes—
210C23×C4#4118yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221#126—2256no—
222C2×C4⋊C4—232no—
223C2×C4⋊C4—232no—
224C2×C4⋊C4—232no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—216no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—216no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—216no—
238C2×C4⋊C4—48no—
239C2×C4⋊C4—48no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—432no—
242C2×C4⋊C4—432no—
243C2×C4⋊C4—416no—
244C2×C4⋊C4C2×D4116yes—
245C2×C4⋊C4C2×D4116yes—
246C2×C22⋊C4—232no—
247C2×C22⋊C4—232no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—232no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—216no—
258C2×C22⋊C4—216no—
259C2×C22⋊C4—216no—
260C2×C22⋊C4—216no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—432no—
264C2×C22⋊C4—432no—
265C2×C22⋊C4C2×D4116yes—
266C2×C42—232no—
267C2×C42—232no—
268C2×C42—28no—
269C2×C42—28no—
270C2×C42—28no—
271C2×C42—28no—
272C2×C42—216no—
273#95—216no—
274#95—216no—
275#95—28no—
276#95—216no—
277#95—28no—
278#95—216no—
279#95—216no—
280#95—216no—
281#95—216no—
282#95—48no—
283C24—28no—
284C24—28no—
285C24—28no—
286C24—216no—
287C24—216no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—44no—
297C24—48no—
298C24—44no—
299C24—44no—
300C24—44no—
301C24—48no—
302C24—48no—
303C24—48no—
304C24—48no—
305C24—48no—
306C24—48no—
307C24—48no—
308C24—48no—
309C24—48no—
310C24—48no—
311C24—48no—
312C24C22×D414yes—
313C24C22×D412yes—
314C24C22×D412yes—
315C24C22×D412yes—
316C24C2×#4114yes—
317C24C2×#4114yes—
318C24C2×#4114yes—
319C24H(5,𝔽2)14yes—
320C242−1+414yes—
321C242−1+414yes—
322C24C2511yescharacteristic
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—216no—
326C22×C4—216no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—216no—
335C22×C4—216no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—216no—
339C22×C4—216no—
340C22×C4—216no—
341C22×C4—216no—
342C22×C4—216no—
343C22×C4—216no—
344C22×C4—216no—
345C22×C4—216no—
346C22×C4—216no—
347C22×C4—216no—
348C22×C4—216no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
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—44no—
395C22×C4—48no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
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#120116yes—
428C22×C4C22×D414yes—
429C22×C4C22×D414yes—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D414yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C2×#4118yes—
437C22×C4C2×#4118yes—
438C22×C4C2×#4118yes—
439C4⋊C4—232no—
440C4⋊C4—232no—
441C4⋊C4—232no—
442C4⋊C4—432no—
443C4⋊C4—432no—
444C4⋊C4—416no—
445C22⋊C4—232no—
446C22⋊C4—232no—
447C22⋊C4—232no—
448C22⋊C4—432no—
449C22⋊C4—432no—
450C42—232no—
451C42—232no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—28no—
456C23—28no—
457C23—416no—
458C23—42no—
459C23—44no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—416no—
467C23—416no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—432no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—432no—
478C23—816no—
479C23—816no—
480C23—816no—
481C23—816no—
482C23C2×#12014yes—
483C23#52514yes—
484C23#52918yes—
485C23#53214yes—
486C23#53214yes—
487C23#53514yes—
488C23#53614yes—
489C23#53614yes—
490C23#53918yes—
491C23#54118yes—
492C23#54218yes—
493C23#54618yes—
494C23#54718yes—
495C23#54718yes—
496C23#54718yes—
497C23#54818yes—
498C23#55018yes—
499C23#55118yes—
500C23#55218yes—
501C23#55218yes—
502C23#56318yes—
503C23C23×D411yescharacteristic
504C23C23×D411yescharacteristic
505C23C23×D411yescharacteristic
506C23C22×#4112yes—
507C23C22×#4112yes—
508C23C22×#4112yes—
509C23C2×H(5,𝔽2)12yes—
510C23C2×2−1+412yes—
511C23C2×2−1+412yes—
512C2×C4—232no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—416no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—416no—
553C2×C4—416no—
554C2×C4—416no—
555C2×C4—416no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—432no—
568C2×C4—432no—
569C2×C4—44no—
570C2×C4—432no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—816no—
579C2×C4C2×#12018yes—
580C22—48no—
581C22—44no—
582C22—48no—
583C22—44no—
584C22—44no—
585C22—832no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22#240618yes—
591C22#243118yes—
592C22#243218yes—
593C22#2479116yes—
594C22#2519116yes—
595C22#2553116yes—
596C22#2698116yes—
597C22#2721116yes—
598C22#2723116yes—
599C22C22×#12011yescharacteristic
600C22C2×#52511yescharacteristic
601C22C2×#52912yes—
602C22C2×#53211yescharacteristic
603C22C2×#53211yescharacteristic
604C22C2×#53511yescharacteristic
605C22C2×#53611yescharacteristic
606C22C2×#53611yescharacteristic
607C22C2×#53912yes—
608C22C2×#54112yes—
609C22C2×#54212yes—
610C22C2×#54612yes—
611C22C2×#54712yes—
612C22C2×#54712yes—
613C22C2×#54712yes—
614C22C2×#54812yes—
615C22C2×#55012yes—
616C22C2×#55112yes—
617C22C2×#55212yes—
618C22C2×#55212yes—
619C22C2×#56312yes—
620C4—28no—
621C4—48no—
622C4—44no—
623C4—44no—
624C4—44no—
625C4—44no—
626C4—48no—
627C4—44no—
628C4—48no—
629C4—84no—
630C2—84no—
631C2#15927116yesdirect product factor
632C2C2×#243111yescharacteristic
633C2C2×#243211yescharacteristic
634C2C2×#247912yes—
635C2C2×#240611yescharacteristic
636C2C2×#251912yes—
637C2C2×#255312yes—
638C2C2×#269812yes—
639C2C2×#272112yes—
640C2C2×#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
3. C22×#398C2semidirect
4. C22×#382C2semidirect
5. #15927C2direct (N × H)
6. C2×#1445C2semidirect
7. C2×#1444C2semidirect
11. C2×#1436C2semidirect
12. C2×#1436C2semidirect
33. C2×#398C22semidirect
35. C2×#382C22semidirect
46. #1445C22semidirect
47. #1444C22semidirect
51. #1436C22semidirect
52. #1436C22semidirect
631. C2#15927direct (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×#3992
C22×#3982
C22×#3822
#159272
C2×#14452
C2×#14442
C2×#14402
C2×#14392
C2×#14372
C2×#14362