Finite Group Databasebeta

← C2×#16393 #610902

Subgroups — C2×#16393 #610902

Order 512.

4678 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#398C211yesmaximal, characteristic, complemented
2C22×#385C211yesmaximal, characteristic
3C22×#381C211yesmaximal, characteristic
4C22×#380C211yesmaximal, characteristic
5#16393C2116yesmaximal, direct product factor
6C2×#1445C211yesmaximal, characteristic, complemented
7C2×#1445C212yesmaximal, complemented
8C2×#1442C211yesmaximal, characteristic, complemented
9C2×#1442C212yesmaximal, complemented
10C2×#1441C211yesmaximal, characteristic, complemented
11C2×#1440C212yesmaximal
12C2×#1439C212yesmaximal
13C25×C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2212yes—
23C2×#398—28no—
24C2×#398C2218yescomplemented
25C2×#385—28no—
26C2×#385C2218yes—
27C2×#381—28no—
28C2×#381C2218yes—
29C2×#380—28no—
30C2×#380C2218yes—
31C22×#95C2212yes—
32C22×#95C2212yes—
33C22×#95C2211yescharacteristic
34C22×#95C2212yes—
35C22×#95C2212yes—
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38C22×#95C2212yes—
39C22×#95C2212yes—
40C22×#95C2212yes—
41C22×#95C2212yes—
42C22×#95C2212yes—
43C22×#95C2211yescharacteristic
44#1445C2218yescomplemented
45#1445C22116yescomplemented
46#1442C2218yescomplemented
47#1442C22116yescomplemented
48#1441C2218yescomplemented
49#1440C22116yes—
50#1439C22116yes—
51C26C2311yescharacteristic
52C24×C4—28no—
53C24×C4—24no—
54C24×C4—24no—
55C24×C4—44no—
56C24×C4D418yes—
57C24×C4C2311yescharacteristic
58C24×C4C2312yes—
59C24×C4C2312yes—
60C24×C4C2311yescharacteristic
61C24×C4C2312yes—
62C24×C4C2311yescharacteristic
63C24×C4C2312yes—
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C24×C4C2314yes—
67C24×C4C2311yescharacteristic
68C22×C4⋊C4—28no—
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—24no—
78C22×C4⋊C4—44no—
79C22×C4⋊C4—44no—
80C22×C4⋊C4—44no—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2318yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C22⋊C4—24no—
87C22×C22⋊C4—24no—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—28no—
90C22×C22⋊C4—24no—
91C22×C22⋊C4—28no—
92C22×C22⋊C4—28no—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—44no—
95C22×C22⋊C4—44no—
96C22×C22⋊C4—44no—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2318yes—
100C22×C22⋊C4C2318yes—
101#398—232no—
102#385—232no—
103#381—232no—
104#380—232no—
105C2×#95—28no—
106C2×#95—28no—
107C2×#95—28no—
108C2×#95—24no—
109C2×#95—24no—
110C2×#95—24no—
111C2×#95—28no—
112C2×#95—44no—
113C2×#95—44no—
114C2×#95—44no—
115C2×#95—44no—
116C2×#95—44no—
117C2×#95D418yes—
118C2×#95C2318yes—
119C2×#95C2318yes—
120C2×#95C2314yes—
121C2×#95C2318yes—
122C2×#95C2318yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2318yes—
126C2×#95C2318yes—
127C2×#95C2318yes—
128C2×#95C2318yes—
129C2×#95C2318yes—
130C2×#95C2314yes—
131C25—22no—
132C25—22no—
133C25—22no—
134C25—22no—
135C25—22no—
136C25—22no—
137C25—24no—
138C25—24no—
139C25—24no—
140C25C2×D414yes—
141C25#4118yes—
142C25C2412yes—
143C25C2411yescharacteristic
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—232no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—216no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—232no—
169C23×C4—28no—
170C23×C4—28no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
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—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—416no—
203C23×C4—48no—
204C23×C4—432no—
205C23×C4—432no—
206C23×C4C2×D418yes—
207C23×C4C2×D414yes—
208C23×C4C2×D4116yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4#4114yes—
212C23×C4#4114yes—
213C23×C4#4114yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—216no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—216no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—48no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—432no—
238C2×C4⋊C4—416no—
239C2×C4⋊C4—416no—
240C2×C4⋊C4—416no—
241C2×C4⋊C4—416no—
242C2×C22⋊C4—232no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—216no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—216no—
249C2×C22⋊C4—216no—
250C2×C22⋊C4—216no—
251C2×C22⋊C4—48no—
252C2×C22⋊C4—48no—
253C2×C22⋊C4—48no—
254C2×C22⋊C4—432no—
255C2×C22⋊C4—432no—
256C2×C22⋊C4—416no—
257#95—216no—
258#95—216no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—216no—
263#95—216no—
264#95—48no—
265#95—48no—
266#95—48no—
267#95—48no—
268#95—48no—
269#95C2×D4116yes—
270C24—28no—
271C24—28no—
272C24—28no—
273C24—28no—
274C24—28no—
275C24—28no—
276C24—216no—
277C24—216no—
278C24—216no—
279C24—216no—
280C24—216no—
281C24—216no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—48no—
292C24—48no—
293C24—48no—
294C24—48no—
295C24—44no—
296C24—44no—
297C24—44no—
298C24—44no—
299C24—48no—
300C24—48no—
301C24#123116yes—
302C24C22×D414yes—
303C24C22×D412yes—
304C24C22×D412yes—
305C24C2×#4118yes—
306C24C2×#4112yes—
307C24C2×#4114yes—
308C24C2×#4112yes—
309C24C2×#4112yes—
310C24H(5,𝔽2)14yes—
311C24H(5,𝔽2)12yes—
312C24H(5,𝔽2)12yes—
313C242−1+414yes—
314C242−1+414yes—
315C24C2511yescharacteristic
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—216no—
319C22×C4—216no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—28no—
323C22×C4—232no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—216no—
329C22×C4—216no—
330C22×C4—216no—
331C22×C4—28no—
332C22×C4—216no—
333C22×C4—28no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—232no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
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—416no—
387C22×C4—4128no—
388C22×C4—432no—
389C22×C4—432no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—44no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—44no—
402C22×C4—48no—
403C22×C4—44no—
404C22×C4—48no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—44no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—48no—
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#121116yes—
435C22×C4C22×D414yes—
436C22×C4C22×D418yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4114yes—
442C4⋊C4—432no—
443C4⋊C4—416no—
444C4⋊C4—416no—
445C4⋊C4—416no—
446C4⋊C4—416no—
447C22⋊C4—232no—
448C22⋊C4—432no—
449C22⋊C4—432no—
450C22⋊C4—416no—
451C23—232no—
452C23—24no—
453C23—24no—
454C23—232no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—28no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—28no—
464C23—28no—
465C23—232no—
466C23—416no—
467C23—416no—
468C23—416no—
469C23—416no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—432no—
475C23—432no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—432no—
481C23—432no—
482C23—44no—
483C23—44no—
484C23C2×#12114yes—
485C23C2×#12318yes—
486C23C2×#12314yes—
487C23#52514yes—
488C23#53414yes—
489C23#53514yes—
490C23#53614yes—
491C23#53614yes—
492C23#53618yes—
493C23#53918yes—
494C23#53918yes—
495C23#53918yes—
496C23#54118yes—
497C23#54218yes—
498C23#54618yes—
499C23#54614yes—
500C23#55014yes—
501C23#55118yes—
502C23#55218yes—
503C23#55218yes—
504C23#55314yes—
505C23#55514yes—
506C23#56218yes—
507C23#56218yes—
508C23C23×D411yescharacteristic
509C23C23×D411yescharacteristic
510C23C22×#4111yescharacteristic
511C23C22×#4112yes—
512C23C22×#4111yescharacteristic
513C23C22×#4111yescharacteristic
514C23C2×H(5,𝔽2)12yes—
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×H(5,𝔽2)11yescharacteristic
517C23C2×2−1+412yes—
518C23C2×2−1+412yes—
519C2×C4—24no—
520C2×C4—232no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—464no—
536C2×C4—416no—
537C2×C4—42no—
538C2×C4—416no—
539C2×C4—416no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—432no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
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—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—816no—
585C2×C4—88no—
586C2×C4—88no—
587C2×C4—816no—
588C2×C4—816no—
589C2×C4C2×#12118yes—
590C22—28no—
591C22—28no—
592C22—28no—
593C22—44no—
594C22—464no—
595C22—44no—
596C22—44no—
597C22—48no—
598C22—48no—
599C22—44no—
600C22—48no—
601C22—48no—
602C22#241218yes—
603C22#241418yes—
604C22#248418yes—
605C22#262118yes—
606C22#262318yes—
607C22#263818yes—
608C22#2687116yes—
609C22#2694116yes—
610C22#2739116yes—
611C22#2744116yes—
612C22#281618yes—
613C22C22×#12111yescharacteristic
614C22C22×#12311yescharacteristic
615C22C2×#52511yescharacteristic
616C22C2×#53411yescharacteristic
617C22C2×#53511yescharacteristic
618C22C2×#53611yescharacteristic
619C22C2×#53611yescharacteristic
620C22C2×#53612yes—
621C22C2×#53912yes—
622C22C2×#53912yes—
623C22C2×#53912yes—
624C22C2×#54112yes—
625C22C2×#54212yes—
626C22C2×#54612yes—
627C22C2×#54611yescharacteristic
628C22C2×#55011yescharacteristic
629C22C2×#55112yes—
630C22C2×#55212yes—
631C22C2×#55212yes—
632C22C2×#55311yescharacteristic
633C22C2×#55511yescharacteristic
634C22C2×#56212yes—
635C22C2×#56212yes—
636C4—28no—
637C4—44no—
638C4—44no—
639C4—48no—
640C4—44no—
641C4—82no—
642C4—84no—
643C4—82no—
644C4—84no—
645C4—84no—
646C2—48no—
647C2#16393116yesdirect product factor
648C2C2×#263811yescharacteristic
649C2C2×#262311yescharacteristic
650C2C2×#262111yescharacteristic
651C2C2×#241211yescharacteristic
652C2C2×#241411yescharacteristic
653C2C2×#248411yescharacteristic
654C2C2×#281611yescharacteristic
655C2C2×#268712yes—
656C2C2×#269412yes—
657C2C2×#273912yes—
658C2C2×#274412yes—
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
1. C22×#398C2semidirect
5. #16393C2direct (N × H)
6. C2×#1445C2semidirect
7. C2×#1445C2semidirect
8. C2×#1442C2semidirect
9. C2×#1442C2semidirect
10. C2×#1441C2semidirect
24. C2×#398C22semidirect
44. #1445C22semidirect
45. #1445C22semidirect
46. #1442C22semidirect
47. #1442C22semidirect
48. #1441C22semidirect
647. C2#16393direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3982
C22×#3852
C22×#3812
C22×#3802
#163932
C2×#14452
C2×#14422
C2×#14412
C2×#14402
C2×#14392