Finite Group Databasebeta

← C2×#17040 #611998

Subgroups — C2×#17040 #611998

Order 512.

4122 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#398C211yesmaximal, characteristic, complemented
2C22×#397C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic, complemented
4C22×#388C211yesmaximal, characteristic
5#17040C2116yesmaximal, direct product factor
6C2×#1443C212yesmaximal, complemented
7C2×#1441C211yesmaximal, characteristic, complemented
8C2×#1441C212yesmaximal, complemented
9C2×#1439C212yesmaximal
10C2×#1439C211yesmaximal, characteristic
11C2×#1439C212yesmaximal
12C2×#1434C211yesmaximal, characteristic, complemented
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24C2×#398—28no—
25C2×#398C2218yescomplemented
26C2×#397—28no—
27C2×#397C2218yes—
28C2×#389—28no—
29C2×#389C2218yescomplemented
30C2×#388—28no—
31C2×#388C2218yes—
32C22×#95C2212yes—
33C22×#95C2212yes—
34C22×#95C2212yes—
35C22×#95C2212yes—
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38C22×#95C2212yes—
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2212yes—
42C22×#95C2212yes—
43C22×#95C2212yes—
44#1443C22116yescomplemented
45#1441C2218yescomplemented
46#1441C22116yescomplemented
47#1439C2218yes—
48#1439C22116yes—
49#1439C22116yes—
50#1434C2218yescomplemented
51C26C2311yescharacteristic
52C24×C4C2312yes—
53C24×C4C2311yescharacteristic
54C24×C4C2311yescharacteristic
55C24×C4C2312yes—
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2312yes—
59C24×C4C2311yescharacteristic
60C24×C4C2312yes—
61C24×C4C2311yescharacteristic
62C22×C4⋊C4—24no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—28no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—28no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—28no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—44no—
72C22×C4⋊C4—44no—
73C22×C4⋊C4—44no—
74C22×C4⋊C4—44no—
75C22×C4⋊C4D418yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2318yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2318yes—
81C22×C22⋊C4—24no—
82C22×C22⋊C4—24no—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—28no—
85C22×C22⋊C4—28no—
86C22×C22⋊C4—24no—
87C22×C22⋊C4—24no—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—28no—
90C22×C22⋊C4—24no—
91C22×C22⋊C4—44no—
92C22×C22⋊C4—44no—
93C22×C22⋊C4—44no—
94C22×C22⋊C4—44no—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2318yes—
98C22×C22⋊C4C2318yes—
99C22×C22⋊C4C2314yes—
100C22×C42—44no—
101C22×C42D418yes—
102C22×C42C2314yes—
103#398—232no—
104#397—232no—
105#389—232no—
106#388—232no—
107C2×#95—28no—
108C2×#95—24no—
109C2×#95—24no—
110C2×#95—28no—
111C2×#95—24no—
112C2×#95—24no—
113C2×#95—28no—
114C2×#95—28no—
115C2×#95—28no—
116C2×#95—44no—
117C2×#95—44no—
118C2×#95—44no—
119C2×#95C2314yes—
120C2×#95C2318yes—
121C2×#95C2314yes—
122C2×#95C2318yes—
123C2×#95C2318yes—
124C2×#95C2318yes—
125C2×#95C2318yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2318yes—
129C2×#95C2318yes—
130C2×#95C2318yes—
131C25—22no—
132C25—22no—
133C25—22no—
134C25—24no—
135C25—22no—
136C25—24no—
137C25—22no—
138C25—22no—
139C25—24no—
140C25—24no—
141C25#4114yes—
142C25C2412yes—
143C25C2411yescharacteristic
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
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—24no—
195C23×C4—24no—
196C23×C4C2×D414yes—
197C23×C4C2×D418yes—
198C23×C4C2×D414yes—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4#4114yes—
202C23×C4#4118yes—
203C23×C4#4114yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C2×C4⋊C4—232no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—216no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—216no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—216no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—48no—
226C2×C4⋊C4—48no—
227C2×C4⋊C4—48no—
228C2×C4⋊C4—432no—
229C2×C4⋊C4—416no—
230C2×C4⋊C4—432no—
231C2×C4⋊C4—416no—
232C2×C4⋊C4C2×D4116yes—
233C2×C22⋊C4—232no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—216no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—216no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—216no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—48no—
245C2×C22⋊C4—48no—
246C2×C22⋊C4—48no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—432no—
249C2×C22⋊C4—416no—
250C2×C22⋊C4—432no—
251C2×C22⋊C4—416no—
252C2×C42—416no—
253C2×C42—48no—
254C2×C42C2×D4116yes—
255#95—28no—
256#95—28no—
257#95—216no—
258#95—216no—
259#95—28no—
260#95—28no—
261#95—216no—
262#95—216no—
263#95—216no—
264#95—48no—
265#95—48no—
266#95—48no—
267C24—28no—
268C24—216no—
269C24—28no—
270C24—216no—
271C24—28no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—24no—
276C24—22no—
277C24—24no—
278C24—22no—
279C24—22no—
280C24—24no—
281C24—24no—
282C24—44no—
283C24—44no—
284C24—44no—
285C24—44no—
286C24—44no—
287C24—44no—
288C24—48no—
289C24—48no—
290C24—48no—
291C24—48no—
292C24—48no—
293C24—44no—
294C24—48no—
295C24—48no—
296C24—44no—
297C24—48no—
298C24—48no—
299C24—48no—
300C24C22×D412yes—
301C24C22×D412yes—
302C24C2×#4114yes—
303C24C2×#4112yes—
304C24C2×#4112yes—
305C24C2×#4112yes—
306C24C2×#4114yes—
307C24H(5,𝔽2)14yes—
308C24H(5,𝔽2)12yes—
309C24H(5,𝔽2)12yes—
310C242−1+414yes—
311C242−1+414yes—
312C24C2511yescharacteristic
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—216no—
316C22×C4—216no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—216no—
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—216no—
328C22×C4—216no—
329C22×C4—28no—
330C22×C4—216no—
331C22×C4—216no—
332C22×C4—216no—
333C22×C4—28no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
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—44no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—44no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—44no—
399C22×C4—48no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—48no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—44no—
408C22×C4—44no—
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#121116yes—
425C22×C4C22×D414yes—
426C22×C4C22×D418yes—
427C22×C4C22×D414yes—
428C22×C4C22×D414yes—
429C22×C4C22×D414yes—
430C22×C4C2×#4118yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4114yes—
433C4⋊C4—232no—
434C4⋊C4—432no—
435C4⋊C4—416no—
436C4⋊C4—432no—
437C4⋊C4—416no—
438C22⋊C4—232no—
439C22⋊C4—432no—
440C22⋊C4—416no—
441C22⋊C4—432no—
442C22⋊C4—416no—
443C42—416no—
444C23—24no—
445C23—28no—
446C23—24no—
447C23—28no—
448C23—24no—
449C23—416no—
450C23—416no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—44no—
464C23—44no—
465C23—42no—
466C23—44no—
467C23—432no—
468C23—44no—
469C23—44no—
470C23—432no—
471C23—416no—
472C23—88no—
473C23—88no—
474C23—88no—
475C23—816no—
476C23—816no—
477C23—88no—
478C23C2×#12114yes—
479C23#52514yes—
480C23#53214yes—
481C23#53414yes—
482C23#53514yes—
483C23#53614yes—
484C23#53614yes—
485C23#53718yes—
486C23#53818yes—
487C23#53918yes—
488C23#53918yes—
489C23#54018yes—
490C23#54218yes—
491C23#54614yes—
492C23#54614yes—
493C23#54818yes—
494C23#55218yes—
495C23#55218yes—
496C23#55314yes—
497C23#55314yes—
498C23#55618yes—
499C23#56318yes—
500C23#56318yes—
501C23C23×D411yescharacteristic
502C23C23×D411yescharacteristic
503C23C22×#4111yescharacteristic
504C23C22×#4111yescharacteristic
505C23C22×#4111yescharacteristic
506C23C22×#4112yes—
507C23C2×H(5,𝔽2)12yes—
508C23C2×H(5,𝔽2)11yescharacteristic
509C23C2×H(5,𝔽2)11yescharacteristic
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—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—416no—
533C2×C4—416no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—432no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—432no—
552C2×C4—416no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
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—88no—
578C2×C4—816no—
579C2×C4—816no—
580C2×C4—88no—
581C2×C4C2×#12118yes—
582C22—44no—
583C22—48no—
584C22—44no—
585C22—44no—
586C22—48no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—832no—
591C22—84no—
592C22—84no—
593C22—82no—
594C22#253518yes—
595C22#254518yes—
596C22#261118yes—
597C22#261818yes—
598C22#263818yes—
599C22#266118yes—
600C22#2702116yes—
601C22#2707116yes—
602C22#2777116yes—
603C22#2787116yes—
604C22#282118yes—
605C22C22×#12111yescharacteristic
606C22C2×#52511yescharacteristic
607C22C2×#53211yescharacteristic
608C22C2×#53411yescharacteristic
609C22C2×#53511yescharacteristic
610C22C2×#53611yescharacteristic
611C22C2×#53611yescharacteristic
612C22C2×#53712yes—
613C22C2×#53812yes—
614C22C2×#53912yes—
615C22C2×#53912yes—
616C22C2×#54012yes—
617C22C2×#54212yes—
618C22C2×#54611yescharacteristic
619C22C2×#54611yescharacteristic
620C22C2×#54812yes—
621C22C2×#55212yes—
622C22C2×#55212yes—
623C22C2×#55311yescharacteristic
624C22C2×#55311yescharacteristic
625C22C2×#55612yes—
626C22C2×#56312yes—
627C22C2×#56312yes—
628C4—28no—
629C4—44no—
630C4—44no—
631C4—48no—
632C4—44no—
633C4—48no—
634C4—84no—
635C4—82no—
636C4—84no—
637C4—82no—
638C2—84no—
639C2#17040116yesdirect product factor
640C2C2×#261811yescharacteristic
641C2C2×#263811yescharacteristic
642C2C2×#266111yescharacteristic
643C2C2×#261111yescharacteristic
644C2C2×#253511yescharacteristic
645C2C2×#254511yescharacteristic
646C2C2×#282111yescharacteristic
647C2C2×#270212yes—
648C2C2×#270712yes—
649C2C2×#277712yes—
650C2C2×#278712yes—
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
3. C22×#389C2semidirect
5. #17040C2direct (N × H)
6. C2×#1443C2semidirect
7. C2×#1441C2semidirect
8. C2×#1441C2semidirect
12. C2×#1434C2semidirect
25. C2×#398C22semidirect
29. C2×#389C22semidirect
44. #1443C22semidirect
45. #1441C22semidirect
46. #1441C22semidirect
50. #1434C22semidirect
639. C2#17040direct (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×#3972
C22×#3892
C22×#3882
#170402
C2×#14432
C2×#14412
C2×#14392
C2×#14342