Finite Group Databasebeta

← C2×#16986 #611865

Subgroups — C2×#16986 #611865

Order 512.

4646 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×#385C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3#16986C2116yesmaximal, direct product factor
4C2×#1444C211yesmaximal, characteristic, complemented
5C2×#1444C211yesmaximal, characteristic, complemented
6C2×#1442C212yesmaximal, complemented
7C2×#1441C211yesmaximal, characteristic, complemented
8C2×#1441C211yesmaximal, characteristic, complemented
9C2×#1441C212yesmaximal, complemented
10C2×#1439C212yesmaximal
11C2×#1439C212yesmaximal
12C2×C22⋊#95C211yesmaximal, characteristic
13C25×C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2212yes—
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2212yes—
24C2×#385—28no—
25C2×#385C2218yes—
26C2×#380—28no—
27C2×#380C2218yes—
28C22×#95C2212yes—
29C22×#95C2212yes—
30C22×#95C2211yescharacteristic
31C22×#95C2211yescharacteristic
32C22×#95C2211yescharacteristic
33C22×#95C2211yescharacteristic
34C22×#95C2212yes—
35C22×#95C2212yes—
36C22×#95C2212yes—
37C22×#95C2212yes—
38C22×#95C2212yes—
39C22×#95C2212yes—
40#1444C2218yescomplemented
41#1444C2218yescomplemented
42#1442C22116yescomplemented
43#1441C2218yescomplemented
44#1441C2218yescomplemented
45#1441C22116yescomplemented
46#1439C22116yes—
47#1439C22116yes—
48C22⋊#95C2218yes—
49C26C2311yescharacteristic
50C24×C4—24no—
51C24×C4—24no—
52C24×C4—44no—
53C24×C4—44no—
54C24×C4D418yes—
55C24×C4C2311yescharacteristic
56C24×C4C2312yes—
57C24×C4C2312yes—
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C24×C4C2312yes—
61C24×C4C2312yes—
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2314yes—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—28no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—44no—
74C22×C4⋊C4—44no—
75C22×C4⋊C4—44no—
76C22×C4⋊C4—44no—
77C22×C4⋊C4—44no—
78C22×C4⋊C4—44no—
79C22×C4⋊C4—44no—
80C22×C4⋊C4—44no—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2318yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2318yes—
87C22×C22⋊C4—24no—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—24no—
90C22×C22⋊C4—28no—
91C22×C22⋊C4—24no—
92C22×C22⋊C4—44no—
93C22×C22⋊C4—44no—
94C22×C22⋊C4—44no—
95C22×C22⋊C4—44no—
96C22×C22⋊C4—44no—
97C22×C22⋊C4Q8116yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2318yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2318yes—
102#385—232no—
103#380—232no—
104C2×#95—28no—
105C2×#95—24no—
106C2×#95—24no—
107C2×#95—24no—
108C2×#95—24no—
109C2×#95—28no—
110C2×#95—44no—
111C2×#95—44no—
112C2×#95—44no—
113C2×#95—44no—
114C2×#95—44no—
115C2×#95—44no—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2318yes—
119C2×#95C2318yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2318yes—
123C2×#95C2318yes—
124C2×#95C2318yes—
125C2×#95C2318yes—
126C2×#95C2318yes—
127C2×#95C2318yes—
128C25—22no—
129C25—22no—
130C25—22no—
131C25—22no—
132C25—24no—
133C25—22no—
134C25—24no—
135C25—24no—
136C25—22no—
137C25C2×D414yes—
138C25C2×Q818yes—
139C25C2412yes—
140C25C2411yescharacteristic
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—232no—
170C23×C4—28no—
171C23×C4—28no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
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—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—416no—
197C23×C4—416no—
198C23×C4—48no—
199C23×C4—48no—
200C23×C4—432no—
201C23×C4—432no—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2×D4116yes—
205C23×C4C2×Q818yes—
206C23×C4C2×Q818yes—
207C23×C4#4114yes—
208C23×C4#4114yes—
209C23×C4#4114yes—
210C23×C4#4114yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—216no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—48no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—48no—
236C2×C4⋊C4—416no—
237C2×C4⋊C4—416no—
238C2×C4⋊C4—416no—
239C2×C4⋊C4—432no—
240C2×C4⋊C4—432no—
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—432no—
249C2×C22⋊C4—48no—
250C2×C22⋊C4—48no—
251C2×C22⋊C4—48no—
252C2×C22⋊C4—48no—
253C2×C22⋊C4—48no—
254C2×C22⋊C4—416no—
255C2×C22⋊C4—432no—
256C2×C22⋊C4C2×Q8132yes—
257#95—216no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—216no—
263#95—48no—
264#95—48no—
265#95—48no—
266#95—48no—
267#95—48no—
268#95—48no—
269C24—28no—
270C24—28no—
271C24—28no—
272C24—28no—
273C24—28no—
274C24—28no—
275C24—216no—
276C24—216no—
277C24—216no—
278C24—216no—
279C24—216no—
280C24—216no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—24no—
286C24—22no—
287C24—24no—
288C24—24no—
289C24—22no—
290C24—48no—
291C24—48no—
292C24—48no—
293C24—48no—
294C24—44no—
295C24—44no—
296C24—44no—
297C24—44no—
298C24—48no—
299C24—48no—
300C24C4⋊Q8116yes—
301C24C22×D414yes—
302C24C22×D412yes—
303C24C22×Q818yes—
304C24C22×Q814yes—
305C24C2×#4112yes—
306C24C2×#4112yes—
307C24C2×#4112yes—
308C24C2×#4112yes—
309C24H(5,𝔽2)14yes—
310C24H(5,𝔽2)14yes—
311C24H(5,𝔽2)14yes—
312C24H(5,𝔽2)12yes—
313C242−1+412yes—
314C24C2511yescharacteristic
315C22×C4—28no—
316C22×C4—28no—
317C22×C4—216no—
318C22×C4—216no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—216no—
327C22×C4—216no—
328C22×C4—216no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—216no—
332C22×C4—216no—
333C22×C4—216no—
334C22×C4—216no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—232no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
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—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—416no—
387C22×C4—416no—
388C22×C4—4128no—
389C22×C4—432no—
390C22×C4—432no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—44no—
404C22×C4—48no—
405C22×C4—44no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—44no—
409C22×C4—48no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—44no—
418C22×C4—48no—
419C22×C4—44no—
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#123116yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×Q818yes—
436C22×C4C22×Q818yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C4⋊C4—416no—
442C4⋊C4—416no—
443C4⋊C4—416no—
444C4⋊C4—432no—
445C4⋊C4—432no—
446C4⋊C4—416no—
447C22⋊C4—232no—
448C22⋊C4—432no—
449C22⋊C4—416no—
450C22⋊C4—432no—
451C23—232no—
452C23—232no—
453C23—24no—
454C23—24no—
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—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—416no—
477C23—44no—
478C23—44no—
479C23—432no—
480C23—432no—
481C23—432no—
482C23—432no—
483C23—416no—
484C23C2×#12314yes—
485C23#52514yes—
486C23C2×C4⋊Q818yes—
487C23C2×C4⋊Q814yes—
488C23#53414yes—
489C23#53414yes—
490C23#53514yes—
491C23#53714yes—
492C23#53818yes—
493C23#53818yes—
494C23#53818yes—
495C23#54018yes—
496C23#54318yes—
497C23#54418yes—
498C23#55214yes—
499C23#55314yes—
500C23#55418yes—
501C23#55418yes—
502C23#55418yes—
503C23#55418yes—
504C23#55514yes—
505C23#55614yes—
506C23#56018yes—
507C23#56218yes—
508C23C23×D411yescharacteristic
509C23C23×Q812yes—
510C23C22×#4111yescharacteristic
511C23C22×#4111yescharacteristic
512C23C22×#4111yescharacteristic
513C23C22×#4111yescharacteristic
514C23C2×H(5,𝔽2)12yes—
515C23C2×H(5,𝔽2)12yes—
516C23C2×H(5,𝔽2)12yes—
517C23C2×H(5,𝔽2)11yescharacteristic
518C23C2×2−1+411yescharacteristic
519C2×C4—232no—
520C2×C4—24no—
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—24no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—464no—
539C2×C4—416no—
540C2×C4—416no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—416no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—432no—
560C2×C4—432no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
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—44no—
585C2×C4—88no—
586C2×C4—816no—
587C2×C4—88no—
588C2×C4—816no—
589C2×C4C2×#12318yes—
590C22—28no—
591C22—28no—
592C22—28no—
593C22—44no—
594C22—48no—
595C22—48no—
596C22—44no—
597C22—464no—
598C22—44no—
599C22—48no—
600C22—44no—
601C22—48no—
602C22#248518yes—
603C22#250218yes—
604C22#258318yes—
605C22#262218yes—
606C22#263518yes—
607C22#265918yes—
608C22#2768116yes—
609C22#2780116yes—
610C22#2784116yes—
611C22#2788116yes—
612C22#281318yes—
613C22C22×#12311yescharacteristic
614C22C2×#52511yescharacteristic
615C22C22×C4⋊Q811yescharacteristic
616C22C2×#53411yescharacteristic
617C22C2×#53411yescharacteristic
618C22C2×#53511yescharacteristic
619C22C2×#53711yescharacteristic
620C22C2×#53812yes—
621C22C2×#53812yes—
622C22C2×#53812yes—
623C22C2×#54012yes—
624C22C2×#54312yes—
625C22C2×#54412yes—
626C22C2×#55211yescharacteristic
627C22C2×#55311yescharacteristic
628C22C2×#55412yes—
629C22C2×#55412yes—
630C22C2×#55412yes—
631C22C2×#55412yes—
632C22C2×#55511yescharacteristic
633C22C2×#55611yescharacteristic
634C22C2×#56012yes—
635C22C2×#56212yes—
636C4—28no—
637C4—44no—
638C4—48no—
639C4—44no—
640C4—44no—
641C4—48no—
642C4—84no—
643C4—82no—
644C4—84no—
645C4—82no—
646C2—48no—
647C2#16986116yesdirect product factor
648C2C2×#262211yescharacteristic
649C2C2×#263511yescharacteristic
650C2C2×#265911yescharacteristic
651C2C2×#250211yescharacteristic
652C2C2×#248511yescharacteristic
653C2C2×#258311yescharacteristic
654C2C2×#281311yescharacteristic
655C2C2×#276812yes—
656C2C2×#278012yes—
657C2C2×#278412yes—
658C2C2×#278812yes—
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. #16986C2direct (N × H)
4. C2×#1444C2semidirect
5. C2×#1444C2semidirect
6. C2×#1442C2semidirect
7. C2×#1441C2semidirect
8. C2×#1441C2semidirect
9. C2×#1441C2semidirect
40. #1444C22semidirect
41. #1444C22semidirect
42. #1442C22semidirect
43. #1441C22semidirect
44. #1441C22semidirect
45. #1441C22semidirect
647. C2#16986direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3852
C22×#3802
#169862
C2×#14442
C2×#14422
C2×#14412
C2×#14392
C2×C22⋊#952