Finite Group Databasebeta

← C2×#15670 #609306

Subgroups — C2×#15670 #609306

Order 512.

5946 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C22×#400C211yesmaximal, characteristic, complemented
3C22×#397C211yesmaximal, characteristic
4C22×#382C211yesmaximal, characteristic, complemented
5#15670C2116yesmaximal, direct product factor
6C2×#1446C212yesmaximal, complemented
7C2×#1445C211yesmaximal, characteristic, complemented
8C2×#1442C212yesmaximal, complemented
9C2×#1440C212yesmaximal
10C2×#1437C211yesmaximal, characteristic
11C2×#1436C211yesmaximal, characteristic, complemented
12C2×C22⋊#95C212yesmaximal
13C24×Q8C2211yescharacteristic
14C25×C4C2211yescharacteristic
15C22×C22⋊Q8—28no—
16C22×C22⋊Q8—28no—
17C22×C22⋊Q8—28no—
18C22×C22⋊Q8C2218yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2212yes—
25C23×C22⋊C4C2212yes—
26C23×C22⋊C4C2211yescharacteristic
27C23×C42C2211yescharacteristic
28C2×#400—28no—
29C2×#400C2218yescomplemented
30C2×#397—28no—
31C2×#397C2218yes—
32C2×#382—28no—
33C2×#382C2218yescomplemented
34C22×#95C2212yes—
35C22×#95C2212yes—
36C22×#95C2212yes—
37C22×#95C2212yes—
38C22×#95C2212yes—
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2212yes—
42C22×#95C2212yes—
43C22×#95C2212yes—
44C22×#95C2212yes—
45C22×#95C2212yes—
46#1446C22116yescomplemented
47#1445C2218yescomplemented
48#1442C22116yescomplemented
49#1440C22116yes—
50#1437C2218yes—
51#1436C2218yescomplemented
52C22⋊#95C22116yes—
53C26C2311yescharacteristic
54C23×Q8—24no—
55C23×Q8—24no—
56C23×Q8—24no—
57C23×Q8—28no—
58C23×Q8—44no—
59C23×Q8C2314yes—
60C24×C4—24no—
61C24×C4—24no—
62C24×C4—24no—
63C24×C4—28no—
64C24×C4—44no—
65C24×C4C2312yes—
66C24×C4C2312yes—
67C24×C4C2312yes—
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2314yes—
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C2×C22⋊Q8—2128no—
77C2×C22⋊Q8—232no—
78C2×C22⋊Q8—232no—
79C2×C22⋊Q8—232no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4D418yes—
91C22×C4⋊C4D418yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4D418yes—
108C22×C22⋊C4C2318yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2318yes—
111C22×C22⋊C4C2314yes—
112C22×C42—24no—
113C22×C42—28no—
114C22×C42C2314yes—
115#400—232no—
116#397—232no—
117#382—232no—
118C2×#95—28no—
119C2×#95—24no—
120C2×#95—28no—
121C2×#95—28no—
122C2×#95—24no—
123C2×#95—28no—
124C2×#95—28no—
125C2×#95—28no—
126C2×#95—28no—
127C2×#95—44no—
128C2×#95—44no—
129C2×#95—44no—
130C2×#95C2314yes—
131C2×#95C2318yes—
132C2×#95C2318yes—
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—
149C25C2×D414yes—
150C25#4118yes—
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—22no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—232no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—28no—
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—28no—
193C23×C4—28no—
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×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—216no—
216C23×C4—48no—
217C23×C4—432no—
218C23×C4—432no—
219C23×C4—416no—
220C23×C4—416no—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4#4118yes—
230C23×C4#4118yes—
231C23×C4C2414yes—
232C23×C4C2414yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C22⋊Q8—2256no—
242C2×C4⋊C4—232no—
243C2×C4⋊C4—232no—
244C2×C4⋊C4—232no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4C2×D4116yes—
257C2×C4⋊C4C2×D4116yes—
258C2×C22⋊C4—232no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—232no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—216no—
267C2×C22⋊C4—216no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—216no—
270C2×C22⋊C4—48no—
271C2×C22⋊C4—432no—
272C2×C22⋊C4—432no—
273C2×C22⋊C4C2×D4116yes—
274C2×C42—232no—
275C2×C42—28no—
276C2×C42—216no—
277#95—216no—
278#95—216no—
279#95—28no—
280#95—216no—
281#95—28no—
282#95—216no—
283#95—216no—
284#95—216no—
285#95—216no—
286#95—48no—
287#95—48no—
288#95—48no—
289C24—28no—
290C24—216no—
291C24—28no—
292C24—216no—
293C24—216no—
294C24—216no—
295C24—216no—
296C24—216no—
297C24—216no—
298C24—216no—
299C24—22no—
300C24—22no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—44no—
307C24—44no—
308C24—44no—
309C24—44no—
310C24—48no—
311C24—48no—
312C24—48no—
313C24—48no—
314C24—48no—
315C24—48no—
316C24#124116yes—
317C24C22×D414yes—
318C24C22×D412yes—
319C24C22×D412yes—
320C24C22×D412yes—
321C24C2×#4118yes—
322C24C2×#4114yes—
323C24C2×#4114yes—
324C24C2×#4114yes—
325C24H(5,𝔽2)14yes—
326C242−1+414yes—
327C242−1+414yes—
328C24C2511yescharacteristic
329C2×Q8—232no—
330C2×Q8—432no—
331C2×Q8—432no—
332C2×Q8—416no—
333C2×Q8—4128no—
334C2×Q8—416no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—216no—
340C22×C4—216no—
341C22×C4—216no—
342C22×C4—216no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—232no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—216no—
354C22×C4—216no—
355C22×C4—216no—
356C22×C4—216no—
357C22×C4—216no—
358C22×C4—216no—
359C22×C4—216no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
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—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—432no—
404C22×C4—432no—
405C22×C4—416no—
406C22×C4—4128no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—416no—
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—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4#120116yes—
446C22×C4#124116yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D414yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C2×#4118yes—
456C22×C4C2×#4118yes—
457C4⋊C4—232no—
458C4⋊C4—232no—
459C4⋊C4—232no—
460C4⋊C4—416no—
461C22⋊C4—232no—
462C22⋊C4—232no—
463C22⋊C4—432no—
464C22⋊C4—432no—
465C42—232no—
466C23—24no—
467C23—28no—
468C23—24no—
469C23—28no—
470C23—232no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—264no—
478C23—416no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—44no—
484C23—44no—
485C23—432no—
486C23—416no—
487C23—44no—
488C23—44no—
489C23—432no—
490C23—44no—
491C23—432no—
492C23—432no—
493C23—432no—
494C23—44no—
495C23C2×#12014yes—
496C23#52514yes—
497C23C2×#12418yes—
498C23C2×#12414yes—
499C23C2×#12414yes—
500C23#52918yes—
501C23#53514yes—
502C23#53614yes—
503C23#53614yes—
504C23#53918yes—
505C23#54118yes—
506C23#54218yes—
507C23#54618yes—
508C23#54618yes—
509C23#54718yes—
510C23#54718yes—
511C23#54818yes—
512C23#54818yes—
513C23#55118yes—
514C23#55218yes—
515C23#55218yes—
516C23#56318yes—
517C23C23×D411yescharacteristic
518C23C23×D411yescharacteristic
519C23C23×D411yescharacteristic
520C23C22×#4112yes—
521C23C22×#4112yes—
522C23C22×#4112yes—
523C23C2×H(5,𝔽2)12yes—
524C23C2×2−1+412yes—
525C23C2×2−1+412yes—
526Q8—464no—
527C2×C4—232no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—232no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—464no—
549C2×C4—416no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—416no—
561C2×C4—416no—
562C2×C4—416no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—432no—
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—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—816no—
592C2×C4—816no—
593C2×C4—816no—
594C2×C4C2×#12018yes—
595C2×C4C2×#12418yes—
596C22—28no—
597C22—216no—
598C22—464no—
599C22—44no—
600C22—48no—
601C22—48no—
602C22—48no—
603C22—44no—
604C22—48no—
605C22—48no—
606C22#240618yes—
607C22#243518yes—
608C22#243618yes—
609C22#2477116yes—
610C22#2488116yes—
611C22#2489116yes—
612C22#2690116yes—
613C22#2698116yes—
614C22#2723116yes—
615C22C22×#12011yescharacteristic
616C22C2×#52511yescharacteristic
617C22C22×#12411yescharacteristic
618C22C22×#12411yescharacteristic
619C22C2×#52912yes—
620C22C2×#53511yescharacteristic
621C22C2×#53611yescharacteristic
622C22C2×#53611yescharacteristic
623C22C2×#53912yes—
624C22C2×#54112yes—
625C22C2×#54212yes—
626C22C2×#54612yes—
627C22C2×#54612yes—
628C22C2×#54712yes—
629C22C2×#54712yes—
630C22C2×#54812yes—
631C22C2×#54812yes—
632C22C2×#55112yes—
633C22C2×#55212yes—
634C22C2×#55212yes—
635C22C2×#56312yes—
636C4—28no—
637C4—28no—
638C4—44no—
639C4—44no—
640C4—44no—
641C4—48no—
642C4—44no—
643C4—84no—
644C4—84no—
645C4—84no—
646C2—48no—
647C2#15670116yesdirect product factor
648C2C2×#243511yescharacteristic
649C2C2×#243611yescharacteristic
650C2C2×#247712yes—
651C2C2×#240611yescharacteristic
652C2C2×#248812yes—
653C2C2×#248912yes—
654C2C2×#269012yes—
655C2C2×#269812yes—
656C2C2×#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×#400C2semidirect
4. C22×#382C2semidirect
5. #15670C2direct (N × H)
6. C2×#1446C2semidirect
7. C2×#1445C2semidirect
8. C2×#1442C2semidirect
11. C2×#1436C2semidirect
29. C2×#400C22semidirect
33. C2×#382C22semidirect
46. #1446C22semidirect
47. #1445C22semidirect
48. #1442C22semidirect
51. #1436C22semidirect
647. C2#15670direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C22×#4002
C22×#3972
C22×#3822
#156702
C2×#14462
C2×#14452
C2×#14422
C2×#14402
C2×#14372
C2×#14362
C2×C22⋊#952