Finite Group Databasebeta

← C2×#15114 #608294

Subgroups — C2×#15114 #608294

Order 512.

6174 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#117C211yesmaximal, characteristic
2C22×#387C212yesmaximal, complemented
3C22×#385C211yesmaximal, characteristic
4#15114C2116yesmaximal, direct product factor
5C2×#1444C211yesmaximal, characteristic, complemented
6C2×#1442C211yesmaximal, characteristic, complemented
7C2×#1441C211yesmaximal, characteristic, complemented
8C2×#1441C211yesmaximal, characteristic, complemented
9C2×#1440C212yesmaximal
10C2×#1439C212yesmaximal
11C2×#1434C212yesmaximal, complemented
12C2×C22⋊#95C211yesmaximal, characteristic
13C25×C4C2211yescharacteristic
14C22×#117—216no—
15C22×#117—28no—
16C22×#117C2218yes—
17C23×C4⋊C4C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2212yes—
24C23×C22⋊C4C2211yescharacteristic
25C23×C22⋊C4C2212yes—
26C23×C42C2212yes—
27C2×#387—216no—
28C2×#387C22116yescomplemented
29C2×#385—28no—
30C2×#385C2218yes—
31C22×#95C2212yes—
32C22×#95C2212yes—
33C22×#95C2212yes—
34C22×#95C2212yes—
35C22×#95C2212yes—
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38C22×#95C2211yescharacteristic
39C22×#95C2212yes—
40C22×#95C2211yescharacteristic
41C22×#95C2212yes—
42C22×#95C2212yes—
43#1444C2218yescomplemented
44#1442C2218yescomplemented
45#1441C2218yescomplemented
46#1441C2218yescomplemented
47#1440C22116yes—
48#1439C22116yes—
49#1434C22116yescomplemented
50C22⋊#95C2218yes—
51C26C2311yescharacteristic
52C24×C4—28no—
53C24×C4—28no—
54C24×C4—24no—
55C24×C4—24no—
56C24×C4D418yes—
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C24×C4C2312yes—
60C24×C4C2314yes—
61C24×C4C2312yes—
62C24×C4C2312yes—
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2312yes—
66C24×C4C2311yescharacteristic
67C24×C4C2311yescharacteristic
68C2×#117—2128no—
69C2×#117—232no—
70C2×#117—264no—
71C22×C4⋊C4—28no—
72C22×C4⋊C4—28no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—28no—
76C22×C4⋊C4—28no—
77C22×C4⋊C4—28no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—28no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2318yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C22⋊C4—28no—
89C22×C22⋊C4—24no—
90C22×C22⋊C4—28no—
91C22×C22⋊C4—28no—
92C22×C22⋊C4—24no—
93C22×C22⋊C4—24no—
94C22×C22⋊C4—24no—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—28no—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4C2318yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2318yes—
103C22×C42—28no—
104C22×C42—28no—
105C22×C42D4116yes—
106C22×C42C2318yes—
107#387—264no—
108#385—232no—
109C2×#95—24no—
110C2×#95—24no—
111C2×#95—28no—
112C2×#95—28no—
113C2×#95—28no—
114C2×#95—28no—
115C2×#95—24no—
116C2×#95—28no—
117C2×#95—28no—
118C2×#95—24no—
119C2×#95—44no—
120C2×#95—44no—
121C2×#95C2318yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2318yes—
126C2×#95C2318yes—
127C2×#95C2318yes—
128C2×#95C2318yes—
129C2×#95C2318yes—
130C2×#95C2314yes—
131C2×#95C2318yes—
132C2×#95C2318yes—
133C25—24no—
134C25—22no—
135C25—22no—
136C25—24no—
137C25—22no—
138C25—22no—
139C25—22no—
140C25—24no—
141C25—22no—
142C25C2×D414yes—
143C25#4118yes—
144C25C2412yes—
145C25C2411yescharacteristic
146C23×C4—264no—
147C23×C4—264no—
148C23×C4—232no—
149C23×C4—24no—
150C23×C4—216no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—232no—
156C23×C4—232no—
157C23×C4—216no—
158C23×C4—28no—
159C23×C4—28no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4C2×D418yes—
206C23×C4C2×D418yes—
207C23×C4C2×D4116yes—
208C23×C4C2×D418yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4#4114yes—
212C23×C4#4118yes—
213C23×C4#4114yes—
214C23×C4#4114yes—
215C23×C4#4114yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226#117—2256no—
227C2×C4⋊C4—264no—
228C2×C4⋊C4—232no—
229C2×C4⋊C4—232no—
230C2×C4⋊C4—216no—
231C2×C4⋊C4—216no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—216no—
235C2×C4⋊C4—216no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—216no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—416no—
243C2×C4⋊C4—416no—
244C2×C22⋊C4—232no—
245C2×C22⋊C4—264no—
246C2×C22⋊C4—216no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—216no—
249C2×C22⋊C4—216no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—216no—
255C2×C22⋊C4—216no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—432no—
258C2×C22⋊C4—416no—
259C2×C42—264no—
260C2×C42—216no—
261C2×C42—216no—
262C2×C42C2×D4132yes—
263#95—28no—
264#95—28no—
265#95—216no—
266#95—216no—
267#95—216no—
268#95—216no—
269#95—28no—
270#95—216no—
271#95—216no—
272#95—28no—
273#95—48no—
274#95—48no—
275C24—28no—
276C24—28no—
277C24—28no—
278C24—216no—
279C24—216no—
280C24—28no—
281C24—28no—
282C24—28no—
283C24—216no—
284C24—216no—
285C24—216no—
286C24—216no—
287C24—24no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—24no—
295C24—22no—
296C24—48no—
297C24—48no—
298C24—48no—
299C24—48no—
300C24—48no—
301C24—44no—
302C24—44no—
303C24—44no—
304C24—44no—
305C24—48no—
306C24#123116yes—
307C24C22×D414yes—
308C24C22×D412yes—
309C24C22×D414yes—
310C24C2×#4118yes—
311C24C2×#4114yes—
312C24C2×#4112yes—
313C24C2×#4112yes—
314C24C2×#4112yes—
315C24C2×#4114yes—
316C24C2×#4112yes—
317C24H(5,𝔽2)12yes—
318C242−1+412yes—
319C242−1+414yes—
320C24C2511yescharacteristic
321C22×C4—2256no—
322C22×C4—264no—
323C22×C4—264no—
324C22×C4—28no—
325C22×C4—232no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—28no—
329C22×C4—216no—
330C22×C4—216no—
331C22×C4—216no—
332C22×C4—216no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—232no—
339C22×C4—232no—
340C22×C4—216no—
341C22×C4—216no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—216no—
345C22×C4—216no—
346C22×C4—216no—
347C22×C4—216no—
348C22×C4—216no—
349C22×C4—216no—
350C22×C4—216no—
351C22×C4—216no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—44no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4#120116yes—
436C22×C4#123116yes—
437C22×C4C22×D418yes—
438C22×C4C22×D418yes—
439C22×C4C22×D414yes—
440C22×C4C22×D418yes—
441C22×C4C22×D414yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4118yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4114yes—
447C4⋊C4—264no—
448C4⋊C4—232no—
449C4⋊C4—232no—
450C4⋊C4—416no—
451C4⋊C4—416no—
452C22⋊C4—232no—
453C22⋊C4—264no—
454C22⋊C4—432no—
455C22⋊C4—416no—
456C42—264no—
457C23—232no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—28no—
462C23—28no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—232no—
471C23—232no—
472C23—432no—
473C23—432no—
474C23—432no—
475C23—432no—
476C23—416no—
477C23—416no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—42no—
488C23—416no—
489C23—416no—
490C23C2×#12014yes—
491C23C2×#12318yes—
492C23C2×#12314yes—
493C23C2×#12314yes—
494C23#52518yes—
495C23#52518yes—
496C23#52518yes—
497C23#52514yes—
498C23#53218yes—
499C23#53214yes—
500C23#53618yes—
501C23#53614yes—
502C23#53718yes—
503C23#53714yes—
504C23#55118yes—
505C23#55218yes—
506C23#55218yes—
507C23#55218yes—
508C23#55218yes—
509C23#55214yes—
510C23#55314yes—
511C23#55514yes—
512C23#55614yes—
513C23#55618yes—
514C23C23×D411yescharacteristic
515C23C23×D412yes—
516C23C22×#4112yes—
517C23C22×#4111yescharacteristic
518C23C22×#4111yescharacteristic
519C23C22×#4111yescharacteristic
520C23C22×#4112yes—
521C23C22×#4111yescharacteristic
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×2−1+411yescharacteristic
524C23C2×2−1+412yes—
525C2×C4—2128no—
526C2×C4—232no—
527C2×C4—232no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—432no—
551C2×C4—432no—
552C2×C4—432no—
553C2×C4—416no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—416no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—88no—
592C2×C4—816no—
593C2×C4—88no—
594C2×C4C2×#12018yes—
595C2×C4C2×#12318yes—
596C22—28no—
597C22—28no—
598C22—28no—
599C22—464no—
600C22—48no—
601C22—48no—
602C22—44no—
603C22—48no—
604C22—48no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22#238818yes—
609C22#248418yes—
610C22#248518yes—
611C22#249818yes—
612C22#249918yes—
613C22#2599116yes—
614C22#2601116yes—
615C22#2609116yes—
616C22#2611116yes—
617C22#261618yes—
618C22#284918yes—
619C22C22×#12011yescharacteristic
620C22C22×#12311yescharacteristic
621C22C22×#12311yescharacteristic
622C22C2×#52512yes—
623C22C2×#52512yes—
624C22C2×#52512yes—
625C22C2×#52511yescharacteristic
626C22C2×#53212yes—
627C22C2×#53211yescharacteristic
628C22C2×#53612yes—
629C22C2×#53611yescharacteristic
630C22C2×#53712yes—
631C22C2×#53711yescharacteristic
632C22C2×#55112yes—
633C22C2×#55212yes—
634C22C2×#55212yes—
635C22C2×#55212yes—
636C22C2×#55212yes—
637C22C2×#55211yescharacteristic
638C22C2×#55311yescharacteristic
639C22C2×#55511yescharacteristic
640C22C2×#55612yes—
641C22C2×#55611yescharacteristic
642C4—28no—
643C4—28no—
644C4—48no—
645C4—48no—
646C4—44no—
647C4—48no—
648C4—44no—
649C4—82no—
650C4—84no—
651C4—82no—
652C2—48no—
653C2#15114116yesdirect product factor
654C2C2×#238811yescharacteristic
655C2C2×#261611yescharacteristic
656C2C2×#248511yescharacteristic
657C2C2×#249911yescharacteristic
658C2C2×#249811yescharacteristic
659C2C2×#248411yescharacteristic
660C2C2×#260112yes—
661C2C2×#259912yes—
662C2C2×#260912yes—
663C2C2×#261112yes—
664C2C2×#284911yescharacteristic
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×#387C2semidirect
4. #15114C2direct (N × H)
5. C2×#1444C2semidirect
6. C2×#1442C2semidirect
7. C2×#1441C2semidirect
8. C2×#1441C2semidirect
11. C2×#1434C2semidirect
28. C2×#387C22semidirect
43. #1444C22semidirect
44. #1442C22semidirect
45. #1441C22semidirect
46. #1441C22semidirect
49. #1434C22semidirect
653. C2#15114direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#1172
C22×#3872
C22×#3852
#151142
C2×#14442
C2×#14422
C2×#14412
C2×#14402
C2×#14392
C2×#14342
C2×C22⋊#952