Finite Group Databasebeta

← C2×#15429 #608967

Subgroups — C2×#15429 #608967

Order 512.

7202 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#396C212yesmaximal, complemented
2C22×#386C212yesmaximal
3C22×#385C212yesmaximal, complemented
4C22×#380C212yesmaximal, complemented
5#15429C2116yesmaximal, direct product factor
6C2×#1444C211yesmaximal, characteristic, complemented
7C2×#1442C211yesmaximal, characteristic, complemented
8C2×#1441C212yesmaximal, complemented
9C2×#1439C212yesmaximal, complemented
10C2×#1438C211yesmaximal, characteristic
11C24×D4C2211yescharacteristic
12C25×C4C2212yes—
13C23×C4⋊C4C2212yes—
14C23×C4⋊C4C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2212yes—
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2212yes—
23C2×#396—216no—
24C2×#396C22116yescomplemented
25C2×#386—216no—
26C2×#386C22116yes—
27C2×#385—216no—
28C2×#385C22116yescomplemented
29C2×#380—216no—
30C2×#380C22116yescomplemented
31C22×#95C2212yes—
32C22×#95C2212yes—
33C22×#95C2212yes—
34C22×#95C2212yes—
35C22×#95C2212yes—
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38C22×#95C2211yescharacteristic
39C22×#95C2211yescharacteristic
40#1444C2218yescomplemented
41#1442C2218yescomplemented
42#1441C22116yescomplemented
43#1439C22116yescomplemented
44#1438C2218yes—
45C26C2312yes—
46C23×D4—28no—
47C23×D4—28no—
48C23×D4—42no—
49C23×D4—42no—
50C23×D4Q818yes—
51C23×D4C2314yes—
52C24×C4—28no—
53C24×C4—28no—
54C24×C4—28no—
55C24×C4—28no—
56C24×C4—28no—
57C24×C4—44no—
58C24×C4—44no—
59C24×C4C2312yes—
60C24×C4C2312yes—
61C24×C4C2312yes—
62C24×C4C2311yescharacteristic
63C24×C4C2312yes—
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C24×C4C2318yes—
67C24×C4C2312yes—
68C22×C4⋊C4—28no—
69C22×C4⋊C4—28no—
70C22×C4⋊C4—28no—
71C22×C4⋊C4—28no—
72C22×C4⋊C4—28no—
73C22×C4⋊C4—28no—
74C22×C4⋊C4—28no—
75C22×C4⋊C4—28no—
76C22×C4⋊C4—44no—
77C22×C4⋊C4—44no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4—44no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4C2318yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2318yes—
84C22×C4⋊C4C2318yes—
85C22×C4⋊C4C2314yes—
86C22×C22⋊C4—28no—
87C22×C22⋊C4—28no—
88C22×C22⋊C4—28no—
89C22×C22⋊C4—28no—
90C22×C22⋊C4—28no—
91C22×C22⋊C4—28no—
92C22×C22⋊C4—28no—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—28no—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—44no—
97C22×C22⋊C4—44no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4—44no—
100C22×C22⋊C4D4116yes—
101C22×C22⋊C4C2318yes—
102C22×C22⋊C4C2318yes—
103C22×C22⋊C4C2318yes—
104C22×C22⋊C4C2318yes—
105C22×C22⋊C4C2318yes—
106#396—264no—
107#386—264no—
108#385—264no—
109#380—264no—
110C2×#95—28no—
111C2×#95—28no—
112C2×#95—28no—
113C2×#95—28no—
114C2×#95—28no—
115C2×#95—42no—
116C2×#95—42no—
117C2×#95D418yescomplemented
118C2×#95D418yescomplemented
119C2×#95C2318yes—
120C2×#95C2318yes—
121C2×#95C2318yes—
122C2×#95C2318yes—
123C2×#95C2318yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C25—24no—
129C25—24no—
130C25—24no—
131C25—24no—
132C25—24no—
133C25—24no—
134C25—24no—
135C25—24no—
136C25—24no—
137C25—24no—
138C25—24no—
139C25—24no—
140C25C2×D418yes—
141C25C2×Q818yes—
142C25#4118yes—
143C25C2414yes—
144C25C2411yescharacteristic
145C22×D4—264no—
146C22×D4—216no—
147C22×D4—216no—
148C22×D4—416no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4—416no—
152C22×D4—432no—
153C22×D4—416no—
154C22×D4C2×Q8116yes—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—264no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—216no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—216no—
186C23×C4—216no—
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×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—216no—
202C23×C4—216no—
203C23×C4—48no—
204C23×C4—48no—
205C23×C4—432no—
206C23×C4—432no—
207C23×C4—432no—
208C23×C4—432no—
209C23×C4—432no—
210C23×C4—432no—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×D418yes—
214C23×C4C2×D418yes—
215C23×C4C2×D418yes—
216C23×C4C2×D418yes—
217C23×C4C2×Q814yes—
218C23×C4#4118yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C2×C4⋊C4—216no—
228C2×C4⋊C4—216no—
229C2×C4⋊C4—216no—
230C2×C4⋊C4—216no—
231C2×C4⋊C4—216no—
232C2×C4⋊C4—216no—
233C2×C4⋊C4—216no—
234C2×C4⋊C4—216no—
235C2×C4⋊C4—48no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—48no—
240C2×C4⋊C4—432no—
241C2×C4⋊C4—432no—
242C2×C4⋊C4—416no—
243C2×C4⋊C4—416no—
244C2×C4⋊C4—432no—
245C2×C22⋊C4—264no—
246C2×C22⋊C4—216no—
247C2×C22⋊C4—216no—
248C2×C22⋊C4—216no—
249C2×C22⋊C4—216no—
250C2×C22⋊C4—216no—
251C2×C22⋊C4—216no—
252C2×C22⋊C4—216no—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—216no—
255C2×C22⋊C4—216no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—48no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—432no—
261C2×C22⋊C4—432no—
262C2×C22⋊C4—432no—
263C2×C22⋊C4—432no—
264C2×C22⋊C4C2×D4132yes—
265#95—216no—
266#95—216no—
267#95—216no—
268#95—216no—
269#95—216no—
270#95—44no—
271#95—44no—
272#95C2×D4116yescomplemented
273#95C2×D4116yescomplemented
274C24—216no—
275C24—216no—
276C24—216no—
277C24—216no—
278C24—216no—
279C24—216no—
280C24—216no—
281C24—216no—
282C24—216no—
283C24—216no—
284C24—216no—
285C24—216no—
286C24—216no—
287C24—216no—
288C24—216no—
289C24—216no—
290C24—216no—
291C24—216no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—48no—
305C24—48no—
306C24—48no—
307C24—48no—
308C24—48no—
309C24—48no—
310C24—48no—
311C24—48no—
312C24—48no—
313C24—48no—
314C24—48no—
315C24—48no—
316C24—48no—
317C24—48no—
318C24—48no—
319C24—48no—
320C24C22⋊Q8132yes—
321C24C22×D418yes—
322C24C22×D412yes—
323C24C22×D414yes—
324C24C22×D412yes—
325C24C22×Q818yes—
326C24C22×Q812yes—
327C24C2×#4118yes—
328C24C2×#4114yes—
329C24C2×#4114yes—
330C24H(5,𝔽2)12yes—
331C24H(5,𝔽2)14yes—
332C24H(5,𝔽2)12yes—
333C242−1+412yes—
334C242−1+412yes—
335C24C2511yescharacteristic
336C2×D4—264no—
337C2×D4—416no—
338C2×D4—4128no—
339C2×D4—416no—
340C2×D4—432no—
341C2×D4—416no—
342C22×C4—216no—
343C22×C4—216no—
344C22×C4—216no—
345C22×C4—216no—
346C22×C4—216no—
347C22×C4—216no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—264no—
353C22×C4—216no—
354C22×C4—216no—
355C22×C4—216no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—216no—
360C22×C4—216no—
361C22×C4—216no—
362C22×C4—216no—
363C22×C4—216no—
364C22×C4—216no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
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—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
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—432no—
406C22×C4—432no—
407C22×C4—4256no—
408C22×C4—432no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—432no—
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#121132yes—
442C22×C4C22×D414yes—
443C22×C4C22×D418yes—
444C22×C4C22×D418yes—
445C22×C4C22×D414yes—
446C22×C4C22×D418yes—
447C22×C4C22×D418yes—
448C22×C4C22×Q814yes—
449C22×C4C2×#4118yes—
450C4⋊C4—432no—
451C4⋊C4—432no—
452C4⋊C4—416no—
453C4⋊C4—416no—
454C4⋊C4—432no—
455C22⋊C4—264no—
456C22⋊C4—432no—
457C22⋊C4—432no—
458C22⋊C4—432no—
459C22⋊C4—432no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—264no—
464C23—28no—
465C23—28no—
466C23—28no—
467C23—264no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—264no—
481C23—432no—
482C23—44no—
483C23—44no—
484C23—432no—
485C23—432no—
486C23—432no—
487C23—432no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—432no—
494C23—44no—
495C23—432no—
496C23—44no—
497C23—44no—
498C23—432no—
499C23—432no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—432no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—432no—
508C23—432no—
509C23C2×#12118yes—
510C23C2×C22⋊Q8116yes—
511C23C2×C22⋊Q818yes—
512C23#52518yes—
513C23#53714yes—
514C23#53718yes—
515C23#53714yes—
516C23#53818yes—
517C23#53918yes—
518C23#54314yes—
519C23D4218yes—
520C23D4214yes—
521C23D4214yes—
522C23#54618yes—
523C23#54618yes—
524C23#54718yes—
525C23D4×Q814yes—
526C23D4×Q814yes—
527C23#55114yes—
528C23#55218yes—
529C23#55418yes—
530C23#55514yes—
531C23#56014yes—
532C23#56314yes—
533C23C23×D411yescharacteristic
534C23C23×D412yes—
535C23C23×D411yescharacteristic
536C23C23×Q811yescharacteristic
537C23C22×#4112yes—
538C23C22×#4112yes—
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×H(5,𝔽2)12yes—
541C23C2×H(5,𝔽2)11yescharacteristic
542C23C2×2−1+411yescharacteristic
543C23C2×2−1+411yescharacteristic
544D4—464no—
545C2×C4—28no—
546C2×C4—264no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—4128no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—416no—
568C2×C4—432no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—432no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—816no—
602C2×C4—816no—
603C2×C4C2×#121116yes—
604C22—216no—
605C22—216no—
606C22—216no—
607C22—48no—
608C22—48no—
609C22—4128no—
610C22—48no—
611C22—48no—
612C22—48no—
613C22—48no—
614C22—48no—
615C22—48no—
616C22—48no—
617C22—48no—
618C22—48no—
619C22—48no—
620C22#2417116yes—
621C22#2424116yes—
622C22#243718yes—
623C22#2449116yes—
624C22#253718yes—
625C22#260918yes—
626C22#267518yes—
627C22#268318yes—
628C22#269318yes—
629C22#2695116yes—
630C22#278418yes—
631C22C22×#12112yes—
632C22C22×C22⋊Q812yes—
633C22C2×#52512yes—
634C22C2×#53711yescharacteristic
635C22C2×#53711yescharacteristic
636C22C2×#53712yes—
637C22C2×#53812yes—
638C22C2×#53912yes—
639C22C2×#54311yescharacteristic
640C22C2×D4211yescharacteristic
641C22C2×D4211yescharacteristic
642C22C2×D4212yes—
643C22C2×#54612yes—
644C22C2×#54612yes—
645C22C2×#54712yes—
646C22C2×D4×Q811yescharacteristic
647C22C2×D4×Q811yescharacteristic
648C22C2×#55111yescharacteristic
649C22C2×#55212yes—
650C22C2×#55412yes—
651C22C2×#55511yescharacteristic
652C22C2×#56011yescharacteristic
653C22C2×#56311yescharacteristic
654C4—216no—
655C4—44no—
656C4—48no—
657C4—48no—
658C4—44no—
659C4—44no—
660C4—84no—
661C4—84no—
662C2—416no—
663C2#15429116yesdirect product factor
664C2C2×#243711yescharacteristic
665C2C2×#242412yes—
666C2C2×#244912yes—
667C2C2×#241712yes—
668C2C2×#260911yescharacteristic
669C2C2×#253711yescharacteristic
670C2C2×#267511yescharacteristic
671C2C2×#268311yescharacteristic
672C2C2×#269311yescharacteristic
673C2C2×#269512yes—
674C2C2×#278411yescharacteristic
subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC24×D4C22
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×#396C2semidirect
3. C22×#385C2semidirect
4. C22×#380C2semidirect
5. #15429C2direct (N × H)
6. C2×#1444C2semidirect
7. C2×#1442C2semidirect
8. C2×#1441C2semidirect
9. C2×#1439C2semidirect
24. C2×#396C22semidirect
28. C2×#385C22semidirect
30. C2×#380C22semidirect
40. #1444C22semidirect
41. #1442C22semidirect
42. #1441C22semidirect
43. #1439C22semidirect
117. C2×#95D4semidirect
118. C2×#95D4semidirect
272. #95C2×D4semidirect
273. #95C2×D4semidirect
663. C2#15429direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3962
C22×#3862
C22×#3852
C22×#3802
#154292
C2×#14442
C2×#14422
C2×#14412
C2×#14392
C2×#14382