Finite Group Databasebeta

← C2×#15430 #608968

Subgroups — C2×#15430 #608968

Order 512.

6922 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C22×#399C211yesmaximal, characteristic
3C22×#398C211yesmaximal, characteristic, complemented
4C22×#395C212yesmaximal, complemented
5C22×#385C212yesmaximal
6C22×#384C211yesmaximal, characteristic, complemented
7#15430C2116yesmaximal, direct product factor
8C2×#1445C211yesmaximal, characteristic, complemented
9C2×#1444C212yesmaximal, complemented
10C2×#1439C212yesmaximal
11C2×#1437C211yesmaximal, characteristic
12C2×#1436C211yesmaximal, characteristic, complemented
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⋊C4C2212yes—
22C23×C4⋊C4C2212yes—
23C23×C4⋊C4C2212yes—
24C23×C4⋊C4C2212yes—
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4C2211yescharacteristic
28C23×C22⋊C4C2212yes—
29C23×C22⋊C4C2211yescharacteristic
30C23×C22⋊C4C2212yes—
31C23×C42C2211yescharacteristic
32C2×#399—28no—
33C2×#399C2218yes—
34C2×#398—28no—
35C2×#398C2218yescomplemented
36C2×#395—216no—
37C2×#395C22116yescomplemented
38C2×#385—216no—
39C2×#385C22116yes—
40C2×#384—28no—
41C2×#384C2218yescomplemented
42C22×#95C2212yes—
43C22×#95C2212yes—
44C22×#95C2212yes—
45C22×#95C2212yes—
46C22×#95C2212yes—
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49C22×#95C2212yes—
50#1445C2218yescomplemented
51#1444C22116yescomplemented
52#1439C22116yes—
53#1437C2218yes—
54#1436C2218yescomplemented
55C26C2311yescharacteristic
56C23×Q8—24no—
57C23×Q8—24no—
58C23×Q8—24no—
59C23×Q8—28no—
60C23×Q8—44no—
61C23×Q8C2314yes—
62C24×C4—24no—
63C24×C4—24no—
64C24×C4—24no—
65C24×C4—28no—
66C24×C4—44no—
67C24×C4C2312yes—
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2312yes—
74C24×C4C2314yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C2×C22⋊Q8—2128no—
79C2×C22⋊Q8—232no—
80C2×C22⋊Q8—232no—
81C2×C22⋊Q8—232no—
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—28no—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—44no—
102C22×C4⋊C4—44no—
103C22×C4⋊C4—44no—
104C22×C4⋊C4D418yes—
105C22×C4⋊C4D418yes—
106C22×C4⋊C4C2318yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2318yes—
110C22×C4⋊C4C2318yes—
111C22×C4⋊C4C2318yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—24no—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—24no—
118C22×C22⋊C4—24no—
119C22×C22⋊C4—28no—
120C22×C22⋊C4—28no—
121C22×C22⋊C4—28no—
122C22×C22⋊C4—28no—
123C22×C22⋊C4—44no—
124C22×C22⋊C4D418yes—
125C22×C22⋊C4Q8116yescomplemented
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2318yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2318yes—
130C22×C42—24no—
131C22×C42—28no—
132C22×C42C2314yes—
133#399—232no—
134#398—232no—
135#395—264no—
136#385—264no—
137#384—232no—
138C2×#95—28no—
139C2×#95—28no—
140C2×#95—24no—
141C2×#95—28no—
142C2×#95—24no—
143C2×#95—28no—
144C2×#95—28no—
145C2×#95D4116yes—
146C2×#95C2318yes—
147C2×#95C2318yes—
148C2×#95C2314yes—
149C2×#95C2318yes—
150C2×#95C2314yes—
151C2×#95C2318yes—
152C2×#95C2318yes—
153C2×#95C2318yes—
154C25—22no—
155C25—22no—
156C25—24no—
157C25—24no—
158C25—24no—
159C25—24no—
160C25—24no—
161C25C2×D414yes—
162C25C2×Q818yes—
163C25C2412yes—
164C25C2411yescharacteristic
165C22×Q8—232no—
166C22×Q8—28no—
167C22×Q8—28no—
168C22×Q8—28no—
169C22×Q8—216no—
170C22×Q8—48no—
171C22×Q8—432no—
172C22×Q8—432no—
173C22×Q8—416no—
174C22×Q8—416no—
175C23×C4—24no—
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—232no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
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—28no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—216no—
224C23×C4—48no—
225C23×C4—432no—
226C23×C4—432no—
227C23×C4—416no—
228C23×C4—416no—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D418yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D418yes—
238C23×C4C2×D414yes—
239C23×C4C2×D418yes—
240C23×C4C2×Q818yes—
241C23×C4C2×Q818yes—
242C23×C4#4118yes—
243C23×C4C2414yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2414yes—
248C23×C4C2412yes—
249C23×C4C2414yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2414yes—
253C22⋊Q8—2256no—
254C2×C4⋊C4—232no—
255C2×C4⋊C4—232no—
256C2×C4⋊C4—232no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—216no—
265C2×C4⋊C4—216no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—216no—
268C2×C4⋊C4—216no—
269C2×C4⋊C4—216no—
270C2×C4⋊C4—216no—
271C2×C4⋊C4—216no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—216no—
274C2×C4⋊C4—216no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—48no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4—432no—
280C2×C4⋊C4—432no—
281C2×C4⋊C4—432no—
282C2×C4⋊C4—416no—
283C2×C4⋊C4—432no—
284C2×C4⋊C4C2×D4116yes—
285C2×C4⋊C4C2×D4116yes—
286C2×C22⋊C4—232no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—232no—
289C2×C22⋊C4—264no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—216no—
295C2×C22⋊C4—216no—
296C2×C22⋊C4—216no—
297C2×C22⋊C4—216no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4—432no—
300C2×C22⋊C4C2×D4116yes—
301C2×C22⋊C4C2×Q8132yescomplemented
302C2×C42—232no—
303C2×C42—28no—
304C2×C42—216no—
305#95—216no—
306#95—216no—
307#95—216no—
308#95—28no—
309#95—216no—
310#95—28no—
311#95—216no—
312#95C2×D4132yes—
313C24—28no—
314C24—216no—
315C24—28no—
316C24—216no—
317C24—216no—
318C24—216no—
319C24—216no—
320C24—216no—
321C24—216no—
322C24—216no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—44no—
331C24—44no—
332C24—44no—
333C24—44no—
334C24—48no—
335C24—48no—
336C24—48no—
337C24—48no—
338C24—48no—
339C24—48no—
340C24C4⋊Q8116yes—
341C24C22×D414yes—
342C24C22×D412yes—
343C24C22×D412yes—
344C24C22×D412yes—
345C24C22×D414yes—
346C24C22×Q818yes—
347C24C22×Q814yes—
348C24C2×#4114yes—
349C24H(5,𝔽2)14yes—
350C242−1+414yes—
351C242−1+414yes—
352C24C2511yescharacteristic
353C2×Q8—232no—
354C2×Q8—432no—
355C2×Q8—432no—
356C2×Q8—416no—
357C2×Q8—4128no—
358C2×Q8—416no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—216no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—216no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—232no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—216no—
388C22×C4—216no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—216no—
392C22×C4—216no—
393C22×C4—216no—
394C22×C4—216no—
395C22×C4—216no—
396C22×C4—216no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—432no—
433C22×C4—432no—
434C22×C4—416no—
435C22×C4—4128no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4#120116yes—
465C22×C4C22⋊Q8132yescomplemented
466C22×C4#127116yes—
467C22×C4C22×D418yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D414yes—
475C22×C4C22×D418yes—
476C22×C4C22×D418yes—
477C22×C4C22×D414yes—
478C22×C4C22×Q818yes—
479C22×C4C22×Q818yes—
480C22×C4C2×#4118yes—
481C4⋊C4—232no—
482C4⋊C4—232no—
483C4⋊C4—232no—
484C4⋊C4—432no—
485C4⋊C4—432no—
486C4⋊C4—432no—
487C4⋊C4—416no—
488C4⋊C4—432no—
489C22⋊C4—232no—
490C22⋊C4—232no—
491C22⋊C4—264no—
492C22⋊C4—432no—
493C42—232no—
494C23—24no—
495C23—28no—
496C23—24no—
497C23—28no—
498C23—232no—
499C23—28no—
500C23—264no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—416no—
507C23—42no—
508C23—42no—
509C23—432no—
510C23—42no—
511C23—44no—
512C23—42no—
513C23—432no—
514C23—44no—
515C23—416no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—432no—
520C23—432no—
521C23—44no—
522C23—432no—
523C23C2×#12014yes—
524C23C2×C22⋊Q818yes—
525C23#52514yes—
526C23C2×#12714yes—
527C23C2×C4⋊Q818yes—
528C23C2×C4⋊Q814yes—
529C23#53514yes—
530C23#53614yes—
531C23#53618yes—
532C23#53614yes—
533C23#53918yes—
534C23#54418yes—
535C23D4218yes—
536C23D4218yes—
537C23#54718yes—
538C23#54718yes—
539C23#54718yes—
540C23D4×Q818yes—
541C23D4×Q818yes—
542C23D4×Q818yes—
543C23#55418yes—
544C23#56318yes—
545C23C23×D411yescharacteristic
546C23C23×D411yescharacteristic
547C23C23×D411yescharacteristic
548C23C23×D412yes—
549C23C23×Q812yes—
550C23C22×#4112yes—
551C23C2×H(5,𝔽2)12yes—
552C23C2×2−1+412yes—
553C23C2×2−1+412yes—
554Q8—464no—
555C2×C4—232no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—232no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—264no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—464no—
587C2×C4—416no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—432no—
600C2×C4—44no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—416no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—432no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—816no—
621C2×C4C2×#12018yes—
622C2×C4C2×C22⋊Q8116yescomplemented
623C2×C4C2×#12718yes—
624C22—216no—
625C22—28no—
626C22—464no—
627C22—48no—
628C22—48no—
629C22—44no—
630C22—48no—
631C22—48no—
632C22—44no—
633C22—48no—
634C22#240618yes—
635C22#243318yes—
636C22#243418yes—
637C22#2440116yes—
638C22#2449116yes—
639C22#2542116yes—
640C22#2677116yes—
641C22#2680116yes—
642C22#2720116yes—
643C22C22×#12011yescharacteristic
644C22C22×C22⋊Q812yes—
645C22C2×#52511yescharacteristic
646C22C22×#12711yescharacteristic
647C22C22×C4⋊Q811yescharacteristic
648C22C2×#53511yescharacteristic
649C22C2×#53612yes—
650C22C2×#53611yescharacteristic
651C22C2×#53611yescharacteristic
652C22C2×#53912yes—
653C22C2×#54412yes—
654C22C2×D4212yes—
655C22C2×D4212yes—
656C22C2×#54712yes—
657C22C2×#54712yes—
658C22C2×#54712yes—
659C22C2×D4×Q812yes—
660C22C2×D4×Q812yes—
661C22C2×D4×Q812yes—
662C22C2×#55412yes—
663C22C2×#56312yes—
664C4—216no—
665C4—28no—
666C4—28no—
667C4—48no—
668C4—44no—
669C4—44no—
670C4—44no—
671C4—48no—
672C4—44no—
673C4—84no—
674C2—48no—
675C2#15430116yesdirect product factor
676C2C2×#244912yes—
677C2C2×#243311yescharacteristic
678C2C2×#244012yes—
679C2C2×#243411yescharacteristic
680C2C2×#240611yescharacteristic
681C2C2×#254212yes—
682C2C2×#267712yes—
683C2C2×#268012yes—
684C2C2×#272012yes—
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. C22×#398C2semidirect
4. C22×#395C2semidirect
6. C22×#384C2semidirect
7. #15430C2direct (N × H)
8. C2×#1445C2semidirect
9. C2×#1444C2semidirect
12. C2×#1436C2semidirect
35. C2×#398C22semidirect
37. C2×#395C22semidirect
41. C2×#384C22semidirect
50. #1445C22semidirect
51. #1444C22semidirect
54. #1436C22semidirect
125. C22×C22⋊C4Q8semidirect
301. C2×C22⋊C4C2×Q8semidirect
465. C22×C4C22⋊Q8semidirect
622. C2×C4C2×C22⋊Q8semidirect
675. C2#15430direct (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×#3992
C22×#3982
C22×#3952
C22×#3852
C22×#3842
#154302
C2×#14452
C2×#14442
C2×#14392
C2×#14372
C2×#14362