Finite Group Databasebeta

← C2×#15671 #609307

Subgroups — C2×#15671 #609307

Order 512.

6570 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C22×#395C211yesmaximal, characteristic, complemented
3C22×#393C211yesmaximal, characteristic
4C22×#391C211yesmaximal, characteristic, complemented
5#15671C2116yesmaximal, direct product factor
6C2×#1446C212yesmaximal, complemented
7C2×#1445C211yesmaximal, characteristic, complemented
8C2×#1441C212yesmaximal, complemented
9C2×#1439C212yesmaximal
10C2×#1437C211yesmaximal, characteristic
11C2×#1436C211yesmaximal, characteristic, complemented
12C2×C22⋊#95C212yesmaximal
13C24×Q8C2211yescharacteristic
14C24×Q8C2211yescharacteristic
15C25×C4C2211yescharacteristic
16C22×C22⋊Q8—28no—
17C22×C22⋊Q8—28no—
18C22×C22⋊Q8—28no—
19C22×C22⋊Q8C2218yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25C23×C22⋊C4C2212yes—
26C23×C22⋊C4C2211yescharacteristic
27C23×C22⋊C4C2212yes—
28C23×C42C2211yescharacteristic
29C2×#395—28no—
30C2×#395C2218yescomplemented
31C2×#393—28no—
32C2×#393C2218yes—
33C2×#391—28no—
34C2×#391C2218yescomplemented
35C22×#95C2212yes—
36C22×#95C2212yes—
37C22×#95C2212yes—
38C22×#95C2211yescharacteristic
39C22×#95C2211yescharacteristic
40C22×#95C2212yes—
41C22×#95C2212yes—
42C22×#95C2212yes—
43C22×#95C2212yes—
44C22×#95C2212yes—
45C22×#95C2212yes—
46#1446C22116yescomplemented
47#1445C2218yescomplemented
48#1441C22116yescomplemented
49#1439C22116yes—
50#1437C2218yes—
51#1436C2218yescomplemented
52C22⋊#95C22116yes—
53C26C2311yescharacteristic
54C23×Q8—24no—
55C23×Q8—24no—
56C23×Q8—24no—
57C23×Q8—24no—
58C23×Q8—24no—
59C23×Q8—24no—
60C23×Q8—28no—
61C23×Q8—28no—
62C23×Q8—44no—
63C23×Q8—44no—
64C23×Q8C2314yes—
65C23×Q8C2314yes—
66C24×C4—24no—
67C24×C4—24no—
68C24×C4—24no—
69C24×C4—28no—
70C24×C4—44no—
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2312yes—
78C24×C4C2312yes—
79C24×C4C2314yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C2×C22⋊Q8—2128no—
83C2×C22⋊Q8—232no—
84C2×C22⋊Q8—232no—
85C2×C22⋊Q8—232no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4D418yes—
97C22×C4⋊C4D418yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2318yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—28no—
109C22×C22⋊C4—44no—
110C22×C22⋊C4—44no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4D418yes—
113C22×C22⋊C4Q8116yescomplemented
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2318yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2318yes—
118C22×C42—24no—
119C22×C42—44no—
120C22×C42C2314yes—
121#395—232no—
122#393—232no—
123#391—232no—
124C2×#95—28no—
125C2×#95—24no—
126C2×#95—28no—
127C2×#95—24no—
128C2×#95—28no—
129C2×#95—28no—
130C2×#95—28no—
131C2×#95—44no—
132C2×#95—44no—
133C2×#95—44no—
134C2×#95—44no—
135C2×#95C2314yes—
136C2×#95C2318yes—
137C2×#95C2318yes—
138C2×#95C2314yes—
139C2×#95C2318yes—
140C2×#95C2318yes—
141C2×#95C2318yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C2×#95C2318yes—
146C25—22no—
147C25—22no—
148C25—24no—
149C25—24no—
150C25—24no—
151C25—24no—
152C25—24no—
153C25C2×D414yes—
154C25C2×Q818yes—
155C25C2412yes—
156C25C2411yescharacteristic
157C22×Q8—232no—
158C22×Q8—232no—
159C22×Q8—28no—
160C22×Q8—28no—
161C22×Q8—28no—
162C22×Q8—28no—
163C22×Q8—28no—
164C22×Q8—28no—
165C22×Q8—216no—
166C22×Q8—216no—
167C22×Q8—48no—
168C22×Q8—48no—
169C22×Q8—432no—
170C22×Q8—432no—
171C22×Q8—432no—
172C22×Q8—432no—
173C22×Q8—416no—
174C22×Q8—416no—
175C22×Q8—416no—
176C22×Q8—416no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—232no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—28no—
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—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—28no—
219C23×C4—28no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—216no—
228C23×C4—48no—
229C23×C4—432no—
230C23×C4—432no—
231C23×C4—416no—
232C23×C4—416no—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4C2×Q818yes—
242C23×C4C2×Q818yes—
243C23×C4#4118yes—
244C23×C4#4118yes—
245C23×C4C2414yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2414yes—
251C23×C4C2414yes—
252C23×C4C2414yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C22⋊Q8—2256no—
256C2×C4⋊C4—232no—
257C2×C4⋊C4—232no—
258C2×C4⋊C4—232no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—216no—
267C2×C4⋊C4—216no—
268C2×C4⋊C4—48no—
269C2×C4⋊C4—432no—
270C2×C4⋊C4C2×D4116yes—
271C2×C4⋊C4C2×D4116yes—
272C2×C22⋊C4—232no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—232no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—216no—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—432no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—432no—
286C2×C22⋊C4C2×D4116yes—
287C2×C22⋊C4C2×Q8132yescomplemented
288C2×C42—28no—
289C2×C42—48no—
290C2×C42—416no—
291#95—216no—
292#95—216no—
293#95—216no—
294#95—28no—
295#95—28no—
296#95—216no—
297#95—216no—
298#95—48no—
299#95—48no—
300#95—48no—
301#95—48no—
302C24—28no—
303C24—216no—
304C24—28no—
305C24—216no—
306C24—216no—
307C24—216no—
308C24—216no—
309C24—216no—
310C24—216no—
311C24—216no—
312C24—22no—
313C24—22no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—44no—
320C24—44no—
321C24—44no—
322C24—44no—
323C24—48no—
324C24—48no—
325C24—48no—
326C24—48no—
327C24—48no—
328C24—48no—
329C24C4⋊Q8116yes—
330C24C22×D414yes—
331C24C22×D412yes—
332C24C22×D412yes—
333C24C22×D412yes—
334C24C22×Q818yes—
335C24C22×Q814yes—
336C24C2×#4114yes—
337C24C2×#4114yes—
338C24H(5,𝔽2)14yes—
339C24H(5,𝔽2)14yes—
340C242−1+414yes—
341C24C2511yescharacteristic
342C2×Q8—232no—
343C2×Q8—232no—
344C2×Q8—432no—
345C2×Q8—432no—
346C2×Q8—432no—
347C2×Q8—432no—
348C2×Q8—4128no—
349C2×Q8—416no—
350C2×Q8—416no—
351C2×Q8—4128no—
352C2×Q8—416no—
353C2×Q8—416no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—216no—
359C22×C4—216no—
360C22×C4—216no—
361C22×C4—216no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—232no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—216no—
373C22×C4—216no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
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—432no—
427C22×C4—432no—
428C22×C4—416no—
429C22×C4—4128no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
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—416no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4#120116yes—
463C22×C4#124116yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×Q818yes—
473C22×C4C22×Q818yes—
474C22×C4C2×#4118yes—
475C22×C4C2×#4118yes—
476C4⋊C4—232no—
477C4⋊C4—232no—
478C4⋊C4—232no—
479C4⋊C4—432no—
480C22⋊C4—232no—
481C22⋊C4—232no—
482C22⋊C4—432no—
483C22⋊C4—432no—
484C42—416no—
485C23—24no—
486C23—28no—
487C23—24no—
488C23—28no—
489C23—232no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—264no—
497C23—416no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—416no—
505C23—432no—
506C23—432no—
507C23—44no—
508C23—44no—
509C23—432no—
510C23—44no—
511C23—432no—
512C23—432no—
513C23—44no—
514C23C2×#12014yes—
515C23#52514yes—
516C23C2×#12414yes—
517C23C2×C4⋊Q818yes—
518C23C2×C4⋊Q814yes—
519C23#53318yes—
520C23#53514yes—
521C23#53514yes—
522C23#53614yes—
523C23#54218yes—
524C23#54218yes—
525C23#54318yes—
526C23#54618yes—
527C23#54618yes—
528C23#54618yes—
529C23#54618yes—
530C23D4×Q818yes—
531C23D4×Q818yes—
532C23#55418yes—
533C23#55418yes—
534C23#55618yes—
535C23#56218yes—
536C23C23×D411yescharacteristic
537C23C23×D411yescharacteristic
538C23C23×D411yescharacteristic
539C23C23×Q812yes—
540C23C22×#4112yes—
541C23C22×#4112yes—
542C23C2×H(5,𝔽2)12yes—
543C23C2×H(5,𝔽2)12yes—
544C23C2×2−1+412yes—
545Q8—464no—
546Q8—464no—
547C2×C4—232no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—232no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
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—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—464no—
575C2×C4—416no—
576C2×C4—432no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—416no—
592C2×C4—416no—
593C2×C4—416no—
594C2×C4—432no—
595C2×C4—432no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
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—44no—
613C2×C4—816no—
614C2×C4C2×#12018yes—
615C2×C4C2×#12418yes—
616C22—216no—
617C22—28no—
618C22—464no—
619C22—48no—
620C22—44no—
621C22—48no—
622C22—48no—
623C22—48no—
624C22—44no—
625C22—48no—
626C22#240418yes—
627C22#243418yes—
628C22#243518yes—
629C22#2465116yes—
630C22#2492116yes—
631C22#2549116yes—
632C22#2692116yes—
633C22#2692116yes—
634C22#2700116yes—
635C22C22×#12011yescharacteristic
636C22C2×#52511yescharacteristic
637C22C22×#12411yescharacteristic
638C22C22×C4⋊Q811yescharacteristic
639C22C2×#53312yes—
640C22C2×#53511yescharacteristic
641C22C2×#53511yescharacteristic
642C22C2×#53611yescharacteristic
643C22C2×#54212yes—
644C22C2×#54212yes—
645C22C2×#54312yes—
646C22C2×#54612yes—
647C22C2×#54612yes—
648C22C2×#54612yes—
649C22C2×#54612yes—
650C22C2×D4×Q812yes—
651C22C2×D4×Q812yes—
652C22C2×#55412yes—
653C22C2×#55412yes—
654C22C2×#55612yes—
655C22C2×#56212yes—
656C4—28no—
657C4—28no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—48no—
662C4—48no—
663C4—44no—
664C4—48no—
665C4—84no—
666C2—48no—
667C2#15671116yesdirect product factor
668C2C2×#243411yescharacteristic
669C2C2×#246512yes—
670C2C2×#243511yescharacteristic
671C2C2×#240411yescharacteristic
672C2C2×#249212yes—
673C2C2×#254912yes—
674C2C2×#269212yes—
675C2C2×#269212yes—
676C2C2×#270012yes—
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×#395C2semidirect
4. C22×#391C2semidirect
5. #15671C2direct (N × H)
6. C2×#1446C2semidirect
7. C2×#1445C2semidirect
8. C2×#1441C2semidirect
11. C2×#1436C2semidirect
30. C2×#395C22semidirect
34. C2×#391C22semidirect
46. #1446C22semidirect
47. #1445C22semidirect
48. #1441C22semidirect
51. #1436C22semidirect
113. C22×C22⋊C4Q8semidirect
287. C2×C22⋊C4C2×Q8semidirect
667. C2#15671direct (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×#3952
C22×#3932
C22×#3912
#156712
C2×#14462
C2×#14452
C2×#14412
C2×#14392
C2×#14372
C2×#14362
C2×C22⋊#952