Finite Group Databasebeta

← C4×#2622 #515190

Subgroups — C4×#2622 #515190

Order 512.

4532 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2622C211yesmaximal, characteristic
2C2×C4×#120C211yesmaximal, characteristic
3C22×C4×C22⋊C4C211yesmaximal, characteristic, complemented
4#14341C211yesmaximal, characteristic, complemented
5#14304C214yesmaximal, complemented
6#14303C214yesmaximal, complemented
7#14265C212yesmaximal, complemented
8#14262C211yesmaximal, characteristic, complemented
9#14259C212yesmaximal, complemented
10#14255C211yesmaximal, characteristic
11C4×#397C214yesmaximal, complemented
12C4×#394C214yesmaximal, complemented
13C4×#379C211yesmaximal, characteristic, complemented
14C4×#385C212yesmaximal, complemented
15C4×#386C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#120C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C42C2212yes—
23#2622C4116yesdirect product factor
24C2×#397C2214yes—
25C2×#394C2214yes—
26C4×#120—28no—
27C2×#386C2212yes—
28C2×#386C2211yescharacteristic
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#385C2212yes—
32C2×#385C2211yescharacteristic
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#379C2211yescharacteristic
36C2×#379C2211yescharacteristic
37C2×#379C2211yescharacteristic
38C2×C4×C4⋊C4C2214yescomplemented
39C2×C4×C22⋊C4—28no—
40C2×C4×C22⋊C4C2214yes—
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2214yes—
45C2×C4×C22⋊C4C2214yes—
46C2×C4×C22⋊C4C2214yescomplemented
47C22×#95C2212yes—
48#1445C2214yescomplemented
49#1443C2214yescomplemented
50#1443C2214yescomplemented
51#1442C2214yescomplemented
52#1442C2214yescomplemented
53#1441C2214yescomplemented
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1440C2214yescomplemented
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yescomplemented
61#1438C2214yes—
62#1437C2214yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2214yes—
72C4×#95C2214yescomplemented
73C4×#95C2214yescomplemented
74C26C2311yescharacteristic
75C23×D4C2311yescharacteristic
76C23×D4C2312yes—
77C24×C4—28no—
78C24×C4—48no—
79C24×C4D418yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2312yes—
87C2×#120—24no—
88C2×#120—24no—
89C2×#120C2×C418yes—
90C2×C4×D4—28no—
91C2×C4×D4—24no—
92C2×C4×D4—28no—
93C2×C4×D4—48no—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—28no—
102C22×C22⋊C4C2×C418yescomplemented
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C42—28no—
120C22×C42—48no—
121C22×C42C2314yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2314yes—
126#397C2×C4132yescomplemented
127#394C2×C4132yescomplemented
128#386—24no—
129#386—28no—
130#386—28no—
131#386C2×C4116yescomplemented
132#385—24no—
133#385—28no—
134#385—28no—
135#385C2×C4116yescomplemented
136#379—24no—
137#379—24no—
138#379C2×C418yescomplemented
139C4×C4⋊C4—48no—
140C4×C22⋊C4—216no—
141C4×C22⋊C4—216no—
142C4×C22⋊C4—28no—
143C4×C22⋊C4—24no—
144C4×C22⋊C4—28no—
145C4×C22⋊C4—48no—
146C4×C22⋊C4—48no—
147C2×#95—28no—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25—24no—
169C25—24no—
170C25—44no—
171C25—44no—
172C25C22×C414yes—
173C25C2×D414yes—
174C25#4114yes—
175C25C2411yescharacteristic
176C25C2412yes—
177C25C2411yescharacteristic
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—22no—
183C22×D4—24no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4C22×C414yes—
187C22×D4C22×C418yes—
188C23×C4—28no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—232no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—432no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—464no—
213C23×C4C22×C418yes—
214C23×C4C2×D418yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D418yes—
218C23×C4#4118yes—
219C23×C4#4118yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232#120—232no—
233C4×D4—432no—
234C4×D4—416no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4C22×C4116yescomplemented
245C2×C22⋊C4—232no—
246C2×C22⋊C4—232no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4C22×C4116yes—
284C2×C22⋊C4C22×C414yes—
285C2×C22⋊C4C22×C418yes—
286C2×C22⋊C4C22×C418yes—
287C2×C22⋊C4C22×C4116yes—
288C2×C22⋊C4C22×C4116yescomplemented
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—28no—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—28no—
297C2×C42—28no—
298C2×C42—28no—
299C2×C42—28no—
300C2×C42—44no—
301C2×C42—44no—
302C2×C42—432no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95C22×C4116yescomplemented
322#95C22×C4116yescomplemented
323C24—24no—
324C24—24no—
325C24—216no—
326C24—28no—
327C24—28no—
328C24—216no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—21no—
341C24—24no—
342C24—24no—
343C24—48no—
344C24—416no—
345C24—48no—
346C24—416no—
347C24—416no—
348C24—416no—
349C24—416no—
350C24—416no—
351C24C4×D4116yes—
352C24C23×C412yes—
353C24C23×C414yes—
354C24C23×C412yes—
355C24C22×D414yes—
356C24C22×D412yes—
357C24C2×#4114yes—
358C24C2×#4112yes—
359C24C2×#4114yes—
360C24C2×#4114yes—
361C24C2511yescharacteristic
362C2×D4—216no—
363C2×D4—28no—
364C2×D4—216no—
365C2×D4—416no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—416no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—48no—
372C22×C4—24no—
373C22×C4—216no—
374C22×C4—232no—
375C22×C4—216no—
376C22×C4—28no—
377C22×C4—216no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
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—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—216no—
411C22×C4—216no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
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—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—4128no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—44no—
448C22×C4—48no—
449C22×C4—416no—
450C22×C4#123116yes—
451C22×C4C23×C412yes—
452C22×C4C23×C418yes—
453C22×C4C23×C418yes—
454C22×C4C23×C414yes—
455C22×C4C22×D414yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C2×#4118yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4114yes—
461C22×C4H(5,𝔽2)18yes—
462C4⋊C4—416no—
463C4⋊C4—416no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C22⋊C4—232no—
467C22⋊C4—232no—
468C22⋊C4—216no—
469C22⋊C4—28no—
470C22⋊C4—216no—
471C22⋊C4—416no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—416no—
477C22⋊C4—48no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—48no—
481C22⋊C4—44no—
482C42—416no—
483C42—44no—
484C42—416no—
485C42—48no—
486C42—416no—
487C23—28no—
488C23—24no—
489C23—24no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—24no—
496C23—24no—
497C23—28no—
498C23—28no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—28no—
508C23—28no—
509C23—232no—
510C23—22no—
511C23—24no—
512C23—28no—
513C23—432no—
514C23—416no—
515C23—432no—
516C23—464no—
517C23—464no—
518C23—416no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—41no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23C2×C4×D418yes—
538C23C2×C4×D414yes—
539C23C4×#4118yes—
540C23C2×#12314yes—
541C23#52514yes—
542C23#52918yes—
543C23C24×C411yescharacteristic
544C23C23×D411yescharacteristic
545C23C22×#4111yescharacteristic
546C23C22×#4112yes—
547C23C22×#4112yes—
548C23C2×H(5,𝔽2)14yes—
549C23#58514yes—
550D4—432no—
551D4—416no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—232no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—432no—
572C2×C4—432no—
573C2×C4—432no—
574C2×C4—416no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
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—432no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
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—44no—
614C2×C4—88no—
615C2×C4C2×#12314yes—
616C2×C4C2×#12314yes—
617C2×C4#53418yes—
618C2×C4#538116yes—
619C22—28no—
620C22—216no—
621C22—28no—
622C22—216no—
623C22—24no—
624C22—24no—
625C22—28no—
626C22—28no—
627C22—28no—
628C22—464no—
629C22—44no—
630C22—44no—
631C22—48no—
632C22—48no—
633C22—48no—
634C22—48no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—48no—
639C22—42no—
640C22—82no—
641C22—84no—
642C22—81no—
643C22—81no—
644C22C4×#12318yes—
645C22C22×C4×D411yescharacteristic
646C22C2×C4×#4112yes—
647C22C4×H(5,𝔽2)14yes—
648C22C22×#12311yescharacteristic
649C22C2×#52511yescharacteristic
650C22C2×#52912yes—
651C22C2×#53412yes—
652C22#344912yes—
653C22C2×#53814yes—
654C22#345514yes—
655C22#345914yes—
656C22#345914yes—
657C22#346014yes—
658C4—24no—
659C4—48no—
660C4—48no—
661C4—44no—
662C4—48no—
663C4—42no—
664C4—44no—
665C4—82no—
666C4—84no—
667C4—81no—
668C4—81no—
669C4#262218yesdirect product factor
670C2—28no—
671C2—48no—
672C2—82no—
673C2C2×C4×#12311yescharacteristic
674C2C4×#53412yes—
675C2C4×#53814yes—
676C2C2×#262211yescharacteristic
677C2#3456212yes—
678C2#3457014yes—
679C2#3463511yescharacteristic
subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C22×C4×C22⋊C4C2semidirect
4. #14341C2semidirect
5. #14304C2semidirect
6. #14303C2semidirect
7. #14265C2semidirect
8. #14262C2semidirect
9. #14259C2semidirect
11. C4×#397C2semidirect
12. C4×#394C2semidirect
13. C4×#379C2semidirect
14. C4×#385C2semidirect
15. C4×#386C2semidirect
23. #2622C4direct (N × H)
38. C2×C4×C4⋊C4C22semidirect
46. C2×C4×C22⋊C4C22semidirect
48. #1445C22semidirect
49. #1443C22semidirect
50. #1443C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
57. #1440C22semidirect
60. #1439C22semidirect
62. #1437C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
72. C4×#95C22semidirect
73. C4×#95C22semidirect
102. C22×C22⋊C4C2×C4semidirect
126. #397C2×C4semidirect
127. #394C2×C4semidirect
131. #386C2×C4semidirect
135. #385C2×C4semidirect
138. #379C2×C4semidirect
244. C2×C4⋊C4C22×C4semidirect
288. C2×C22⋊C4C22×C4semidirect
321. #95C22×C4semidirect
322. #95C22×C4semidirect
669. C4#2622direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26222
C2×C4×#1202
C22×C4×C22⋊C42
#143412
#143042
#143032
#142652
#142622
#142592
#142552
C4×#3972
C4×#3942
C4×#3792
C4×#3852
C4×#3862