Finite Group Databasebeta

← #772880

Subgroups — #772880

Order 512.

3262 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C212yesmaximal, complemented
2C2×Q8×C22⋊C4C212yesmaximal
3#16305C212yesmaximal
4#15964C212yesmaximal, complemented
5#15511C212yesmaximal, complemented
6#15333C214yesmaximal, complemented
7#15135C211yesmaximal, characteristic
8#15131C212yesmaximal
9#14771C214yesmaximal
10#14213C212yesmaximal
11#13976C212yesmaximal, complemented
12#13976C212yesmaximal, complemented
13#13935C212yesmaximal, complemented
14#13922C211yesmaximal, characteristic
15C4×#393C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2212yes—
19C22×C4×Q8C2212yes—
20#2582—216no—
21C2×#399C2212yes—
22C2×#398C2214yes—
23C2×#398C2214yes—
24C2×#396C2214yes—
25C2×#393C2211yescharacteristic
26Q8×C22⋊C4—216no—
27C2×#391C2214yes—
28C2×#391C2212yes—
29C2×#391C2214yes—
30C2×#391C2214yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2214yes—
37C2×#389C2212yes—
38C2×#384C2212yes—
39C2×#384C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2214yes—
42C2×#380C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2214yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46#1446C2211yescharacteristic
47#1446C2212yes—
48#1446C2212yes—
49#1445C2212yes—
50#1444C2212yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1444C2214yes—
55#1442C2211yescharacteristic
56#1442C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1439C2214yes—
62#1439C2212yes—
63#1439C2211yescharacteristic
64#1439C2212yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2214yes—
72#1434C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C4×#95C2211yescharacteristic
76C4×#95C2212yes—
77C23×Q8—28no—
78C23×Q8—28no—
79C23×Q8—42no—
80C23×Q8—42no—
81C23×Q8Q818yescomplemented
82C23×Q8C2312yes—
83C23×Q8C2311yescharacteristic
84C23×Q8C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C2×C4×Q8—28no—
88C2×C4×Q8—28no—
89C2×C4×Q8—28no—
90C2×C4×Q8—28no—
91C2×C4×Q8—28no—
92C2×C4×Q8—42no—
93C2×C4×Q8—44no—
94C2×C4×Q8—44no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2314yes—
119C22×C42C2314yes—
120C22×C42C2312yes—
121#399—44no—
122#398—216no—
123#398—216no—
124#396—48no—
125#393Q818yescomplemented
126#391—28no—
127#391—28no—
128#391—28no—
129#391—216no—
130#391—216no—
131#391—216no—
132#391D4116yes—
133#389—28no—
134#389—28no—
135#389—216no—
136#389—44no—
137#384—28no—
138#384—42no—
139#380—28no—
140#380—216no—
141#380—42no—
142C4×C22⋊C4—28no—
143C4×C22⋊C4—216no—
144C4×C22⋊C4—42no—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C25C2411yescharacteristic
174C22×Q8—264no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—28no—
178C22×Q8—24no—
179C22×Q8—28no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—41no—
184C22×Q8—42no—
185C22×Q8—42no—
186C22×Q8—41no—
187C22×Q8—416no—
188C22×Q8—416no—
189C22×Q8—432no—
190C22×Q8—416no—
191C22×Q8C2×D418yes—
192C22×Q8C2×Q818yes—
193C22×Q8C2×Q818yes—
194C22×Q8C2×Q814yescomplemented
195C22×Q8#4118yes—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4C2×Q814yes—
205C23×C4C2×Q818yes—
206C23×C4C2414yes—
207C23×C4C2411yescharacteristic
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C4×Q8—416no—
221C4×Q8—432no—
222C4×Q8—432no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C2×D418yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4C2×Q818yes—
280C2×C22⋊C4C2×Q814yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—28no—
284C2×C42—28no—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—24no—
288C2×C42—28no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42C2×D418yes—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—42no—
311#95—44no—
312#95—42no—
313#95—41no—
314#95—42no—
315#95—41no—
316#95—41no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—42no—
321#95C2×D4116yes—
322#95C2×D4116yes—
323#95C2×Q814yescomplemented
324#95C2×Q818yescomplemented
325C24—22no—
326C24—22no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—21no—
331C24—22no—
332C24—21no—
333C24C22×Q812yes—
334C24C22×Q812yes—
335C24H(5,𝔽2)12yes—
336C24C2511yescharacteristic
337C2×Q8—216no—
338C2×Q8—232no—
339C2×Q8—216no—
340C2×Q8—216no—
341C2×Q8—216no—
342C2×Q8—216no—
343C2×Q8—216no—
344C2×Q8—216no—
345C2×Q8—48no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—48no—
349C2×Q8—44no—
350C2×Q8—48no—
351C2×Q8—416no—
352C2×Q8—48no—
353C2×Q8—4128no—
354C2×Q8C22⋊Q8132yes—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—232no—
398C22×C4—216no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—416no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4C22×D418yes—
434C22×C4C22×D414yes—
435C22×C4C22×D418yes—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×Q814yes—
439C22×C4C22×Q812yes—
440C22×C4C22×Q812yes—
441C22×C4C22×Q812yes—
442C22×C4C22×Q818yes—
443C22×C4C22×Q814yes—
444C22×C4C22×Q812yes—
445C22×C4C2×#4118yes—
446C22×C4C2×#4114yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C42−1+412yes—
450C22×C42−1+414yes—
451C4⋊C4—216no—
452C4⋊C4—416no—
453C4⋊C4—48no—
454C4⋊C4—416no—
455C4⋊C4—416no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—88no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C4⋊C4—88no—
465C22⋊C4—216no—
466C22⋊C4—44no—
467C22⋊C4—48no—
468C22⋊C4—416no—
469C22⋊C4—44no—
470C22⋊C4—84no—
471C42—216no—
472C42—416no—
473C42—416no—
474C42—44no—
475C42—48no—
476C42—44no—
477C23—24no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—41no—
490C23—42no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23Q8214yes—
500C23C23×D412yes—
501C23C23×Q811yescharacteristic
502C23C23×Q811yescharacteristic
503C23C22×#4112yes—
504C23C2×H(5,𝔽2)11yescharacteristic
505C23C2×H(5,𝔽2)11yescharacteristic
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×2−1+412yes—
508C23C2×2−1+411yescharacteristic
509C23#58512yes—
510C23#58511yescharacteristic
511Q8—264no—
512Q8—416no—
513Q8—432no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—432no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—416no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4C2×C22⋊Q8116yes—
596C2×C4C2×C22⋊Q818yes—
597C2×C4#54314yes—
598C2×C4D4214yes—
599C2×C4#54618yes—
600C2×C4D4×Q818yes—
601C2×C4#55114yes—
602C22—28no—
603C22—24no—
604C22—44no—
605C22—44no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22C22×C22⋊Q812yes—
614C22C2×#54311yescharacteristic
615C22C2×D4211yescharacteristic
616C22C2×#54612yes—
617C22C2×D4×Q812yes—
618C22C2×#55111yescharacteristic
619C22C2×Q8211yescharacteristic
620C22Q8×#4112yes—
621C22#348812yes—
622C22#348912yes—
623C22#349712yes—
624C22#350111yescharacteristic
625C22#350212yes—
626C22#350211yescharacteristic
627C22#350311yescharacteristic
628C22#352212yes—
629C22#352312yes—
630C22#352412yes—
631C22#353611yescharacteristic
632C22#353811yescharacteristic
633C22#355111yescharacteristic
634C22#355412yes—
635C22#355811yescharacteristic
636C4—216no—
637C4—28no—
638C4—48no—
639C4—44no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—44no—
645C4—82no—
646C4—84no—
647C4—84no—
648C4—81no—
649C4#243718yes—
650C2—44no—
651C2C2×#243711yescharacteristic
652C2#3578612yes—
653C2Q8×C22⋊Q812yes—
654C2#3962411yescharacteristic
655C2#4281811yescharacteristic
656C2#4524411yescharacteristic
657C2#4531912yes—
658C2#4541911yescharacteristic
659C2#5899811yescharacteristic
660C2#5941312yes—
661C2#5993311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2582C2semidirect
4. #15964C2semidirect
5. #15511C2semidirect
6. #15333C2semidirect
11. #13976C2semidirect
12. #13976C2semidirect
13. #13935C2semidirect
81. C23×Q8Q8semidirect
125. #393Q8semidirect
194. C22×Q8C2×Q8semidirect
323. #95C2×Q8semidirect
324. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×Q8×C22⋊C42
#163052
#159642
#155112
#153332
#151352
#151312
#147712
#142132
#139762
#139352
#139222
C4×#3932