Finite Group Databasebeta

← #949108

Subgroups — #949108

Order 512.

2746 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C212yesmaximal, complemented
2#17218C212yesmaximal, complemented
3#17197C212yesmaximal, complemented
4#16606C214yesmaximal, complemented
5#16343C214yesmaximal, complemented
6#15961C212yesmaximal
7#15951C212yesmaximal
8#15223C212yesmaximal, complemented
9#15220C212yesmaximal, complemented
10#15133C212yesmaximal, complemented
11#14282C211yesmaximal, characteristic, complemented
12#14277C211yesmaximal, characteristic, complemented
13#13880C212yesmaximal, complemented
14#13872C212yesmaximal
15#13847C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2214yes—
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20#2440—216no—
21C2×#398C2214yes—
22C2×#398C2214yes—
23C2×#395C2214yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#386C2214yes—
28C2×#385C2214yes—
29C2×#385C2214yes—
30C2×#385C2214yes—
31C2×#384C2212yes—
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×#376C2211yescharacteristic
42#1446C2212yes—
43#1446C2212yes—
44#1445C2211yescharacteristic
45#1445C2211yescharacteristic
46#1445C2212yes—
47#1445C2214yes—
48#1445C2214yes—
49#1445C2212yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2212yes—
53#1444C2212yes—
54#1443C2214yes—
55#1442C2212yes—
56#1442C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1439C2212yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2211yescharacteristic
70#1438C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C22⋊#95C2214yes—
76C23×Q8C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2314yes—
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8D418yescomplemented
83C2×C22⋊Q8—216no—
84C2×C22⋊Q8—48no—
85C2×C22⋊Q8—48no—
86C2×C4×D4—42no—
87C2×C4×D4—42no—
88C2×C4×D4Q818yescomplemented
89C22×C4⋊C4—28no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#398—216no—
120#398—48no—
121#395—216no—
122#389—28no—
123#389—28no—
124#389—44no—
125#386—48no—
126#385—216no—
127#385—48no—
128#385—48no—
129#384—28no—
130#384—42no—
131#382—24no—
132#382—42no—
133#380—28no—
134#380—44no—
135#380—44no—
136C4×C4⋊C4—42no—
137C4×C22⋊C4—28no—
138C4×C22⋊C4—28no—
139#376—42no—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C25C2412yes—
169C22×Q8—44no—
170C22×Q8—44no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8C2×D418yes—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4C2×Q814yes—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—28no—
186C23×C4—28no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4C2×Q818yes—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2414yes—
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C4⋊Q8—416no—
208C22⋊Q8—464no—
209C4×D4—416no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—432no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2×D418yes—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4C2×Q814yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4C2×Q818yes—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42C2×D414yes—
288C2×C42C2×Q814yes—
289C2×C42#4114yes—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—41no—
301#95—41no—
302#95—41no—
303#95—41no—
304#95—41no—
305#95—41no—
306#95—44no—
307#95—44no—
308#95—42no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24C22×Q814yes—
331C24H(5,𝔽2)14yes—
332C242−1+414yes—
333C24C2511yescharacteristic
334C2×Q8—416no—
335C2×Q8—48no—
336C2×Q8—84no—
337C2×Q8—88no—
338C2×Q8—84no—
339C2×D4—216no—
340C2×D4—44no—
341C2×D4—48no—
342C2×D4—44no—
343C2×D4—44no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—216no—
393C22×C4—216no—
394C22×C4—216no—
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—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—416no—
417C22×C4—416no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—88no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D418yes—
430C22×C4C22×D412yes—
431C22×C4C22×Q812yes—
432C22×C4C22×Q812yes—
433C22×C4C22×Q814yes—
434C22×C4C22×Q812yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4H(5,𝔽2)14yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C42−1+414yes—
443C22×C42−1+412yes—
444C4⋊C4—216no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—416no—
449C4⋊C4—416no—
450C4⋊C4—416no—
451C4⋊C4—416no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—88no—
456C4⋊C4—88no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C22⋊C4—216no—
460C22⋊C4—216no—
461C22⋊C4—216no—
462C22⋊C4—216no—
463C22⋊C4—48no—
464C22⋊C4—416no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C42—28no—
468C42—28no—
469C42—28no—
470C42—48no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23#54418yes—
506C23C23×D411yescharacteristic
507C23C23×Q811yescharacteristic
508C23C22×#4111yescharacteristic
509C23C2×H(5,𝔽2)11yescharacteristic
510C23C2×H(5,𝔽2)11yescharacteristic
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×H(5,𝔽2)12yes—
513C23C2×H(5,𝔽2)12yes—
514C23C2×2−1+411yescharacteristic
515C23C2×2−1+412yes—
516C23C2×2−1+412yes—
517Q8—816no—
518D4—416no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—432no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
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—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—832no—
600C2×C4#53414yes—
601C2×C4#53914yes—
602C2×C4#54314yes—
603C2×C4D4×Q814yes—
604C2×C4#55014yes—
605C2×C4#55414yes—
606C2×C4#56214yes—
607C22—28no—
608C22—28no—
609C22—28no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22C2×#53411yescharacteristic
623C22C2×#53911yescharacteristic
624C22C2×#54311yescharacteristic
625C22C2×#54412yes—
626C22C2×D4×Q811yescharacteristic
627C22C2×#55011yescharacteristic
628C22C2×#55411yescharacteristic
629C22#348412yes—
630C22#348612yes—
631C22#349212yes—
632C22#349212yes—
633C22#350312yes—
634C22C2×#56211yescharacteristic
635C22#353612yes—
636C22#353612yes—
637C22#353711yescharacteristic
638C22#353811yescharacteristic
639C22#354312yes—
640C22#354412yes—
641C22#354911yescharacteristic
642C22#355011yescharacteristic
643C22#355212yes—
644C22#355212yes—
645C4—42no—
646C4—42no—
647C4—44no—
648C4—44no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—44no—
653C4—48no—
654C4—42no—
655C4—84no—
656C4—84no—
657C4—84no—
658C4#273218yes—
659C2—48no—
660C2#3730611yescharacteristic
661C2#3747711yescharacteristic
662C2#3768612yes—
663C2#3787412yes—
664C2#4247511yescharacteristic
665C2#4250211yescharacteristic
666C2#4250611yescharacteristic
667C2#4280711yescharacteristic
668C2#4535612yes—
669C2C2×#273211yescharacteristic
670C2#5299212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2440C2semidirect
2. #17218C2semidirect
3. #17197C2semidirect
4. #16606C2semidirect
5. #16343C2semidirect
8. #15223C2semidirect
9. #15220C2semidirect
10. #15133C2semidirect
11. #14282C2semidirect
12. #14277C2semidirect
13. #13880C2semidirect
82. C2×C4⋊Q8D4semidirect
88. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
#172182
#171972
#166062
#163432
#159612
#159512
#152232
#152202
#151332
#142822
#142772
#138802
#138722
#138472