Finite Group Databasebeta

← #556009

Subgroups — #556009

Order 512.

3006 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#388C211yesmaximal, characteristic
2#14683C211yesmaximal, characteristic, complemented
3#14682C211yesmaximal, characteristic, complemented
4#14500C214yesmaximal, complemented
5#14496C214yesmaximal, complemented
6#14399C214yesmaximal, complemented
7#14396C214yesmaximal, complemented
8#14341C211yesmaximal, characteristic
9#14338C211yesmaximal, characteristic
10#13883C212yesmaximal, complemented
11#13878C212yesmaximal, complemented
12#13873C212yesmaximal
13#13868C212yesmaximal
14#13795C211yesmaximal, characteristic, complemented
15#13776C211yesmaximal, characteristic, complemented
16C22×#117C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C2×#388C4116yescomplemented
23C2×#388C2214yes—
24C2×#388C2212yes—
25C2×#388C2212yes—
26C2×#388C2212yes—
27C2×#388C2212yes—
28C2×#383C2212yescomplemented
29C2×#382C2212yescomplemented
30C2×#381C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#379C2212yes—
34C2×#379C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38#1444C2214yescomplemented
39#1443C2214yescomplemented
40#1442C2214yescomplemented
41#1442C2214yescomplemented
42#1440C2214yes—
43#1440C2214yes—
44#1440C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1437C2214yes—
51#1437C2214yes—
52#1437C2214yes—
53#1436C2214yescomplemented
54#1436C2214yescomplemented
55#1436C2214yescomplemented
56#1436C2214yescomplemented
57#1436C2214yescomplemented
58C4⋊#95C2212yescomplemented
59C4⋊#95C2214yescomplemented
60C4⋊#95C2214yescomplemented
61#1434C2214yescomplemented
62#1434C2212yescomplemented
63#1434C2212yescomplemented
64#1434C2212yescomplemented
65#1434C2212yescomplemented
66#1434C2212yescomplemented
67C22⋊#95C2211yescharacteristic
68C22⋊#95C2211yescharacteristic
69C22⋊#95C2212yes—
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C26C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×#117—28no—
84C2×#117—44no—
85C2×#117—44no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4C2×C418yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—44no—
95C22×C22⋊C4—44no—
96C22×C22⋊C4C2×C418yes—
97C22×C22⋊C4C2×C418yes—
98C22×C22⋊C4C2×C418yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C42—44no—
114C22×C42C2×C418yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2314yes—
118C22×C42C2314yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#388—48no—
122#388C2×C4116yes—
123#388C2×C4116yes—
124#388C2×C4116yes—
125#388C2×C4116yes—
126#383—44no—
127#382—44no—
128#381—28no—
129#380—44no—
130#380—44no—
131#379—44no—
132#379—44no—
133C4×C22⋊C4—44no—
134C2×#95C2×C418yes—
135C2×#95C2×C418yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C25—24no—
164C25—24no—
165C25—22no—
166C25—22no—
167C25—42no—
168C25—42no—
169C25—42no—
170C25—42no—
171C25C22×C414yes—
172C25C2412yes—
173C25C2411yescharacteristic
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4C22×C414yes—
194C23×C4C22×C414yes—
195C23×C4C22×C414yes—
196C23×C4C22×C414yes—
197C23×C4C22×C414yes—
198C23×C4C22×C414yes—
199C23×C4#4118yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213#117—432no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—416no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4C22×C418yes—
229C2×C4⋊C4C22×C418yes—
230C2×C22⋊C4—232no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—416no—
249C2×C22⋊C4—416no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4C22×C418yes—
268C2×C22⋊C4C22×C418yes—
269C2×C22⋊C4C22×C418yes—
270C2×C22⋊C4C22×C418yes—
271C2×C22⋊C4C22×C418yes—
272C2×C22⋊C4C22×C418yes—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—28no—
276C2×C42—416no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42C22×C414yes—
285C2×C42C22×C414yes—
286C2×C42C22×C414yes—
287C2×C42C22×C414yes—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—42no—
292#95—42no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—42no—
297#95—42no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—44no—
306#95—42no—
307#95—42no—
308#95—44no—
309#95—42no—
310#95—44no—
311#95C22×C418yes—
312#95C22×C418yes—
313#95C22×C418yes—
314#95C22×C418yes—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—28no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—28no—
331C24—28no—
332C24—44no—
333C24—44no—
334C24—44no—
335C24—44no—
336C24—416no—
337C24—416no—
338C24—48no—
339C24—48no—
340C24—84no—
341C24—84no—
342C24—88no—
343C24—88no—
344C24—84no—
345C24—84no—
346C24C23×C414yes—
347C24C23×C412yes—
348C24C2×#4114yes—
349C24H(5,𝔽2)14yes—
350C24H(5,𝔽2)12yes—
351C242−1+412yes—
352C24C2511yescharacteristic
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—216no—
356C22×C4—216no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
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—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—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4#117116yes—
438C22×C4C23×C414yes—
439C22×C4C23×C412yes—
440C22×C4C23×C412yes—
441C22×C4C23×C412yes—
442C22×C4C23×C414yes—
443C22×C4C23×C412yes—
444C22×C4C23×C412yes—
445C22×C4C23×C414yes—
446C22×C4C23×C412yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C22×C4H(5,𝔽2)14yes—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—416no—
453C4⋊C4—88no—
454C4⋊C4—88no—
455C22⋊C4—216no—
456C22⋊C4—216no—
457C22⋊C4—216no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C42—28no—
470C42—28no—
471C42—44no—
472C42—44no—
473C42—416no—
474C42—88no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—28no—
480C23—28no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—28no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—416no—
495C23—416no—
496C23—416no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—88no—
514C23—88no—
515C23—88no—
516C23—88no—
517C23—832no—
518C23—832no—
519C23C2×#11714yes—
520C23#51814yes—
521C23#52014yes—
522C23#53818yes—
523C23#53918yes—
524C23C24×C411yescharacteristic
525C23C22×#4112yes—
526C23C2×H(5,𝔽2)12yes—
527C23C2×H(5,𝔽2)12yes—
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×2−1+411yescharacteristic
530C23#58512yes—
531C23#58512yes—
532C23#58512yes—
533C2×C4—232no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—416no—
566C2×C4—416no—
567C2×C4—416no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
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—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—88no—
607C2×C4—88no—
608C2×C4C2×#11718yes—
609C2×C4#51814yes—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—216no—
617C22—216no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—832no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22#231818yes—
631C22C22×#11711yescharacteristic
632C22C2×#51811yescharacteristic
633C22C2×#51811yescharacteristic
634C22C2×#52011yescharacteristic
635C22#342811yescharacteristic
636C22#342811yescharacteristic
637C22#342811yescharacteristic
638C22C2×#53812yes—
639C22C2×#53912yes—
640C22#349314yes—
641C22#350614yes—
642C22#353412yes—
643C22#353512yes—
644C22#354214yes—
645C22#354514yes—
646C22#356012yes—
647C22#356412yes—
648C4—28no—
649C4—24no—
650C4—42no—
651C4—42no—
652C4—44no—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—84no—
658C4—84no—
659C4—84no—
660C4—84no—
661C2—28no—
662C2—84no—
663C2C2×#231811yescharacteristic
664C2#3417911yescharacteristic
665C2#3418611yescharacteristic
666C2#3436311yescharacteristic
667C2#3436611yescharacteristic
668C2#3437611yescharacteristic
669C2#3438111yescharacteristic
670C2#3679514yes—
671C2#3679614yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14683C2semidirect
3. #14682C2semidirect
4. #14500C2semidirect
5. #14496C2semidirect
6. #14399C2semidirect
7. #14396C2semidirect
10. #13883C2semidirect
11. #13878C2semidirect
14. #13795C2semidirect
15. #13776C2semidirect
22. C2×#388C4semidirect
28. C2×#383C22semidirect
29. C2×#382C22semidirect
38. #1444C22semidirect
39. #1443C22semidirect
40. #1442C22semidirect
41. #1442C22semidirect
53. #1436C22semidirect
54. #1436C22semidirect
55. #1436C22semidirect
56. #1436C22semidirect
57. #1436C22semidirect
58. C4⋊#95C22semidirect
59. C4⋊#95C22semidirect
60. C4⋊#95C22semidirect
61. #1434C22semidirect
62. #1434C22semidirect
63. #1434C22semidirect
64. #1434C22semidirect
65. #1434C22semidirect
66. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3882
#146832
#146822
#145002
#144962
#143992
#143962
#143412
#143382
#138832
#138782
#138732
#138682
#137952
#137762