Finite Group Databasebeta

← #536187

Subgroups — #536187

Order 512.

3000 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#397C211yesmaximal, characteristic
2#14615C214yesmaximal, complemented
3#14613C212yesmaximal, complemented
4#14577C212yesmaximal, complemented
5#14253C212yesmaximal, complemented
6#14248C212yesmaximal, complemented
7#14197C214yesmaximal, complemented
8#13903C211yesmaximal, characteristic
9#13883C212yesmaximal, complemented
10#13874C212yesmaximal, complemented
11#13873C212yesmaximal
12#13869C211yesmaximal, characteristic
13#13802C212yesmaximal, complemented
14C22⋊C42C212yesmaximal
15#13776C212yesmaximal, complemented
16C23×C4⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C2×#397C4116yescomplemented
21C2×#397C2212yes—
22C2×#397C2211yescharacteristic
23C2×#397C2211yescharacteristic
24C2×#397C2211yescharacteristic
25C2×#397C2212yes—
26C2×#397C2211yescharacteristic
27C2×#388C2214yes—
28C2×#381C2212yes—
29C2×#381C2212yes—
30C2×#380C2212yes—
31C2×#380C2211yescharacteristic
32C2×#379C2212yes—
33C2×#379C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2214yes—
39C22×#95C2211yescharacteristic
40C22×#95C2212yes—
41#1443C2214yescomplemented
42#1443C2214yescomplemented
43#1442C2214yescomplemented
44#1442C2214yescomplemented
45#1440C2212yes—
46#1440C2212yes—
47#1440C2214yes—
48#1439C2214yes—
49#1439C2212yes—
50#1439C2212yes—
51#1437C2214yes—
52#1437C2212yes—
53#1437C2214yes—
54#1437C2214yes—
55#1437C2212yes—
56#1436C2214yescomplemented
57#1436C2214yescomplemented
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60#1436C2214yescomplemented
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1436C2214yescomplemented
64#1434C2214yescomplemented
65#1434C2214yescomplemented
66#1434C2214yescomplemented
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C4×#95C2214yescomplemented
76C26C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2312yes—
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C22×C4⋊C4—42no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4C2×C418yescomplemented
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4—28no—
91C22×C22⋊C4—42no—
92C22×C22⋊C4—48no—
93C22×C22⋊C4—42no—
94C22×C22⋊C4—42no—
95C22×C22⋊C4C2×C4116yes—
96C22×C22⋊C4C2×C418yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C42C2314yes—
115C22×C42C2312yes—
116C22×C42C2314yes—
117C22×C42C2314yes—
118#397C2×C4116yes—
119#397C2×C418yes—
120#397C2×C418yes—
121#397C2×C418yescomplemented
122#397C2×C4116yescomplemented
123#397C2×C418yescomplemented
124#388—48no—
125#381—28no—
126#381—28no—
127#380—28no—
128#380—42no—
129#379—28no—
130#379—42no—
131C4×C22⋊C4—28no—
132C4×C22⋊C4—44no—
133C4×C22⋊C4—48no—
134C4×C22⋊C4—42no—
135C4×C22⋊C4—44no—
136C2×#95C2×C418yescomplemented
137C2×#95C2×C4116yescomplemented
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C25—24no—
166C25—24no—
167C25—24no—
168C25—22no—
169C25—22no—
170C25—44no—
171C25—42no—
172C25C22×C414yes—
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C25C2411yescharacteristic
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—28no—
188C23×C4—44no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—42no—
196C23×C4C22×C414yes—
197C23×C4C22×C418yes—
198C23×C4C22×C418yes—
199C23×C4C22×C414yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2414yes—
211C23×C4C2414yes—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—416no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4C22×C418yes—
225C2×C4⋊C4C22×C414yes—
226C2×C4⋊C4C22×C414yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—432no—
245C2×C22⋊C4—416no—
246C2×C22⋊C4—416no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4C22×C418yes—
266C2×C22⋊C4C22×C418yes—
267C2×C22⋊C4C22×C414yes—
268C2×C22⋊C4C22×C414yes—
269C2×C22⋊C4C22×C418yes—
270C2×C22⋊C4C22×C418yes—
271C2×C22⋊C4C22×C414yes—
272C2×C22⋊C4C22×C414yes—
273C2×C22⋊C4#4118yes—
274C2×C22⋊C4#4118yes—
275C2×C42—28no—
276C2×C42—44no—
277C2×C42—42no—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42—44no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—44no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—42no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—42no—
304#95—44no—
305#95—44no—
306#95C22×C418yes—
307#95C22×C414yes—
308#95C22×C418yes—
309#95C22×C418yes—
310#95C22×C418yes—
311#95C22×C418yes—
312#95C22×C414yes—
313C24—22no—
314C24—24no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—21no—
319C24—21no—
320C24—24no—
321C24—22no—
322C24—28no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—21no—
327C24—21no—
328C24—24no—
329C24—22no—
330C24—216no—
331C24—28no—
332C24—44no—
333C24—48no—
334C24—416no—
335C24—416no—
336C24—48no—
337C24—48no—
338C24—44no—
339C24—48no—
340C24—44no—
341C24—88no—
342C24—88no—
343C24—84no—
344C24—84no—
345C24C23×C412yes—
346C24C23×C412yes—
347C24C23×C412yes—
348C24C2×#4114yes—
349C24C2×#4114yes—
350C24H(5,𝔽2)14yes—
351C24H(5,𝔽2)12yes—
352C24H(5,𝔽2)12yes—
353C24C2511yescharacteristic
354C22×C4—216no—
355C22×C4—28no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—216no—
399C22×C4—28no—
400C22×C4—216no—
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—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—416no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—44no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4—84no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4—84no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4C23×C414yes—
435C22×C4C23×C414yes—
436C22×C4C23×C412yes—
437C22×C4C23×C412yes—
438C22×C4C23×C412yes—
439C22×C4C23×C412yes—
440C22×C4C23×C414yes—
441C22×C4C23×C414yes—
442C22×C4C2×#4118yes—
443C22×C4C2×#4118yes—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—88no—
448C4⋊C4—84no—
449C22⋊C4—216no—
450C22⋊C4—216no—
451C22⋊C4—216no—
452C22⋊C4—216no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—48no—
460C22⋊C4—416no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—88no—
464C22⋊C4—84no—
465C22⋊C4—84no—
466C42—88no—
467C42—88no—
468C42—88no—
469C42—84no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—416no—
495C23—432no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—416no—
504C23—41no—
505C23—42no—
506C23—41no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—88no—
516C23—816no—
517C23—816no—
518C23—832no—
519C23—816no—
520C23C4×#4118yes—
521C23#51814yes—
522C23#51814yes—
523C23#55114yes—
524C23#55318yes—
525C23#55914yes—
526C23C24×C411yescharacteristic
527C23C22×#4112yes—
528C23C22×#4112yes—
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×H(5,𝔽2)11yescharacteristic
531C23C2×H(5,𝔽2)12yes—
532C23#58514yes—
533C23#58512yes—
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—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—416no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
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—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
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—816no—
602C2×C4—88no—
603C2×C4—816no—
604C2×C4—816no—
605C22—28no—
606C22—216no—
607C22—24no—
608C22—28no—
609C22—24no—
610C22—28no—
611C22—28no—
612C22—24no—
613C22—28no—
614C22—44no—
615C22—48no—
616C22—44no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—44no—
624C22—44no—
625C22—832no—
626C22—82no—
627C22—84no—
628C22—82no—
629C22#237418yes—
630C22C2×C4×#4112yes—
631C22C2×#51811yescharacteristic
632C22C2×#51811yescharacteristic
633C22#342811yescharacteristic
634C22#342812yes—
635C22C2×#55111yescharacteristic
636C22C2×#55312yes—
637C22C2×#55911yescharacteristic
638C22#351314yes—
639C22#351314yes—
640C22#352114yes—
641C22#352114yes—
642C22#353412yes—
643C22#353412yes—
644C22#354512yes—
645C22#354512yes—
646C4—48no—
647C4—48no—
648C4—44no—
649C4—44no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—44no—
656C4—84no—
657C4—84no—
658C2—28no—
659C2—44no—
660C2—82no—
661C2C2×#237411yescharacteristic
662C2#3423812yes—
663C2#3424712yes—
664C2#3435811yescharacteristic
665C2#3436611yescharacteristic
666C2#4330712yes—
667C2#4330814yes—
668C2#4332512yes—
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. #14615C2semidirect
3. #14613C2semidirect
4. #14577C2semidirect
5. #14253C2semidirect
6. #14248C2semidirect
7. #14197C2semidirect
9. #13883C2semidirect
10. #13874C2semidirect
13. #13802C2semidirect
15. #13776C2semidirect
20. C2×#397C4semidirect
41. #1443C22semidirect
42. #1443C22semidirect
43. #1442C22semidirect
44. #1442C22semidirect
56. #1436C22semidirect
57. #1436C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1434C22semidirect
65. #1434C22semidirect
66. #1434C22semidirect
75. C4×#95C22semidirect
84. C22×C4⋊C4C2×C4semidirect
121. #397C2×C4semidirect
122. #397C2×C4semidirect
123. #397C2×C4semidirect
136. C2×#95C2×C4semidirect
137. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3972
#146152
#146132
#145772
#142532
#142482
#141972
#139032
#138832
#138742
#138732
#138692
#138022
C22⋊C422
#137762