Finite Group Databasebeta

← #522647

Subgroups — #522647

Order 512.

3170 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
1C2×#2641C211yesmaximal, characteristic
2C2×#2328C211yesmaximal, characteristic
3C2×C4×#125C211yesmaximal, characteristic, complemented
4#14713C211yesmaximal, characteristic
5#14536C214yesmaximal, complemented
6#14523C214yesmaximal, complemented
7#14283C211yesmaximal, characteristic, complemented
8#14201C214yesmaximal, complemented
9#14195C214yesmaximal, complemented
10#14141C212yesmaximal, complemented
11#14022C212yesmaximal, complemented
12#14018C212yesmaximal, complemented
13#13820C212yesmaximal, complemented
14#13779C211yesmaximal, characteristic, complemented
15#13759C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C4×D4C2212yes—
19#2641C4116yescomplemented
20C2×#394C2214yes—
21C2×#394C2214yes—
22#2328—28no—
23C4×#125—28no—
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2214yes—
28C2×#390C2214yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#389C2212yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#387C2214yes—
35C2×#386C2214yes—
36C2×#386C2214yes—
37C2×#385C2214yes—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×C4×C4⋊C4C2214yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2214yes—
46C2×#376C2211yescharacteristic
47C2×#376C2211yescharacteristic
48C2×C43C2211yescharacteristic
49#1441C2214yes—
50#1441C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2212yes—
62#1436C2214yes—
63#1436C2212yes—
64#1436C2214yes—
65#1436C2212yes—
66#1436C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C22⋊#95C2214yes—
71C22⋊#95C2214yes—
72C4×#95C2211yescharacteristic
73C4×#95C2211yescharacteristic
74C23×D4C2312yes—
75C23×D4C2314yes—
76C24×C4C2314yes—
77C2×#127—24no—
78C2×#127—24no—
79C2×#127C2×C418yes—
80C2×#125—24no—
81C2×#125—24no—
82C2×#125C2×C418yescomplemented
83C2×C4×D4—28no—
84C2×C4×D4—28no—
85C2×C4×D4—44no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2314yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2314yes—
116C22×C42C2311yescharacteristic
117#394C2×C4132yescomplemented
118#394C2×C4132yescomplemented
119#390—28no—
120#390—28no—
121#390—48no—
122#390—48no—
123#390—44no—
124#390C2×C4116yescomplemented
125#390C2×C4116yescomplemented
126#389—28no—
127#388—48no—
128#388—48no—
129#387—216no—
130#386—216no—
131#386—216no—
132#385—48no—
133#384—24no—
134#382D418yes—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—216no—
139C4×C4⋊C4—44no—
140C4×C22⋊C4—48no—
141#376—24no—
142#376C2×C418yescomplemented
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2311yescharacteristic
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C43—24no—
165C25C2414yes—
166C22×D4—24no—
167C22×D4—28no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—28no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4C22×C418yes—
179C22×D4C22×C4116yes—
180C23×C4—28no—
181C23×C4—28no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4C2414yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2414yes—
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201#127—232no—
202#125—232no—
203C4×D4—432no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4C22×C418yes—
227C2×C4⋊C4C22×C4116yes—
228C2×C4⋊C4C2×D418yes—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4C22×C4116yes—
259C2×C22⋊C4C22×C4116yes—
260C2×C22⋊C4C22×C4116yes—
261C2×C22⋊C4C22×C4116yes—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—28no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—28no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—42no—
283C2×C42—44no—
284C2×C42—42no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42C22×C414yes—
289C2×C42C22×C414yes—
290C2×C42C22×C414yes—
291C2×C42C2×D414yes—
292C2×C42#4114yes—
293C2×C42#4114yes—
294C2×C42#4114yes—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—28no—
299#95—24no—
300#95—22no—
301#95—22no—
302#95—24no—
303#95—24no—
304#95—42no—
305#95—42no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95C22×C418yes—
314#95C22×C418yes—
315#95C2×D4116yes—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24C23×C418yes—
331C24C2511yescharacteristic
332C2×D4—216no—
333C2×D4—232no—
334C2×D4—216no—
335C2×D4—216no—
336C2×D4—416no—
337C2×D4—416no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—88no—
343C2×D4—88no—
344C2×D4—88no—
345C2×D4—88no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—416no—
413C22×C4—416no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4—88no—
417C22×C4—88no—
418C22×C4—88no—
419C22×C4C23×C412yes—
420C22×C4C23×C412yes—
421C22×C4C23×C414yes—
422C22×C4C23×C418yes—
423C22×C4C23×C414yes—
424C22×C4C23×C412yes—
425C22×C4C22×D418yes—
426C22×C4C22×D414yes—
427C22×C4C22×D412yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4114yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4H(5,𝔽2)14yes—
436C22×C4H(5,𝔽2)14yes—
437C22×C4H(5,𝔽2)12yes—
438C22×C42−1+412yes—
439C4⋊C4—232no—
440C4⋊C4—216no—
441C4⋊C4—216no—
442C4⋊C4—416no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—88no—
448C22⋊C4—416no—
449C22⋊C4—416no—
450C22⋊C4—416no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—416no—
454C22⋊C4—88no—
455C22⋊C4—88no—
456C22⋊C4—88no—
457C22⋊C4—88no—
458C22⋊C4—88no—
459C42—28no—
460C42—28no—
461C42—28no—
462C42—28no—
463C42—48no—
464C42—416no—
465C42—48no—
466C42—44no—
467C42—44no—
468C42—88no—
469C42C4×D4116yes—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23C24×C411yescharacteristic
506C23C23×D411yescharacteristic
507C23C22×#4111yescharacteristic
508C23C22×#4111yescharacteristic
509C23C22×#4111yescharacteristic
510C23C2×H(5,𝔽2)11yescharacteristic
511C23C2×H(5,𝔽2)12yes—
512C23C2×H(5,𝔽2)12yes—
513C23C2×2−1+411yescharacteristic
514C23#58512yes—
515C23#58512yes—
516D4—432no—
517D4—832no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—832no—
596C2×C4C2×C4×D418yes—
597C2×C4C2×C4×D414yes—
598C2×C4C4×#4114yes—
599C2×C4#52014yes—
600C2×C4#54014yes—
601C2×C4#54614yes—
602C2×C4#54814yes—
603C2×C4#55614yes—
604C2×C4#55914yes—
605C22—48no—
606C22—48no—
607C22—48no—
608C22—48no—
609C22—48no—
610C22—48no—
611C22—48no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22C22×C4×D411yescharacteristic
621C22C2×C4×#4111yescharacteristic
622C22C2×#52011yescharacteristic
623C22C4×H(5,𝔽2)12yes—
624C22C4×H(5,𝔽2)12yes—
625C22C2×#54011yescharacteristic
626C22#345511yescharacteristic
627C22#346012yes—
628C22C2×#54611yescharacteristic
629C22C2×#54811yescharacteristic
630C22C2×#55611yescharacteristic
631C22C2×#55911yescharacteristic
632C22#348812yes—
633C22#348812yes—
634C22#349412yes—
635C22#350811yescharacteristic
636C22#350812yes—
637C22#351012yes—
638C22#351211yescharacteristic
639C22#354312yes—
640C22#354712yes—
641C22#355912yes—
642C22#356212yes—
643C4—24no—
644C4—28no—
645C4—42no—
646C4—42no—
647C4—44no—
648C4—48no—
649C4—44no—
650C4—44no—
651C4—48no—
652C4—42no—
653C4—81no—
654C4—81no—
655C4—84no—
656C4#236718yes—
657C2—88no—
658C2C4×#54011yescharacteristic
659C2C2×#236711yescharacteristic
660C2#3426912yes—
661C2#3428011yescharacteristic
662C2#3443112yes—
663C2#3770611yescharacteristic
664C2#3771011yescharacteristic
665C2#3775112yes—
666C2#3976611yescharacteristic
667C2#3977011yescharacteristic
668C2#3980312yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×C4×#125C2semidirect
5. #14536C2semidirect
6. #14523C2semidirect
7. #14283C2semidirect
8. #14201C2semidirect
9. #14195C2semidirect
10. #14141C2semidirect
11. #14022C2semidirect
12. #14018C2semidirect
13. #13820C2semidirect
14. #13779C2semidirect
15. #13759C2semidirect
19. #2641C4semidirect
82. C2×#125C2×C4semidirect
117. #394C2×C4semidirect
118. #394C2×C4semidirect
124. #390C2×C4semidirect
125. #390C2×C4semidirect
142. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26412
C2×#23282
C2×C4×#1252
#147132
#145362
#145232
#142832
#142012
#141952
#141412
#140222
#140182
#138202
#137792
#137592