Finite Group Databasebeta

← #536192

Subgroups — #536192

Order 512.

2516 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2395C211yesmaximal, characteristic
2#14616C214yesmaximal, complemented
3#14592C212yesmaximal, complemented
4#14590C212yesmaximal, complemented
5#14254C212yesmaximal, complemented
6#14248C212yesmaximal, complemented
7#14196C214yesmaximal
8#13906C211yesmaximal, characteristic
9#13884C212yesmaximal, complemented
10#13875C212yesmaximal, complemented
11#13875C212yesmaximal, complemented
12#13872C211yesmaximal, characteristic
13#13804C212yesmaximal, complemented
14C22⋊C4×C4⋊C4C212yesmaximal, complemented
15#13777C212yesmaximal, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#2395C4116yescomplemented
20C2×#400C2211yescharacteristic
21C2×#399C2211yescharacteristic
22C2×#398C2211yescharacteristic
23C2×#397C2211yescharacteristic
24C2×#397C2212yes—
25C2×#394C2212yes—
26C2×#390C2212yes—
27C2×#387C2214yes—
28C2×#386C2212yes—
29C2×#384C2211yescharacteristic
30C2×#382C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C22⋊C4C2214yes—
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43#1443C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1441C2212yes—
48#1440C2214yes—
49#1440C2214yes—
50#1439C2212yes—
51#1439C2212yes—
52#1439C2212yes—
53#1437C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70#1434C2214yes—
71#1434C2214yes—
72#1434C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C4×#95C2214yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4×D4—42no—
81C2×C4×D4—42no—
82C2×C4×D4C2×C418yes—
83C2×#117—42no—
84C2×#117—42no—
85C2×#117C2×C418yescomplemented
86C22×C4⋊C4—28no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2314yes—
115C22×C42C2314yes—
116C22×C42C2314yes—
117#400C2×C418yescomplemented
118#399C2×C418yes—
119#398C2×C418yescomplemented
120#397C2×C418yes—
121#397C2×C4116yescomplemented
122#394C2×C4116yes—
123#390—44no—
124#387—48no—
125#386—44no—
126#384—42no—
127#382—28no—
128#380—28no—
129#380—28no—
130#380—28no—
131#380—42no—
132C4×C4⋊C4—28no—
133C4×C4⋊C4—44no—
134C4×C4⋊C4—44no—
135C4×C4⋊C4—42no—
136C4×C4⋊C4C2×C4116yescomplemented
137C4×C22⋊C4—48no—
138C4×C22⋊C4C2×C4116yes—
139C4×C22⋊C4C2×C418yescomplemented
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×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C25C2411yescharacteristic
167C25C2411yescharacteristic
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4C22×C414yes—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—28no—
185C23×C4—42no—
186C23×C4—44no—
187C23×C4—42no—
188C23×C4C22×C414yes—
189C23×C4C22×C414yes—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C4×D4—416no—
204#117—416no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—416no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4C22×C418yes—
234C2×C4⋊C4C22×C414yes—
235C2×C4⋊C4C22×C414yes—
236C2×C4⋊C4C22×C414yes—
237C2×C4⋊C4C22×C418yes—
238C2×C4⋊C4#4118yes—
239C2×C4⋊C4#4118yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4C22×C418yes—
265C2×C22⋊C4C22×C418yes—
266C2×C22⋊C4C22×C414yes—
267C2×C22⋊C4C22×C414yes—
268C2×C22⋊C4C22×C418yes—
269C2×C22⋊C4C22×C418yes—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—28no—
277C2×C42—24no—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42—42no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42—42no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42C22×C418yes—
289C2×C42C22×C414yes—
290C2×C42C22×C418yes—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
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—44no—
307#95—42no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95C22×C418yes—
312#95C22×C418yes—
313#95C22×C414yes—
314#95C22×C414yes—
315#95C22×C418yes—
316#95C22×C418yes—
317C24—22no—
318C24—22no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—22no—
334C24—22no—
335C24C23×C412yes—
336C24C23×C412yes—
337C24H(5,𝔽2)12yes—
338C242−1+412yes—
339C24C2511yescharacteristic
340C2×D4—28no—
341C2×D4—28no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—44no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—28no—
402C22×C4—28no—
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—48no—
418C22×C4—416no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—88no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4C23×C414yes—
429C22×C4C23×C412yes—
430C22×C4C23×C414yes—
431C22×C4C23×C412yes—
432C22×C4C23×C414yes—
433C22×C4C23×C414yes—
434C22×C4C23×C412yes—
435C22×C4C23×C412yes—
436C22×C4C23×C412yes—
437C22×C4C2×#4118yes—
438C22×C4C2×#4118yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C42−1+412yes—
445C4⋊C4—216no—
446C4⋊C4—216no—
447C4⋊C4—216no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C22⋊C4—216no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—416no—
464C22⋊C4—88no—
465C22⋊C4—88no—
466C22⋊C4—84no—
467C42—216no—
468C42—216no—
469C42—28no—
470C42—84no—
471C42—88no—
472C42—88no—
473C42—88no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—41no—
492C23—42no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—41no—
502C23—42no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23#52014yes—
520C23C24×C411yescharacteristic
521C23C22×#4112yes—
522C23C22×#4112yes—
523C23C2×H(5,𝔽2)11yescharacteristic
524C23C2×H(5,𝔽2)11yescharacteristic
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×H(5,𝔽2)11yescharacteristic
527C23C2×2−1+411yescharacteristic
528C23C2×2−1+411yescharacteristic
529C23#58512yes—
530C23#58512yes—
531D4—416no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—416no—
596C2×C4—416no—
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—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—816no—
612C2×C4C4×#4118yes—
613C2×C4#51814yes—
614C2×C4#52014yes—
615C2×C4#55114yes—
616C2×C4#55318yes—
617C2×C4#55914yes—
618C22—24no—
619C22—28no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—42no—
627C22—44no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—44no—
634C22—81no—
635C22—81no—
636C22—82no—
637C22—82no—
638C22—81no—
639C22—81no—
640C22C2×C4×#4112yes—
641C22C2×#51811yescharacteristic
642C22C2×#52011yescharacteristic
643C22C2×#52011yescharacteristic
644C22#342812yes—
645C22C2×#55111yescharacteristic
646C22C2×#55312yes—
647C22C2×#55911yescharacteristic
648C22#351312yes—
649C22#351312yes—
650C22#351412yes—
651C22#351512yes—
652C22#351612yes—
653C22#351612yes—
654C22#351712yes—
655C22#351812yes—
656C22#353611yescharacteristic
657C22#353611yescharacteristic
658C22#353711yescharacteristic
659C22#353811yescharacteristic
660C22#355111yescharacteristic
661C22#355111yescharacteristic
662C22#355112yes—
663C4—24no—
664C4—44no—
665C4—42no—
666C4—44no—
667C4—42no—
668C4—42no—
669C4—48no—
670C4—48no—
671C4—44no—
672C4—44no—
673C4—84no—
674C4—84no—
675C4#237418yes—
676C2—44no—
677C2—82no—
678C2C2×#237411yescharacteristic
679C2#3424112yes—
680C2#3424212yes—
681C2#3436011yescharacteristic
682C2#3436911yescharacteristic
683C2#4521411yescharacteristic
684C2#4521612yes—
685C2#4522412yes—
686C2#4522611yescharacteristic
687C2#4530811yescharacteristic
688C2#4531511yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14616C2semidirect
3. #14592C2semidirect
4. #14590C2semidirect
5. #14254C2semidirect
6. #14248C2semidirect
9. #13884C2semidirect
10. #13875C2semidirect
11. #13875C2semidirect
13. #13804C2semidirect
14. C22⋊C4×C4⋊C4C2semidirect
15. #13777C2semidirect
19. #2395C4semidirect
85. C2×#117C2×C4semidirect
117. #400C2×C4semidirect
119. #398C2×C4semidirect
121. #397C2×C4semidirect
136. C4×C4⋊C4C2×C4semidirect
139. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23952
#146162
#145922
#145902
#142542
#142482
#141962
#139062
#138842
#138752
#138722
#138042
C22⋊C4×C4⋊C42
#137772