Finite Group Databasebeta

← #641134

Subgroups — #641134

Order 512.

3662 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C212yesmaximal, complemented
2C2×#2618C212yesmaximal
3C2×C4×#121C211yesmaximal, characteristic
4C22×#389C211yesmaximal, characteristic, complemented
5#17070C212yesmaximal, complemented
6#17069C212yesmaximal, complemented
7#16821C214yesmaximal, complemented
8#15149C212yesmaximal, complemented
9#15131C214yesmaximal, complemented
10#15129C212yesmaximal, complemented
11#14927C214yesmaximal, complemented
12#14686C211yesmaximal, characteristic, complemented
13#14339C212yesmaximal, complemented
14#13872C212yesmaximal, complemented
15C22×#123C2214yes—
16C22×#121C2211yescharacteristic
17C22×C4×D4C2212yes—
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2618—216no—
24#2486—216no—
25C2×#398C2214yes—
26C2×#398C2214yes—
27C2×#397C2214yes—
28C2×#396C2218yes—
29C2×#394C2214yes—
30C4×#121—28no—
31C2×#390C2211yescharacteristic
32C2×#389—28no—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#389C2212yes—
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#388C2212yes—
40C2×#386C2214yes—
41C2×#386C2212yes—
42C2×#384C2212yes—
43C2×#381C2214yes—
44C2×#380C2211yescharacteristic
45C2×#380C2212yes—
46C2×#380C2214yes—
47C2×#380C2212yes—
48C2×#380C2214yes—
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52C22×#95C2212yes—
53#1442C2218yes—
54#1441C2214yes—
55#1441C2218yes—
56#1441C2214yes—
57#1441C2218yes—
58#1441C2214yes—
59#1440C2218yes—
60#1439C2218yes—
61#1439C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C22⋊#95C2214yes—
67C23×D4C2312yes—
68C23×D4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2312yes—
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2311yescharacteristic
75C2×#123—216no—
76C2×#123—216no—
77C2×#123—48no—
78C2×#121—24no—
79C2×#121—24no—
80C2×#121D418yes—
81C2×C4×D4—28no—
82C2×C4×D4—28no—
83C2×C4×D4—28no—
84C2×C4×D4—24no—
85C2×C4×D4—44no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—28no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4D418yescomplemented
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2318yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2318yes—
115C22×C22⋊C4C2312yes—
116C22×C42—44no—
117C22×C42D418yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#398—216no—
124#398—216no—
125#397—216no—
126#396—232no—
127#394—216no—
128#390—24no—
129#389—216no—
130#389—28no—
131#389—28no—
132#389—28no—
133#389—28no—
134#389—24no—
135#388—28no—
136#386—216no—
137#386—28no—
138#384—44no—
139#381—216no—
140#380—28no—
141#380—28no—
142#380—24no—
143#380—216no—
144#380—44no—
145#380—48no—
146C4×C4⋊C4—24no—
147C4×C22⋊C4—28no—
148C2×#95—28no—
149C2×#95C2314yes—
150C2×#95C2318yes—
151C2×#95C2314yes—
152C2×#95C2318yes—
153C2×#95C2318yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C2×#95C2314yes—
162C25C2412yes—
163C25C2411yescharacteristic
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4C2×D418yes—
175C22×D4C2×D414yes—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4C2×D418yes—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2×D414yes—
206C23×C4#4114yes—
207C23×C4#4114yes—
208C23×C4C2418yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220#123—464no—
221#121—232no—
222C4×D4—416no—
223C4×D4—432no—
224C2×C4⋊C4—232no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—216no—
241C2×C4⋊C4—216no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—48no—
256C2×C4⋊C4—416no—
257C2×C4⋊C4—432no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C2×D418yes—
263C2×C4⋊C4C2×D418yes—
264C2×C4⋊C4C2×D414yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—216no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—216no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4C2×D418yes—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—416no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42C2×D414yes—
296C2×C42C2×D414yes—
297C2×C42C2×D414yes—
298C2×C42C2×D414yes—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—216no—
305#95—28no—
306#95—28no—
307#95—216no—
308#95—216no—
309#95—48no—
310#95—44no—
311#95—48no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—22no—
325C24—24no—
326C24C22×D414yes—
327C24C22×D412yes—
328C24C22×D412yes—
329C24C22×D412yes—
330C24C2×#4112yes—
331C24C2×#4112yes—
332C24H(5,𝔽2)12yes—
333C242−1+412yes—
334C24C2511yescharacteristic
335C2×D4—216no—
336C2×D4—216no—
337C2×D4—216no—
338C2×D4—28no—
339C2×D4—48no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—48no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—216no—
359C22×C4—216no—
360C22×C4—216no—
361C22×C4—28no—
362C22×C4—216no—
363C22×C4—216no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—28no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—416no—
421C22×C4—416no—
422C22×C4—88no—
423C22×C4—88no—
424C22×C4#121116yes—
425C22×C4#121116yes—
426C22×C4C22×D412yes—
427C22×C4C22×D418yes—
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4H(5,𝔽2)14yes—
443C22×C42−1+414yes—
444C22×C42−1+414yes—
445C4⋊C4—216no—
446C4⋊C4—216no—
447C4⋊C4—28no—
448C4⋊C4—432no—
449C4⋊C4—44no—
450C4⋊C4—48no—
451C4⋊C4—44no—
452C4⋊C4—416no—
453C4⋊C4—416no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—416no—
457C4⋊C4—88no—
458C4⋊C4—88no—
459C22⋊C4—216no—
460C22⋊C4—432no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—416no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—816no—
467C42—28no—
468C42—28no—
469C42—44no—
470C42—44no—
471C42—48no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—22no—
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—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23C2×#12114yes—
512C23C2×#12114yes—
513C23#53714yes—
514C23D4214yes—
515C23#54714yes—
516C23#54714yes—
517C23#55514yes—
518C23C23×D411yescharacteristic
519C23C23×D411yescharacteristic
520C23C23×D411yescharacteristic
521C23C22×#4111yescharacteristic
522C23C22×#4111yescharacteristic
523C23C2×H(5,𝔽2)12yes—
524C23C2×H(5,𝔽2)11yescharacteristic
525C23C2×2−1+412yes—
526C23C2×2−1+412yes—
527C23C2×2−1+411yescharacteristic
528C23#58512yes—
529D4—416no—
530D4—432no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—232no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—416no—
558C2×C4—416no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—432no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—48no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—816no—
593C2×C4—816no—
594C2×C4C2×#12114yes—
595C2×C4C2×#12114yes—
596C2×C4C2×#12114yes—
597C2×C4C2×#12114yes—
598C2×C4#53414yes—
599C2×C4#53614yes—
600C2×C4#53614yes—
601C2×C4#53718yes—
602C2×C4#53918yes—
603C22—28no—
604C22—24no—
605C22—28no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—28no—
610C22—28no—
611C22—24no—
612C22—24no—
613C22—44no—
614C22—44no—
615C22—48no—
616C22—48no—
617C22—44no—
618C22—84no—
619C22—82no—
620C22—84no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22#253718yes—
625C22C22×#12111yescharacteristic
626C22C22×#12111yescharacteristic
627C22C2×#53411yescharacteristic
628C22C2×#53611yescharacteristic
629C22C2×#53611yescharacteristic
630C22C2×#53712yes—
631C22C2×#53711yescharacteristic
632C22#344911yescharacteristic
633C22C2×#53912yes—
634C22C2×D4211yescharacteristic
635C22C2×#54711yescharacteristic
636C22C2×#54711yescharacteristic
637C22C2×#55511yescharacteristic
638C22#349612yes—
639C22#349712yes—
640C22#349912yes—
641C22#350512yes—
642C22#350712yes—
643C22#350912yes—
644C22#353612yes—
645C22#353812yes—
646C22#355912yes—
647C22#356312yes—
648C4—24no—
649C4—24no—
650C4—24no—
651C4—24no—
652C4—42no—
653C4—48no—
654C4—44no—
655C4—82no—
656C4—84no—
657C4—88no—
658C4—82no—
659C4#261818yescomplemented
660C2—28no—
661C2—84no—
662C2C2×#261811yescharacteristic
663C2C2×#253711yescharacteristic
664C2#3500011yescharacteristic
665C2#3517811yescharacteristic
666C2#3531611yescharacteristic
667C2#3744211yescharacteristic
668C2#3754911yescharacteristic
669C2#3935912yes—
670C2#3937612yes—
671C2#3948712yes—
672C2#3950412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2486C2semidirect
4. C22×#389C2semidirect
5. #17070C2semidirect
6. #17069C2semidirect
7. #16821C2semidirect
8. #15149C2semidirect
9. #15131C2semidirect
10. #15129C2semidirect
11. #14927C2semidirect
12. #14686C2semidirect
13. #14339C2semidirect
14. #13872C2semidirect
93. C22×C4⋊C4D4semidirect
659. C4#2618semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#26182
C2×C4×#1212
C22×#3892
#170702
#170692
#168212
#151492
#151312
#151292
#149272
#146862
#143392
#138722