Finite Group Databasebeta

← #536721

Subgroups — #536721

Order 512.

2942 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic
3C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
4#14529C214yesmaximal, complemented
5#14408C214yesmaximal, complemented
6#14188C214yesmaximal, complemented
7#14150C211yesmaximal, characteristic, complemented
8#14141C212yesmaximal
9#14072C214yesmaximal, complemented
10#14001C212yesmaximal
11#13981C212yesmaximal, complemented
12#13850C211yesmaximal, characteristic, complemented
13#13818C212yesmaximal, complemented
14#13785C211yesmaximal, characteristic
15#13779C211yesmaximal, characteristic, complemented
16C22×C4×Q8C2211yescharacteristic
17C22×C4×Q8C2212yes—
18C22×C4×D4C2211yescharacteristic
19C22×#117C2212yes—
20#2580C4116yescomplemented
21C2×#399C2212yes—
22C2×#393C2212yes—
23#2354—28no—
24Q8×C4⋊C4—28no—
25C2×#390C2211yescharacteristic
26C2×#390C2212yes—
27C2×#390C2211yescharacteristic
28C2×#390C2212yes—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#387C2214yes—
38C2×#385C2214yes—
39C2×#384C2212yes—
40C2×#384C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×#376C2211yescharacteristic
49#1446C2212yes—
50#1445C2212yes—
51#1444C2212yes—
52#1442C2212yes—
53#1442C2214yes—
54#1441C2212yes—
55#1441C2212yes—
56#1441C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2211yescharacteristic
74C22⋊#95C2214yes—
75C4×#95C2212yes—
76C4×#95C2212yes—
77C4×#95C2214yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C2×C4×Q8—28no—
82C2×C4×Q8—24no—
83C2×C4×Q8—24no—
84C2×C4×Q8—48no—
85C2×C4×Q8C2×C418yes—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4C2×C418yes—
89C2×#117—28no—
90C2×#117—48no—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2314yes—
116C22×C42C2314yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2314yes—
121C22×C42C2311yescharacteristic
122#399C2×C4116yes—
123#393C2×C4116yes—
124#390—28no—
125#390—24no—
126#390—24no—
127#390C2×C4116yes—
128#389—216no—
129#389—28no—
130#389—28no—
131#389C2×C4116yes—
132#388—216no—
133#388—48no—
134#388—48no—
135#388—48no—
136#387—48no—
137#385C2×C4132yes—
138#384—28no—
139#384D418yes—
140#382C2×C418yes—
141#380—28no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—28no—
147#376—24no—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C25C2412yes—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8C22×C414yes—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4C22×C414yes—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4C22×C418yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C4×Q8—232no—
206C4×Q8—432no—
207C4×D4—232no—
208#117—432no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4C22×C414yes—
242C2×C4⋊C4C22×C418yes—
243C2×C4⋊C4C22×C414yes—
244C2×C4⋊C4C22×C418yes—
245C2×C4⋊C4C22×C4116yes—
246C2×C4⋊C4C2×D418yes—
247C2×C4⋊C4C2×D418yes—
248C2×C4⋊C4C2×D418yes—
249C2×C4⋊C4#4114yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
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⋊C4—44no—
268C2×C22⋊C4C22×C418yes—
269C2×C22⋊C4C22×C4116yes—
270C2×C22⋊C4C22×C4116yes—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—28no—
276C2×C42—28no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—28no—
283C2×C42—28no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—28no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42C22×C418yes—
293C2×C42C22×C414yes—
294C2×C42C2×D414yes—
295C2×C42#4114yes—
296C2×C42#4114yes—
297#95—28no—
298#95—24no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—42no—
313#95—44no—
314#95—42no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95C22×C418yes—
320#95C22×C418yes—
321#95C22×C4116yes—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—22no—
330C24—22no—
331C24—24no—
332C24C23×C414yes—
333C24C2511yescharacteristic
334C2×Q8—28no—
335C2×Q8—28no—
336C2×Q8—216no—
337C2×Q8—28no—
338C2×Q8—216no—
339C2×D4—28no—
340C2×D4—28no—
341C2×D4—28no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—216no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—416no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—416no—
416C22×C4C23×C418yes—
417C22×C4C23×C414yes—
418C22×C4C23×C414yes—
419C22×C4C23×C412yes—
420C22×C4C23×C412yes—
421C22×C4C23×C414yes—
422C22×C4C23×C412yes—
423C22×C4C22×D418yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C2×#4114yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4H(5,𝔽2)14yes—
435C22×C4H(5,𝔽2)14yes—
436C22×C42−1+414yes—
437C22×C42−1+412yes—
438C22×C42−1+412yes—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—216no—
442C4⋊C4—28no—
443C4⋊C4—216no—
444C4⋊C4—216no—
445C4⋊C4—28no—
446C4⋊C4—416no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—416no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—88no—
453C4⋊C4—88no—
454C22⋊C4—216no—
455C22⋊C4—232no—
456C22⋊C4—416no—
457C22⋊C4—416no—
458C22⋊C4—88no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C42—216no—
462C42—28no—
463C42—416no—
464C42—416no—
465C42—48no—
466C42—48no—
467C42—44no—
468C42—88no—
469C42C4×D4116yes—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—28no—
474C23—28no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23C24×C411yescharacteristic
494C23C23×D411yescharacteristic
495C23C22×#4111yescharacteristic
496C23C22×#4111yescharacteristic
497C23C22×#4111yescharacteristic
498C23C2×H(5,𝔽2)12yes—
499C23C2×H(5,𝔽2)12yes—
500C23C2×2−1+411yescharacteristic
501C23C2×2−1+411yescharacteristic
502C23C2×2−1+412yes—
503C23#58512yes—
504Q8—232no—
505D4—416no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—432no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4C2×C4×D414yes—
586C2×C4C2×C4×D414yes—
587C2×C4C2×C4×D414yes—
588C2×C4C4×#4114yes—
589C2×C4#51914yes—
590C2×C4#52018yes—
591C2×C4#54714yes—
592C2×C4#54814yes—
593C2×C4#55214yes—
594C2×C4#55614yes—
595C22—48no—
596C22—44no—
597C22—44no—
598C22—48no—
599C22—44no—
600C22—82no—
601C22—84no—
602C22—84no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22C22×C4×D411yescharacteristic
607C22C2×C4×#4111yescharacteristic
608C22C2×#51911yescharacteristic
609C22C2×#52012yes—
610C22C4×H(5,𝔽2)12yes—
611C22C2×#54711yescharacteristic
612C22C2×#54811yescharacteristic
613C22C2×#55211yescharacteristic
614C22C2×#55611yescharacteristic
615C22#348411yescharacteristic
616C22#348411yescharacteristic
617C22#348611yescharacteristic
618C22#348711yescharacteristic
619C22#349212yes—
620C22#350612yes—
621C22#350712yes—
622C22#350812yes—
623C22#351412yes—
624C22#352212yes—
625C22#354412yes—
626C22#354712yes—
627C22#354812yes—
628C22#355512yes—
629C4—24no—
630C4—28no—
631C4—24no—
632C4—48no—
633C4—44no—
634C4—48no—
635C4—44no—
636C4—44no—
637C4—44no—
638C4—44no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4#236218yes—
643C2—84no—
644C2C2×#236211yescharacteristic
645C2#3420011yescharacteristic
646C2#3426511yescharacteristic
647C2#3433112yes—
648C2#3448412yes—
649C2#4490512yes—
650C2#4491112yes—
651C2#4493111yescharacteristic
652C2#4493711yescharacteristic
653C2#4493811yescharacteristic
654C2#4539411yescharacteristic
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
3. C2×Q8×C4⋊C4C2semidirect
4. #14529C2semidirect
5. #14408C2semidirect
6. #14188C2semidirect
7. #14150C2semidirect
9. #14072C2semidirect
11. #13981C2semidirect
12. #13850C2semidirect
13. #13818C2semidirect
15. #13779C2semidirect
20. #2580C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
C2×#23542
C2×Q8×C4⋊C42
#145292
#144082
#141882
#141502
#141412
#140722
#140012
#139812
#138502
#138182
#137852
#137792