Finite Group Databasebeta

← #554205

Subgroups — #554205

Order 512.

3242 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2434C211yesmaximal, characteristic
2C2×#2329C211yesmaximal, characteristic
3C22×#391C211yesmaximal, characteristic
4#14695C212yesmaximal
5#14574C214yesmaximal, complemented
6#14524C214yesmaximal, complemented
7#14432C214yesmaximal, complemented
8#14420C214yesmaximal, complemented
9#14265C212yesmaximal
10#14159C211yesmaximal, characteristic
11#14040C212yesmaximal
12#14039C212yesmaximal
13#13966C211yesmaximal, characteristic
14#13922C211yesmaximal, characteristic
15#13796C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×C22⋊Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2434C4116yescomplemented
22C2×#395C2214yes—
23#2329—28no—
24C2×#391—28no—
25C2×#391C2212yes—
26C2×#391C2214yes—
27C2×#391C2214yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#389C2212yes—
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#387C2214yes—
35C2×#387C2214yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#380C2211yescharacteristic
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1446C2212yes—
48#1446C2214yes—
49#1445C2212yes—
50#1445C2214yes—
51#1445C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2212yes—
57#1442C2212yes—
58#1441C2214yes—
59#1441C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1434C2214yes—
66#1434C2214yes—
67#1434C2214yes—
68#1434C2214yes—
69#1434C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C23×Q8—24no—
79C23×Q8—24no—
80C23×Q8—44no—
81C23×Q8—44no—
82C23×Q8C2×C418yes—
83C23×Q8C2312yes—
84C23×Q8C2312yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8—44no—
95C2×C4⋊Q8C2×C4116yescomplemented
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8C2×C418yes—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C42—24no—
115C22×C42—44no—
116C22×C42C2314yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121#395C2×C4132yescomplemented
122#391—28no—
123#391—216no—
124#391—216no—
125#391—44no—
126#391—44no—
127#391C2×C4116yescomplemented
128#389—28no—
129#387—28no—
130#387—28no—
131#387—48no—
132#387—48no—
133#385—28no—
134#385C2×C4116yes—
135#385C2×C4116yes—
136#380—24no—
137C4×C22⋊C4—28no—
138C4×C22⋊C4—24no—
139C4×C22⋊C4C2×C418yes—
140C2×#95—24no—
141C2×#95—24no—
142C2×#95—24no—
143C2×#95D418yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C25C2411yescharacteristic
176C22×Q8—232no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—22no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—432no—
194C22×Q8—416no—
195C22×Q8—416no—
196C22×Q8—432no—
197C22×Q8C22×C418yes—
198C22×Q8C22×C418yes—
199C22×Q8C22×C414yes—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4C22×C414yes—
225C23×C4C22×C414yes—
226C23×C4C22×C414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4#4114yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2412yes—
245C4⋊Q8—432no—
246C22⋊Q8—232no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—416no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4C22×C414yes—
268C2×C4⋊C4C22×C418yes—
269C2×C4⋊C4C22×C4116yes—
270C2×C4⋊C4C22×C4116yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4C22×C418yes—
281C2×C22⋊C4C22×C418yes—
282C2×C42—28no—
283C2×C42—28no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—44no—
290C2×C42—416no—
291C2×C42—44no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—44no—
295C2×C42C22×C418yes—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95C22×C418yes—
323#95C22×C4116yes—
324#95C22×C4116yes—
325#95C2×D418yes—
326#95C2×D418yes—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24C23×C412yes—
332C24C22×D412yes—
333C24C2×#4112yes—
334C242−1+414yes—
335C242−1+414yes—
336C24C2511yescharacteristic
337C2×Q8—216no—
338C2×Q8—216no—
339C2×Q8—28no—
340C2×Q8—216no—
341C2×Q8—216no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—4128no—
354C2×Q8—82no—
355C2×Q8—82no—
356C2×Q8—84no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—216no—
360C22×C4—28no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—216no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
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—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—44no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—416no—
434C22×C4—44no—
435C22×C4—416no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4C4×D4116yes—
446C22×C4C23×C414yes—
447C22×C4C23×C418yes—
448C22×C4C23×C414yes—
449C22×C4C23×C414yes—
450C22×C4C23×C412yes—
451C22×C4C23×C412yes—
452C22×C4C23×C414yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4H(5,𝔽2)14yes—
460C22×C4H(5,𝔽2)14yes—
461C22×C42−1+414yes—
462C4⋊C4—28no—
463C4⋊C4—216no—
464C4⋊C4—232no—
465C4⋊C4—416no—
466C4⋊C4—416no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C22⋊C4—216no—
472C22⋊C4—216no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C42—48no—
476C42—416no—
477C42—84no—
478C42—84no—
479C42—84no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23C2×C4×D414yes—
493C23#51918yes—
494C23#53614yes—
495C23#53614yes—
496C23#54118yes—
497C23C24×C411yescharacteristic
498C23C23×D411yescharacteristic
499C23C22×#4111yescharacteristic
500C23C2×H(5,𝔽2)12yes—
501C23C2×H(5,𝔽2)12yes—
502C23C2×2−1+412yes—
503C23C2×2−1+412yes—
504C23C2×2−1+412yes—
505C23#58512yes—
506Q8—432no—
507Q8—432no—
508Q8—88no—
509C2×C4—232no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—416no—
543C2×C4—416no—
544C2×C4—416no—
545C2×C4—416no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—48no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—88no—
586C2×C4—88no—
587C2×C4C2×C4×D414yes—
588C2×C4C2×C4×D414yes—
589C2×C4#51818yes—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—216no—
596C22—216no—
597C22—24no—
598C22#232718yes—
599C22C22×C4×D411yescharacteristic
600C22C2×#51812yes—
601C22C2×#51912yes—
602C22C4×2−1+412yes—
603C22C2×#53611yescharacteristic
604C22C2×#53611yescharacteristic
605C22C2×#54112yes—
606C22#348411yescharacteristic
607C22#348411yescharacteristic
608C22#348511yescharacteristic
609C22#348611yescharacteristic
610C22#349012yes—
611C22#350712yes—
612C22#353814yes—
613C22#354114yes—
614C22#355214yes—
615C22#356114yes—
616C4—28no—
617C4—24no—
618C4—24no—
619C4—44no—
620C4—44no—
621C4—44no—
622C4—44no—
623C4—44no—
624C4—48no—
625C4—44no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—84no—
630C2—28no—
631C2C2×#232711yescharacteristic
632C2#3419811yescharacteristic
633C2#3426611yescharacteristic
634C2#3447514yes—
635C2#3670511yescharacteristic
636C2#3670811yescharacteristic
637C2#3681014yes—
638C2#3696611yescharacteristic
639C2#3723311yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #14574C2semidirect
6. #14524C2semidirect
7. #14432C2semidirect
8. #14420C2semidirect
21. #2434C4semidirect
95. C2×C4⋊Q8C2×C4semidirect
121. #395C2×C4semidirect
127. #391C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24342
C2×#23292
C22×#3912
#146952
#145742
#145242
#144322
#144202
#142652
#141592
#140402
#140392
#139662
#139222
#137962