Finite Group Databasebeta

← #900779

Subgroups — #900779

Order 512.

2766 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic
2C2×#2717C212yesmaximal
3C2×C4×#124C211yesmaximal, characteristic
4#17078C212yesmaximal, complemented
5#16688C212yesmaximal, complemented
6#15298C214yesmaximal, complemented
7#15290C214yesmaximal, complemented
8#14880C212yesmaximal, complemented
9#14830C214yesmaximal, complemented
10#14699C212yesmaximal, complemented
11#14440C211yesmaximal, characteristic
12#14004C214yesmaximal, complemented
13#13968C212yesmaximal
14C22×#124C2211yescharacteristic
15C22×#123C2212yes—
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#2724—28no—
19#2717—216no—
20C2×#398C2212yes—
21C2×#398C2212yes—
22C2×#397C2214yes—
23C2×#397C2214yes—
24C2×#397C2214yes—
25C2×#395C2214yes—
26C4×#124—28no—
27C2×#391C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#388C2214yes—
33C2×#387C2214yes—
34C2×#386C2212yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#382C2211yescharacteristic
39C2×#381C2212yes—
40C2×#380C2214yes—
41C2×C4×C22⋊C4C2214yes—
42C22×#95C2212yes—
43C2×C43C2211yescharacteristic
44#1445C2214yes—
45#1444C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2212yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65#1434C2212yes—
66#1434C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2312yes—
74C24×C4C2312yes—
75C2×#124—24no—
76C2×#124—24no—
77C2×#124D418yes—
78C2×#123—28no—
79C2×#123—28no—
80C2×#123—28no—
81C2×C4×Q8—24no—
82C2×C4×Q8—42no—
83C2×C4×Q8—42no—
84C2×C4×D4—24no—
85C2×C4×D4—42no—
86C2×C4×D4—42no—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113#398—28no—
114#398—44no—
115#397—216no—
116#397—216no—
117#397—216no—
118#395—216no—
119#391—24no—
120#390D418yes—
121#389—28no—
122#389—28no—
123#389—44no—
124#388—216no—
125#387—216no—
126#386—28no—
127#385—216no—
128#385—216no—
129#385—216no—
130#382—24no—
131#381—44no—
132#380—216no—
133C4×C22⋊C4—216no—
134C2×#95—44no—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
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×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C43—24no—
158C25C2412yes—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—41no—
162C22×Q8—41no—
163C22×Q8—41no—
164C22×Q8—41no—
165C22×Q8C2×D414yes—
166C22×D4—41no—
167C22×D4—41no—
168C22×D4—41no—
169C22×D4—41no—
170C22×D4C2×D414yes—
171C22×D4C2×D414yes—
172C22×D4#4114yes—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—28no—
179C23×C4—24no—
180C23×C4—28no—
181C23×C4—42no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202#124—232no—
203#123—432no—
204C4×Q8—416no—
205C4×D4—416no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4#4118yes—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4C2×D4116yes—
264C2×C22⋊C4C2×D4116yes—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—42no—
278C2×C42—41no—
279C2×C42—41no—
280C2×C42—42no—
281C2×C42C2×D414yes—
282C2×C42C2×D414yes—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—42no—
305C24—24no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—22no—
312C24—22no—
313C24C22×D414yes—
314C24C22×D414yes—
315C24C2×#4114yes—
316C24C2511yescharacteristic
317C2×Q8—28no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4#121116yes—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—416no—
397C22×C4—416no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—84no—
403C22×C4—88no—
404C22×C4—88no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4C22×D414yes—
412C22×C4C22×D414yes—
413C22×C4C22×D414yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D414yes—
417C22×C4C2×#4118yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4114yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C42−1+414yes—
423C22×C42−1+414yes—
424C4⋊C4—44no—
425C4⋊C4—48no—
426C4⋊C4—416no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—44no—
430C4⋊C4—416no—
431C4⋊C4—48no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C4⋊C4—88no—
436C4⋊C4—88no—
437C22⋊C4—232no—
438C22⋊C4—232no—
439C22⋊C4—416no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—416no—
443C22⋊C4—84no—
444C42—28no—
445C42—28no—
446C42—44no—
447C42—48no—
448C42—48no—
449C42—48no—
450C23—24no—
451C23—24no—
452C23—28no—
453C23—28no—
454C23—28no—
455C23—24no—
456C23—24no—
457C23—28no—
458C23—24no—
459C23—24no—
460C23—28no—
461C23—28no—
462C23—44no—
463C23—42no—
464C23—44no—
465C23—44no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23C2×#12118yes—
473C23C23×D411yescharacteristic
474C23C23×D411yescharacteristic
475C23C22×#4111yescharacteristic
476C23C22×#4112yes—
477C23C2×H(5,𝔽2)11yescharacteristic
478C23C2×H(5,𝔽2)11yescharacteristic
479C23C2×2−1+412yes—
480C23C2×2−1+412yes—
481C23#58511yescharacteristic
482C23#58511yescharacteristic
483C23#58512yes—
484Q8—416no—
485D4—416no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—816no—
560C2×C4—816no—
561C2×C4—816no—
562C2×C4C2×#12114yes—
563C2×C4#54718yes—
564C2×C4#56314yes—
565C22—216no—
566C22—28no—
567C22—48no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—48no—
573C22—48no—
574C22—44no—
575C22—44no—
576C22C22×#12111yescharacteristic
577C22#345911yescharacteristic
578C22#345911yescharacteristic
579C22#346112yes—
580C22C2×#54712yes—
581C22D4×#4112yes—
582C22#348111yescharacteristic
583C22#348212yes—
584C22#348411yescharacteristic
585C22#348511yescharacteristic
586C22#348912yes—
587C22#349112yes—
588C22#349611yescharacteristic
589C22#349611yescharacteristic
590C22#349912yes—
591C22#351512yes—
592C22#352212yes—
593C22C2×#56311yescharacteristic
594C22#354511yescharacteristic
595C22#355011yescharacteristic
596C22#356112yes—
597C22#356312yes—
598C22#356412yes—
599C4—24no—
600C4—28no—
601C4—24no—
602C4—44no—
603C4—44no—
604C4—48no—
605C4—44no—
606C4—42no—
607C4—84no—
608C4—82no—
609C4—84no—
610C4—84no—
611C2—48no—
612C2#3498411yescharacteristic
613C2#3625912yes—
614C2#3716211yescharacteristic
615C2#3776312yes—
616C2#3878812yes—
617C2#3944612yes—
618C2#4613611yescharacteristic
619C2#4614711yescharacteristic
620C2#5230411yescharacteristic
621C2#5907011yescharacteristic
622C2#5983611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17078C2semidirect
5. #16688C2semidirect
6. #15298C2semidirect
7. #15290C2semidirect
8. #14880C2semidirect
9. #14830C2semidirect
10. #14699C2semidirect
12. #14004C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#27172
C2×C4×#1242
#170782
#166882
#152982
#152902
#148802
#148302
#146992
#144402
#140042
#139682