Finite Group Databasebeta

← #618039

Subgroups — #618039

Order 512.

3710 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic
2C2×#2584C212yesmaximal, complemented
3C2×#2629C211yesmaximal, characteristic
4#15848C214yesmaximal, complemented
5#15614C214yesmaximal
6#15318C214yesmaximal, complemented
7#15276C214yesmaximal
8#15202C212yesmaximal, complemented
9#15158C212yesmaximal, complemented
10#14697C212yesmaximal
11#14264C212yesmaximal
12#13926C212yesmaximal, complemented
13#13777C211yesmaximal, characteristic
14C2×#533—216no—
15C2×#533—28no—
16C22×C4⋊Q8C2212yes—
17C22×#125C2211yescharacteristic
18C22×#125C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C4×Q8C2212yes—
22C22×C4×Q8C2212yes—
23C22×#117C2211yescharacteristic
24C22×#117C2211yescharacteristic
25C23×C42C2211yescharacteristic
26#2629—28no—
27#2584—216no—
28C2×#398C2214yes—
29C2×#398C2214yes—
30C2×#396C2214yes—
31C2×#395C2214yes—
32C2×#395C2214yes—
33C2×#391C2214yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2214yes—
40C2×#389C2214yes—
41C2×#385C2214yes—
42C2×#384C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2212yes—
46#1446C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1444C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1437C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2214yes—
70C22⋊#95C2214yes—
71C23×Q8C2312yes—
72C23×Q8C2312yes—
73C24×C4C2312yes—
74C24×C4C2311yescharacteristic
75#533—2128no—
76C2×C4⋊Q8—28no—
77C2×C4⋊Q8—28no—
78C2×C4⋊Q8—28no—
79C2×#125—28no—
80C2×#125—28no—
81C2×#125—28no—
82C2×#125—28no—
83C2×#125—24no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C4×Q8—28no—
91C2×C4×Q8—28no—
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8D4116yes—
96C2×#117—24no—
97C2×#117—28no—
98C2×#117—28no—
99C2×#117D418yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C42—24no—
125C22×C42—28no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2314yes—
132C22×C42C2311yescharacteristic
133#398—216no—
134#398—216no—
135#396—216no—
136#395—216no—
137#395—48no—
138#391—48no—
139#389—216no—
140#389—216no—
141#389—28no—
142#389—28no—
143#389—28no—
144#389—28no—
145#389—28no—
146#385—48no—
147#384—24no—
148#380—28no—
149#380Q8116yescomplemented
150C4×C4⋊C4—28no—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C25C2411yescharacteristic
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—42no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8C2×D418yes—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4C2×D414yes—
195C23×C4C2×Q818yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C4⋊Q8—264no—
209#125—264no—
210#125—416no—
211C22⋊Q8—264no—
212C22⋊Q8—264no—
213C4×Q8—264no—
214C4×Q8—264no—
215#117—232no—
216#117—232no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×Q818yes—
273C2×C4⋊C4C2×Q818yes—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4C2×D418yes—
286C2×C22⋊C4C2×Q8116yes—
287C2×C42—232no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—28no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42C2×D418yes—
301C2×C42C2×D414yes—
302C2×C42C2×D418yes—
303C2×C42C2×D418yes—
304C2×C42C2×D414yes—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
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#95C2×Q8116yes—
323C24—22no—
324C24—22no—
325C24—21no—
326C24—21no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24C22×D412yes—
331C24C22×Q814yes—
332C24C2511yescharacteristic
333C2×Q8—216no—
334C2×Q8—216no—
335C2×Q8—216no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—216no—
361C22×C4—216no—
362C22×C4—216no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
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—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
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—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4C4⋊Q8116yes—
420C22×C4C22×D414yes—
421C22×C4C22×D414yes—
422C22×C4C22×D414yes—
423C22×C4C22×D414yes—
424C22×C4C22×D414yes—
425C22×C4C22×D414yes—
426C22×C4C22×D414yes—
427C22×C4C22×D412yes—
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4C22×D414yes—
431C22×C4C22×Q818yes—
432C22×C4C22×Q818yes—
433C22×C4C22×Q814yes—
434C22×C4C22×Q814yes—
435C22×C4H(5,𝔽2)14yes—
436C22×C42−1+414yes—
437C22×C42−1+412yes—
438C22×C42−1+412yes—
439C4⋊C4—216no—
440C4⋊C4—216no—
441C4⋊C4—216no—
442C4⋊C4—216no—
443C4⋊C4—216no—
444C4⋊C4—216no—
445C4⋊C4—216no—
446C4⋊C4—216no—
447C4⋊C4—216no—
448C4⋊C4—216no—
449C4⋊C4—416no—
450C4⋊C4—416no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—416no—
454C4⋊C4—88no—
455C4⋊C4—88no—
456C4⋊C4—88no—
457C4⋊C4—88no—
458C22⋊C4—28no—
459C22⋊C4—28no—
460C22⋊C4—216no—
461C22⋊C4—416no—
462C22⋊C4—88no—
463C42—28no—
464C42—28no—
465C42—216no—
466C42—216no—
467C42—28no—
468C42—88no—
469C42#120116yes—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23C2×C4⋊Q814yes—
491C23C23×D411yescharacteristic
492C23C23×D412yes—
493C23C23×Q812yes—
494C23C2×H(5,𝔽2)12yes—
495C23C2×2−1+411yescharacteristic
496C23C2×2−1+411yescharacteristic
497C23C2×2−1+412yes—
498C23#58512yes—
499C23#58512yes—
500Q8—432no—
501Q8—432no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—232no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—432no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
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—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4C2×#12018yes—
577C2×C4C2×#12014yes—
578C2×C4C2×C4⋊Q814yes—
579C2×C4C2×C4⋊Q814yes—
580C2×C4#53614yes—
581C2×C4#53718yes—
582C2×C4D4×Q818yes—
583C2×C4D4×Q818yes—
584C22—24no—
585C22—28no—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22C22×#12011yescharacteristic
594C22C22×C4⋊Q811yescharacteristic
595C22C2×#53611yescharacteristic
596C22C2×#53712yes—
597C22#344912yes—
598C22C2×D4×Q812yes—
599C22C2×D4×Q812yes—
600C22#348812yes—
601C22#348812yes—
602C22#348912yes—
603C22#348912yes—
604C22#350112yes—
605C22#350212yes—
606C22#350312yes—
607C22#350412yes—
608C22#354811yescharacteristic
609C22#355412yes—
610C22#355512yes—
611C22#355611yescharacteristic
612C22#355711yescharacteristic
613C22#355711yescharacteristic
614C4—24no—
615C4—28no—
616C4—28no—
617C4—48no—
618C4—48no—
619C4—44no—
620C4—44no—
621C4—44no—
622C4—44no—
623C4—84no—
624C4—84no—
625C4#243418yes—
626C2—44no—
627C2C2×#243411yescharacteristic
628C2#3525112yes—
629C2#3556612yes—
630C2#3727411yescharacteristic
631C2#3737711yescharacteristic
632C2#3959012yes—
633C2#3959112yes—
634C2#3959812yes—
635C2#3959912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2584C2semidirect
4. #15848C2semidirect
6. #15318C2semidirect
8. #15202C2semidirect
9. #15158C2semidirect
12. #13926C2semidirect
149. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5332
C2×#25842
C2×#26292
#158482
#156142
#153182
#152762
#152022
#151582
#146972
#142642
#139262
#137772