Finite Group Databasebeta

← #1016772

Subgroups — #1016772

Order 512.

2258 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2763C212yesmaximal
2#17219C212yesmaximal, complemented
3#16481C212yesmaximal
4#16389C214yesmaximal, complemented
5#16096C212yesmaximal, complemented
6#16039C214yesmaximal
7#15158C212yesmaximal, complemented
8#15139C212yesmaximal, complemented
9#15125C211yesmaximal, characteristic
10#14130C212yesmaximal
11#14130C211yesmaximal, characteristic
12#14053C212yesmaximal, complemented
13#13974C212yesmaximal
14#13918C212yesmaximal, complemented
15C4×#393C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#2763—216no—
20C2×#400C2214yes—
21C2×#395C2214yes—
22C2×#393C2212yes—
23C2×#393C2211yescharacteristic
24C2×#391C2214yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#385C2212yes—
31C2×#385C2214yes—
32C2×#382C2214yes—
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38#1446C2212yes—
39#1446C2214yes—
40#1446C2211yescharacteristic
41#1444C2214yes—
42#1444C2212yes—
43#1443C2214yes—
44#1443C2214yes—
45#1442C2212yes—
46#1442C2212yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2212yes—
50#1442C2211yescharacteristic
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1440C2214yes—
59#1439C2212yes—
60#1439C2211yescharacteristic
61#1439C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2212yes—
74C4×#95C2212yes—
75C4×#95C2211yescharacteristic
76C23×Q8C2312yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—42no—
82C2×C4×Q8—44no—
83C2×C4×Q8—42no—
84C22×C4⋊C4—44no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2314yes—
109#400—216no—
110#395—216no—
111#393—28no—
112#393D418yes—
113#391—216no—
114#389—44no—
115#389—44no—
116#389—44no—
117#389—44no—
118#388—216no—
119#385—216no—
120#385—28no—
121#382—216no—
122#380—28no—
123#380Q818yescomplemented
124C4×C4⋊C4—28no—
125C4×C22⋊C4—42no—
126C4×C22⋊C4—44no—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C25C2411yescharacteristic
160C22×Q8—24no—
161C22×Q8—28no—
162C22×Q8—42no—
163C22×Q8—41no—
164C22×Q8—41no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8—44no—
168C22×Q8—42no—
169C22×Q8C2×D414yes—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—44no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—44no—
179C23×C4C2×Q814yes—
180C23×C4C2414yes—
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2411yescharacteristic
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2414yes—
190C23×C4C2411yescharacteristic
191C23×C4C2414yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C4⋊Q8—416no—
195C4×Q8—416no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—416no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4C2×Q818yes—
229C2×C4⋊C4#4118yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4C2×D418yes—
246C2×C22⋊C4C2×D414yes—
247C2×C22⋊C4C2×Q818yes—
248C2×C42—28no—
249C2×C42—24no—
250C2×C42—28no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—44no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—24no—
268#95—28no—
269#95—28no—
270#95—24no—
271#95—24no—
272#95—42no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—42no—
278#95—41no—
279#95—44no—
280#95—42no—
281#95—44no—
282#95—41no—
283#95—42no—
284#95—42no—
285#95—41no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95C2×D414yes—
291#95C2×D418yes—
292#95C2×Q818yes—
293C24—22no—
294C24—21no—
295C24—22no—
296C24—21no—
297C24—21no—
298C24—22no—
299C24—21no—
300C24—22no—
301C24C22×D412yes—
302C24C22×Q812yes—
303C242−1+412yes—
304C24C2511yescharacteristic
305C2×Q8—48no—
306C2×Q8—44no—
307C2×Q8—48no—
308C2×Q8—416no—
309C2×Q8—44no—
310C2×Q8—48no—
311C2×Q8—44no—
312C2×Q8—84no—
313C2×Q8—88no—
314C2×Q8—84no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—28no—
356C22×C4—216no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—48no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—48no—
394C22×C4—88no—
395C22×C4—84no—
396C22×C4—88no—
397C22×C4—84no—
398C22×C4C22×D412yes—
399C22×C4C22×D414yes—
400C22×C4C22×D412yes—
401C22×C4C22×D414yes—
402C22×C4C22×Q812yes—
403C22×C4C22×Q818yes—
404C22×C4C22×Q812yes—
405C22×C4C2×#4118yes—
406C22×C4C2×#4114yes—
407C22×C4H(5,𝔽2)14yes—
408C22×C4H(5,𝔽2)12yes—
409C22×C42−1+412yes—
410C22×C42−1+412yes—
411C4⋊C4—216no—
412C4⋊C4—216no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—88no—
420C4⋊C4—88no—
421C4⋊C4—88no—
422C4⋊C4—88no—
423C22⋊C4—216no—
424C22⋊C4—48no—
425C22⋊C4—44no—
426C22⋊C4—416no—
427C22⋊C4—44no—
428C22⋊C4—48no—
429C42—416no—
430C42—48no—
431C42—44no—
432C42—48no—
433C42—44no—
434C23—22no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—24no—
442C23—24no—
443C23—22no—
444C23—24no—
445C23—22no—
446C23—42no—
447C23—41no—
448C23—42no—
449C23—42no—
450C23—41no—
451C23—42no—
452C23—41no—
453C23—42no—
454C23—41no—
455C23—42no—
456C23D4×Q814yes—
457C23C23×D411yescharacteristic
458C23C23×Q811yescharacteristic
459C23C22×#4112yes—
460C23C2×H(5,𝔽2)12yes—
461C23C2×H(5,𝔽2)11yescharacteristic
462C23C2×2−1+411yescharacteristic
463C23C2×2−1+411yescharacteristic
464C23C2×2−1+411yescharacteristic
465C23#58512yes—
466C23#58511yescharacteristic
467C23#58512yes—
468Q8—416no—
469Q8—816no—
470C2×C4—22no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—416no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—816no—
553C2×C4#54218yes—
554C2×C4#54414yes—
555C2×C4#55114yes—
556C2×C4#55418yes—
557C2×C4H(3,𝔽4)14yes—
558C22—28no—
559C22—24no—
560C22—42no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—44no—
567C22—42no—
568C22—44no—
569C22C2×#54212yes—
570C22C2×#54411yescharacteristic
571C22C2×D4×Q811yescharacteristic
572C22D4×#4112yes—
573C22C2×#55111yescharacteristic
574C22C2×#55412yes—
575C22#349612yes—
576C22#349611yescharacteristic
577C22#349911yescharacteristic
578C22#350211yescharacteristic
579C22#350412yes—
580C22#351612yes—
581C22#351612yes—
582C22#352112yes—
583C22C2×H(3,𝔽4)11yescharacteristic
584C22#353811yescharacteristic
585C22#354012yes—
586C22#354011yescharacteristic
587C22#354112yes—
588C22#354712yes—
589C22#355111yescharacteristic
590C22#356311yescharacteristic
591C22#356512yes—
592C4—42no—
593C4—44no—
594C4—42no—
595C4—48no—
596C4—48no—
597C4—42no—
598C4—44no—
599C4—42no—
600C4—81no—
601C4—82no—
602C4—84no—
603C4—84no—
604C4—82no—
605C4#276318yescomplemented
606C2—44no—
607C2#3936811yescharacteristic
608C2#3939812yes—
609C2#4274312yes—
610C2#4295511yescharacteristic
611C2#4541512yes—
612C2#4542011yescharacteristic
613C2#4547911yescharacteristic
614C2C2×#276311yescharacteristic
615C2#5300511yescharacteristic
616C2#5983112yes—
617C2#5984311yescharacteristic
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. #17219C2semidirect
4. #16389C2semidirect
5. #16096C2semidirect
7. #15158C2semidirect
8. #15139C2semidirect
12. #14053C2semidirect
14. #13918C2semidirect
123. #380Q8semidirect
605. C4#2763semidirect

Maximal subgroups

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

subgroupindex
C2×#27632
#172192
#164812
#163892
#160962
#160392
#151582
#151392
#151252
#141302
#140532
#139742
#139182
C4×#3932