Finite Group Databasebeta

← #917554

Subgroups — #917554

Order 512.

2814 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C211yesmaximal, characteristic
2C22×#395C211yesmaximal, characteristic
3#17144C212yesmaximal, complemented
4#17143C212yesmaximal, complemented
5#17032C212yesmaximal, complemented
6#17004C212yesmaximal, complemented
7#16617C214yesmaximal, complemented
8#15862C214yesmaximal, complemented
9#15073C212yesmaximal
10#14935C212yesmaximal
11#14919C212yesmaximal
12#14336C212yesmaximal
13#13972C212yesmaximal
14#13904C211yesmaximal, characteristic
15#13870C212yesmaximal
16C24×Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20#2484—28no—
21C2×#400C2211yescharacteristic
22C2×#400C2211yescharacteristic
23C2×#398C2211yescharacteristic
24C2×#398C2211yescharacteristic
25C2×#395—28no—
26C2×#395C2212yes—
27C2×#395C2212yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#387C2212yes—
31C2×#387C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#381C2212yes—
35C2×#381C2212yes—
36C2×#381C2211yescharacteristic
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#381C2212yes—
42C2×#380C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2212yes—
46#1446C2214yes—
47#1446C2214yes—
48#1446C2214yes—
49#1446C2214yes—
50#1445C2214yes—
51#1445C2212yes—
52#1445C2212yes—
53#1443C2214yes—
54#1443C2214yes—
55#1443C2214yes—
56#1443C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73#1434C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8—24no—
78C23×Q8—28no—
79C23×Q8—42no—
80C23×Q8—44no—
81C23×Q8—42no—
82C23×Q8C2312yes—
83C23×Q8C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C2×#117—24no—
90C2×#117—42no—
91C2×#117—42no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4D418yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116#400—24no—
117#400—24no—
118#398—24no—
119#398—24no—
120#395—28no—
121#395—28no—
122#395—28no—
123#395—28no—
124#387—28no—
125#387—28no—
126#385—28no—
127#385—44no—
128#381—24no—
129#381—28no—
130#381—28no—
131#381—44no—
132#381—44no—
133#381—44no—
134#381—44no—
135#381—44no—
136#380—42no—
137C4×C22⋊C4—42no—
138C2×#95—28no—
139C2×#95—24no—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2311yescharacteristic
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
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×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C25C2411yescharacteristic
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—28no—
178C22×Q8—28no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—432no—
186C22×Q8—416no—
187C22×Q8—416no—
188C22×Q8—432no—
189C22×Q8—416no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4#4118yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2411yescharacteristic
231#117—416no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—416no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—432no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D414yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
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—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—28no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—22no—
296#95—22no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—42no—
308#95—42no—
309#95—44no—
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—
320C24—22no—
321C24—24no—
322C24—22no—
323C24C22×D412yes—
324C24C2×#4114yes—
325C24H(5,𝔽2)12yes—
326C24H(5,𝔽2)12yes—
327C24H(5,𝔽2)14yes—
328C24C2511yescharacteristic
329C2×Q8—416no—
330C2×Q8—416no—
331C2×Q8—48no—
332C2×Q8—48no—
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—416no—
336C2×Q8—416no—
337C2×Q8—4128no—
338C2×Q8—48no—
339C2×Q8—48no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—216no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—216no—
348C22×C4—216no—
349C22×C4—216no—
350C22×C4—216no—
351C22×C4—216no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—28no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
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—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C2×#4118yes—
443C22×C42−1+418yes—
444C22×C42−1+414yes—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—416no—
450C4⋊C4—416no—
451C4⋊C4—48no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—84no—
456C4⋊C4—88no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C42—48no—
462C42—84no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—28no—
467C23—28no—
468C23—24no—
469C23—24no—
470C23—28no—
471C23—28no—
472C23—24no—
473C23—24no—
474C23#53414yes—
475C23#53818yes—
476C23#54618yes—
477C23#55114yes—
478C23#56014yes—
479C23C23×D411yescharacteristic
480C23C22×#4112yes—
481C23C2×H(5,𝔽2)11yescharacteristic
482C23C2×H(5,𝔽2)11yescharacteristic
483C23C2×H(5,𝔽2)12yes—
484C23C2×2−1+412yes—
485C23C2×2−1+414yes—
486C23#58512yes—
487Q8—432no—
488Q8—432no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—416no—
511C2×C4—432no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—416no—
522C2×C4—416no—
523C2×C4—432no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
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#53614yes—
572C22—28no—
573C22—24no—
574C22—216no—
575C22—24no—
576C22—28no—
577C22—28no—
578C22—28no—
579C22#269718yes—
580C22C2×#53411yescharacteristic
581C22C2×#53611yescharacteristic
582C22#344911yescharacteristic
583C22C2×#53812yes—
584C22C2×#54612yes—
585C22C2×#55111yescharacteristic
586C22#348612yes—
587C22#349514yes—
588C22#350914yes—
589C22C2×#56011yescharacteristic
590C22#353512yes—
591C22#353512yes—
592C22#353514yes—
593C22#354414yes—
594C22#354412yes—
595C22#354412yes—
596C4—24no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—48no—
603C4—48no—
604C4—82no—
605C4—82no—
606C4—84no—
607C4—82no—
608C4—84no—
609C4—82no—
610C2—28no—
611C2#3725811yescharacteristic
612C2#3744711yescharacteristic
613C2#3784112yes—
614C2#3946512yes—
615C2#4231914yes—
616C2#4479512yes—
617C2C2×#269711yescharacteristic
618C2#5291312yes—
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
3. #17144C2semidirect
4. #17143C2semidirect
5. #17032C2semidirect
6. #17004C2semidirect
7. #16617C2semidirect
8. #15862C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
C22×#3952
#171442
#171432
#170322
#170042
#166172
#158622
#150732
#149352
#149192
#143362
#139722
#139042
#138702