Finite Group Databasebeta

← #880160

Subgroups — #880160

Order 512.

2694 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16599C214yesmaximal, complemented
2#15331C214yesmaximal, complemented
3#15329C214yesmaximal, complemented
4#15229C212yesmaximal, complemented
5#15157C212yesmaximal, complemented
6#15146C212yesmaximal, complemented
7#14713C211yesmaximal, characteristic
8#14709C211yesmaximal, characteristic, complemented
9#14513C214yesmaximal, complemented
10#14025C212yesmaximal
11#14018C212yesmaximal, complemented
12#13979C212yesmaximal, complemented
13C4×#391C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4C2212yes—
18C2×#400C2214yes—
19C2×#393C2214yes—
20C2×#391C2212yes—
21C2×#390C2212yes—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#390C2214yes—
26C2×#390C2214yes—
27C2×#390C2212yes—
28C2×#389C2212yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#385C2214yes—
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C43C2211yescharacteristic
36#1442C2214yes—
37#1442C2214yes—
38#1441C2214yes—
39#1439C2214yes—
40#1439C2214yes—
41#1439C2214yes—
42#1439C2214yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1438C2214yes—
51#1438C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2212yes—
55#1436C2212yes—
56C4⋊#95C2212yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C22⋊#95C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C4×#95C2212yes—
66C4×#95C2212yes—
67C23×Q8C2311yescharacteristic
68C23×D4C2312yes—
69C23×D4C2314yes—
70C24×C4C2314yes—
71C2×#127—42no—
72C2×#127—42no—
73C2×#127Q818yescomplemented
74C2×#125—24no—
75C2×#125—42no—
76C2×#125—42no—
77C2×C4×Q8—24no—
78C2×C4×Q8—42no—
79C2×C4×Q8—42no—
80C2×C4×D4—44no—
81C2×C4×D4—44no—
82C2×C4×D4—44no—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C42C2312yes—
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2312yes—
110#400—48no—
111#393—48no—
112#391—28no—
113#390—28no—
114#390—28no—
115#390—48no—
116#390—44no—
117#390—44no—
118#390—44no—
119#390—48no—
120#389—44no—
121#386—216no—
122#386—48no—
123#385—48no—
124#384—42no—
125#382—42no—
126C4×C4⋊C4—42no—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2311yescharacteristic
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149C43—42no—
150C25C2414yes—
151C22×Q8—22no—
152C22×Q8—24no—
153C22×Q8—24no—
154C22×Q8—41no—
155C22×Q8—41no—
156C22×D4—24no—
157C22×D4—24no—
158C22×D4—28no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—28no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4C2×Q818yes—
169C22×D4C2×Q8116yes—
170C23×C4—28no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189#127—416no—
190#125—416no—
191C4×Q8—416no—
192C4×D4—432no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
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—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42C2×Q814yes—
272C2×C42#4114yes—
273C2×C42#4114yes—
274#95—28no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—28no—
279#95—24no—
280#95—28no—
281#95—24no—
282#95—42no—
283#95—42no—
284#95—44no—
285#95—41no—
286#95—41no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95#4118yes—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24C22×Q818yes—
311C24C2511yescharacteristic
312C2×Q8—28no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×D4—216no—
318C2×D4—216no—
319C2×D4—216no—
320C2×D4—232no—
321C2×D4—416no—
322C2×D4—416no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—88no—
328C2×D4—88no—
329C2×D4—88no—
330C2×D4—88no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
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—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—416no—
396C22×C4—88no—
397C22×C4—88no—
398C22×C4—88no—
399C22×C4—88no—
400C22×C4—88no—
401C22×C4—88no—
402C22×C4C22×Q814yes—
403C22×C4C22×Q812yes—
404C22×C4C2×#4114yes—
405C22×C4C2×#4114yes—
406C22×C4C2×#4114yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4114yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4H(5,𝔽2)14yes—
413C22×C4H(5,𝔽2)14yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C4H(5,𝔽2)12yes—
417C22×C4H(5,𝔽2)12yes—
418C4⋊C4—48no—
419C4⋊C4—44no—
420C4⋊C4—48no—
421C4⋊C4—416no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—88no—
426C4⋊C4—88no—
427C22⋊C4—416no—
428C22⋊C4—416no—
429C22⋊C4—88no—
430C22⋊C4—88no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C42—28no—
439C42—28no—
440C42—28no—
441C42—48no—
442C42—48no—
443C42—48no—
444C42—48no—
445C23—28no—
446C23—28no—
447C23—28no—
448C23—28no—
449C23—28no—
450C23—28no—
451C23—28no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23C23×Q811yescharacteristic
481C23C22×#4111yescharacteristic
482C23C22×#4112yes—
483C23C22×#4111yescharacteristic
484C23C2×H(5,𝔽2)11yescharacteristic
485C23C2×H(5,𝔽2)11yescharacteristic
486C23C2×H(5,𝔽2)11yescharacteristic
487C23C2×H(5,𝔽2)12yes—
488C23C2×H(5,𝔽2)11yescharacteristic
489C23C2×H(5,𝔽2)12yes—
490C23#58512yes—
491Q8—416no—
492D4—432no—
493D4—832no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—832no—
566C2×C4#53814yes—
567C2×C4#53814yes—
568C2×C4#54014yes—
569C2×C4#54314yes—
570C2×C4#55418yes—
571C2×C4#55414yes—
572C2×C4#55414yes—
573C2×C4#55914yes—
574C2×C4H(3,𝔽4)14yes—
575C22—48no—
576C22—48no—
577C22—48no—
578C22—48no—
579C22—48no—
580C22—48no—
581C22—48no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22#344411yescharacteristic
591C22C2×#53811yescharacteristic
592C22C2×#53811yescharacteristic
593C22C2×#54011yescharacteristic
594C22C2×#54311yescharacteristic
595C22C2×#55411yescharacteristic
596C22C2×#55411yescharacteristic
597C22C2×#55412yes—
598C22C2×#55911yescharacteristic
599C22#348012yes—
600C22#350112yes—
601C22#350812yes—
602C22#350811yescharacteristic
603C22#350812yes—
604C22#351211yescharacteristic
605C22#351412yes—
606C22#351912yes—
607C22#351912yes—
608C22C2×H(3,𝔽4)11yescharacteristic
609C22#354212yes—
610C22#354312yes—
611C22#354612yes—
612C22#354612yes—
613C4—24no—
614C4—28no—
615C4—44no—
616C4—44no—
617C4—42no—
618C4—44no—
619C4—84no—
620C4—84no—
621C4—82no—
622C4—82no—
623C4—84no—
624C4#280418yes—
625C2—88no—
626C2#3508111yescharacteristic
627C2#3657311yescharacteristic
628C2#3660211yescharacteristic
629C2#3731711yescharacteristic
630C2#3896212yes—
631C2#3963411yescharacteristic
632C2#3977011yescharacteristic
633C2#4003012yes—
634C2#4605412yes—
635C2C2×#280411yescharacteristic
636C2#5945712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #16599C2semidirect
2. #15331C2semidirect
3. #15329C2semidirect
4. #15229C2semidirect
5. #15157C2semidirect
6. #15146C2semidirect
8. #14709C2semidirect
9. #14513C2semidirect
11. #14018C2semidirect
12. #13979C2semidirect
13. C4×#391C2semidirect
73. C2×#127Q8semidirect

Maximal subgroups

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

subgroupindex
#165992
#153312
#153292
#152292
#151572
#151462
#147132
#147092
#145132
#140252
#140182
#139792
C4×#3912