Finite Group Databasebeta

← #4123546

Subgroups — #4123546

Order 512.

2884 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2458C214yesmaximal, complemented
2#17137C211yesmaximal, characteristic, complemented
3#15978C212yesmaximal, complemented
4#15966C211yesmaximal, characteristic, complemented
5#15902C214yesmaximal, complemented
6#15662C214yesmaximal, complemented
7#15569C212yesmaximal, complemented
8#15261C212yesmaximal
9#14223C212yesmaximal, complemented
10#14174C212yesmaximal, complemented
11#14154C211yesmaximal, characteristic
12#14145C212yesmaximal, complemented
13#14036C214yesmaximal, complemented
14C22×#127C2211yescharacteristic
15C22×#123C2212yes—
16C22×C4×Q8C2212yes—
17#2458—232no—
18C2×#398C2212yes—
19C2×#395C2212yes—
20C2×#394C2214yes—
21C2×#393C2214yes—
22C2×#393C2214yes—
23C2×#390C2214yes—
24C2×#390C2214yes—
25C2×#388C2214yes—
26C2×#388C2214yes—
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#387C2214yes—
30C2×#387C2214yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#385C2214yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4C2214yes—
38#1446C2212yes—
39#1445C2212yes—
40#1442C2212yes—
41#1442C2212yes—
42#1441C2212yes—
43#1441C2212yes—
44#1440C2214yes—
45#1440C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1438C2212yes—
51#1438C2212yes—
52#1438C2214yes—
53#1438C2211yescharacteristic
54#1438C2212yes—
55#1438C2211yescharacteristic
56#1438C2211yescharacteristic
57#1438C2211yescharacteristic
58#1438C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1434C2212yes—
65#1434C2212yes—
66#1434C2214yes—
67#1434C2214yes—
68C23×Q8C2312yes—
69C23×D4C2312yes—
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C23×D4C2311yescharacteristic
73C24×C4C2312yes—
74C2×#127—24no—
75C2×#127—42no—
76C2×#127—42no—
77C2×#123—216no—
78C2×#123—44no—
79C2×C4×Q8—216no—
80C2×C4×Q8—44no—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104C22×C42C2314yes—
105C22×C42C2314yes—
106#398D4116yescomplemented
107#395—28no—
108#394—216no—
109#393—216no—
110#393—216no—
111#390—216no—
112#390—216no—
113#388—216no—
114#388—216no—
115#388—216no—
116#388—216no—
117#387—216no—
118#387—216no—
119#386—28no—
120#386—44no—
121#385—216no—
122C4×C4⋊C4—44no—
123C4×C4⋊C4—42no—
124C4×C4⋊C4—42no—
125C4×C22⋊C4—216no—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2311yescharacteristic
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2311yescharacteristic
144C2×#95C2311yescharacteristic
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2314yes—
148C25C2412yes—
149C25C2412yes—
150C22×Q8—28no—
151C22×Q8—24no—
152C22×Q8—42no—
153C22×Q8—44no—
154C22×Q8—42no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—28no—
158C22×D4—28no—
159C22×D4—22no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—42no—
163C22×D4—42no—
164C22×D4—41no—
165C22×D4—41no—
166C22×D4—41no—
167C22×D4—41no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4#4114yes—
173C23×C4—24no—
174C23×C4—28no—
175C23×C4—42no—
176C23×C4—44no—
177C23×C4—42no—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189#127—416no—
190#123—432no—
191C4×Q8—432no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4C2×D4116yes—
208C2×C4⋊C4C2×D418yes—
209C2×C4⋊C4C2×D418yes—
210C2×C4⋊C4#4118yes—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
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—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—44no—
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—42no—
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—42no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—28no—
251C2×C42—28no—
252C2×C42—28no—
253C2×C42—28no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—28no—
257C2×C42—44no—
258C2×C42—44no—
259C2×C42—44no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—24no—
270#95—28no—
271#95—44no—
272#95—44no—
273#95—42no—
274#95—44no—
275#95—42no—
276#95—41no—
277#95—41no—
278#95—41no—
279#95—41no—
280#95C2×D4116yes—
281#95C2×D418yes—
282C24—24no—
283C24—24no—
284C24—22no—
285C24—22no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24C2×#4114yes—
300C24C2511yescharacteristic
301C2×Q8—48no—
302C2×Q8—416no—
303C2×Q8—416no—
304C2×Q8—48no—
305C2×Q8—48no—
306C2×D4—44no—
307C2×D4—48no—
308C2×D4—48no—
309C2×D4—48no—
310C2×D4—416no—
311C2×D4—416no—
312C2×D4—44no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—88no—
316C2×D4—88no—
317C2×D4—84no—
318C2×D4—84no—
319C2×D4—84no—
320C2×D4—84no—
321C2×D4—84no—
322C2×D4—84no—
323C2×D4—84no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—48no—
376C22×C4—416no—
377C22×C4—88no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4C22×D414yes—
381C22×C4C22×D414yes—
382C22×C4C22×D418yes—
383C22×C4C22×D418yes—
384C22×C4C22×D414yes—
385C22×C4C2×#4112yes—
386C22×C4C2×#4114yes—
387C22×C4C2×#4114yes—
388C22×C4C2×#4114yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C4H(5,𝔽2)12yes—
391C4⋊C4—232no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—416no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C22⋊C4—416no—
399C22⋊C4—416no—
400C22⋊C4—416no—
401C22⋊C4—416no—
402C22⋊C4—48no—
403C22⋊C4—416no—
404C22⋊C4—416no—
405C22⋊C4—416no—
406C22⋊C4—416no—
407C22⋊C4—88no—
408C22⋊C4—84no—
409C22⋊C4—84no—
410C22⋊C4—88no—
411C42—28no—
412C42—416no—
413C42—48no—
414C42—48no—
415C42—88no—
416C23—28no—
417C23—28no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—42no—
422C23—44no—
423C23—44no—
424C23—42no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—44no—
436C23—44no—
437C23—42no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—42no—
455C23—42no—
456C23—44no—
457C23—44no—
458C23C23×D412yes—
459C23C22×#4111yescharacteristic
460C23C22×#4112yes—
461C23C2×H(5,𝔽2)11yescharacteristic
462C23C2×H(5,𝔽2)11yescharacteristic
463C23#58514yes—
464C23#58512yes—
465C23#58512yes—
466Q8—432no—
467D4—816no—
468D4—816no—
469D4—88no—
470D4—88no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—28no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
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—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—816no—
532C2×C4D4214yes—
533C2×C4#55914yes—
534C22—48no—
535C22—48no—
536C22—44no—
537C22—44no—
538C22—44no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22—84no—
543C22—84no—
544C22—84no—
545C22—84no—
546C22—82no—
547C22—82no—
548C22—84no—
549C22—84no—
550C22—82no—
551C22—84no—
552C22C2×D4211yescharacteristic
553C22D4×#4112yes—
554C22D4×#4114yes—
555C22#347512yes—
556C22C2×#55911yescharacteristic
557C22#348714yes—
558C22#349011yescharacteristic
559C22#349112yes—
560C22#349612yes—
561C22#350512yes—
562C22#352414yes—
563C22#354012yes—
564C22#354012yes—
565C22#354511yescharacteristic
566C22#354511yescharacteristic
567C22#356614yes—
568C4—24no—
569C4—24no—
570C4—44no—
571C4—48no—
572C4—44no—
573C4—42no—
574C4—48no—
575C4—44no—
576C4—44no—
577C4—84no—
578C4—84no—
579C2—84no—
580C2—162no—
581C2#4699811yescharacteristic
582C2#4754612yes—
583C2#4780112yes—
584C2#5900112yes—
585C2#5915914yes—
586C2#5962812yes—
587C2#5981411yescharacteristic
588C2#5986111yescharacteristic
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. C2×#2458C2semidirect
2. #17137C2semidirect
3. #15978C2semidirect
4. #15966C2semidirect
5. #15902C2semidirect
6. #15662C2semidirect
7. #15569C2semidirect
9. #14223C2semidirect
10. #14174C2semidirect
12. #14145C2semidirect
13. #14036C2semidirect
106. #398D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24582
#171372
#159782
#159662
#159022
#156622
#155692
#152612
#142232
#141742
#141542
#141452
#140362