Finite Group Databasebeta

← #4309189

Subgroups — #4309189

Order 512.

3762 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C212yesmaximal
2C2×#2621C211yesmaximal, characteristic, complemented
3C2×#2280C211yesmaximal, characteristic
4#16675C214yesmaximal, complemented
5#16561C218yesmaximal, complemented
6#16391C214yesmaximal
7#15236C212yesmaximal, complemented
8#15233C212yesmaximal, complemented
9#15182C212yesmaximal, complemented
10#15178C212yesmaximal, complemented
11#14340C211yesmaximal, characteristic, complemented
12#14258C212yesmaximal, complemented
13C25×C4C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic, complemented
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2212yes—
17C22×C22⋊Q8C2212yes—
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#2848—216no—
21#2621—28no—
22C2×#398C2214yescomplemented
23C2×#397C2214yes—
24C2×#391C2212yescomplemented
25C2×#391C2212yescomplemented
26C2×#391C2212yescomplemented
27C2×#385C2214yes—
28C2×#385C2214yes—
29C2×#385C2214yes—
30C2×#384C2211yescharacteristic, complemented
31C2×#382C2211yescharacteristic, complemented
32C2×#382C2211yescharacteristic, complemented
33#2280—28no—
34C2×#381C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2214yes—
39C2×#379C2214yes—
40C2×#379C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2212yes—
43#1444C2218yescomplemented
44#1443C2218yescomplemented
45#1442C2218yescomplemented
46#1441C2214yescomplemented
47#1441C2218yescomplemented
48#1441C2218yescomplemented
49#1440C2218yes—
50#1440C2218yes—
51#1440C2218yes—
52#1439C2214yes—
53#1439C2218yes—
54#1439C2218yes—
55C4⋊#95C2212yescomplemented
56C4⋊#95C2212yescomplemented
57C4⋊#95C2212yescomplemented
58C26C2311yescharacteristic
59C23×Q8C2312yes—
60C24×C4—24no—
61C24×C4—24no—
62C24×C4—42no—
63C24×C4—42no—
64C24×C4—42no—
65C24×C4—42no—
66C24×C4D418yes—
67C24×C4C2312yes—
68C24×C4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2311yescharacteristic
71C2×C4⋊Q8—24no—
72C2×C4⋊Q8—42no—
73C2×C4⋊Q8—42no—
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8—28no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—28no—
78C2×C22⋊Q8—28no—
79C2×C22⋊Q8—44no—
80C2×C22⋊Q8—44no—
81C2×C22⋊Q8—44no—
82C2×C22⋊Q8—44no—
83C2×#117—24no—
84C2×#117—42no—
85C2×#117—42no—
86C2×#117—42no—
87C2×#117—42no—
88C2×#117D418yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#398—216no—
113#397—216no—
114#391—28no—
115#391—44no—
116#391—44no—
117#385—216no—
118#385—216no—
119#385—216no—
120#384—42no—
121#382—24no—
122#382—42no—
123#381—28no—
124#380—28no—
125#380—28no—
126#380—28no—
127#380—216no—
128#379—216no—
129#379—28no—
130C4×C22⋊C4—28no—
131C4×C22⋊C4—28no—
132C2×#95C2318yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2318yes—
140C2×#95C2318yes—
141C2×#95C2318yes—
142C2×#95C2318yes—
143C2×#95C2314yes—
144C2×#95C2318yes—
145C25—22no—
146C25—22no—
147C25—28no—
148C25—41no—
149C25—41no—
150C25—41no—
151C25—41no—
152C25—44no—
153C25C2×D414yes—
154C25C2411yescharacteristic
155C25C2412yes—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—232no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—416no—
179C23×C4—416no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—41no—
185C23×C4—41no—
186C23×C4—41no—
187C23×C4—41no—
188C23×C4—416no—
189C23×C4—416no—
190C23×C4—416no—
191C23×C4—416no—
192C23×C4C2×D418yes—
193C23×C4C2×D414yes—
194C23×C4C2×D414yes—
195C23×C4C2×D414yes—
196C23×C4C2×D418yes—
197C23×C4#4114yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C4⋊Q8—416no—
209C22⋊Q8—432no—
210C22⋊Q8—432no—
211C22⋊Q8—432no—
212#117—416no—
213#117—416no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C2×D418yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—216no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4C2×D418yes—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42C2×D414yes—
278C2×C42C2×D414yes—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—216no—
286#95—48no—
287#95—48no—
288#95—48no—
289#95—44no—
290#95—48no—
291#95—48no—
292C24—22no—
293C24—22no—
294C24—21no—
295C24—21no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—21no—
301C24—21no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—28no—
308C24—216no—
309C24—44no—
310C24—44no—
311C24—48no—
312C24—416no—
313C24—416no—
314C24—48no—
315C24—48no—
316C24—44no—
317C24—44no—
318C24—44no—
319C24—44no—
320C24—416no—
321C24—416no—
322C24—48no—
323C24—48no—
324C24C22×D412yes—
325C24C22×D412yes—
326C24C22×D414yes—
327C24C2×#4112yes—
328C242−1+418yes—
329C24C2511yescharacteristic
330C2×Q8—48no—
331C2×Q8—48no—
332C2×Q8—48no—
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—48no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—216no—
363C22×C4—216no—
364C22×C4—216no—
365C22×C4—216no—
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—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—4128no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4#121116yes—
413C22×C4C22×D414yes—
414C22×C4C22×D414yes—
415C22×C4C22×D414yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C4H(5,𝔽2)12yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C4H(5,𝔽2)12yes—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—416no—
429C4⋊C4—416no—
430C4⋊C4—416no—
431C4⋊C4—48no—
432C4⋊C4—416no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—88no—
436C22⋊C4—48no—
437C22⋊C4—416no—
438C22⋊C4—416no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—48no—
442C22⋊C4—416no—
443C22⋊C4—416no—
444C22⋊C4—416no—
445C22⋊C4—416no—
446C42—28no—
447C42—28no—
448C42—48no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—28no—
464C23—28no—
465C23—28no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—416no—
472C23—432no—
473C23—416no—
474C23—432no—
475C23—432no—
476C23—432no—
477C23—416no—
478C23—432no—
479C23—432no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23C2×#12114yes—
497C23#53618yes—
498C23C23×D411yescharacteristic
499C23C23×D411yescharacteristic
500C23C22×#4111yescharacteristic
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×H(5,𝔽2)11yescharacteristic
504C23C2×H(5,𝔽2)11yescharacteristic
505C23C2×2−1+414yes—
506C23#58514yes—
507Q8—816no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—232no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—432no—
523C2×C4—416no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—432no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4C2×#12114yes—
565C2×C4C2×#12114yes—
566C2×C4#53414yes—
567C2×C4#53414yes—
568C2×C4#53414yes—
569C2×C4#53514yes—
570C2×C4#53814yes—
571C2×C4#53814yes—
572C22—28no—
573C22—28no—
574C22—28no—
575C22—28no—
576C22—24no—
577C22—464no—
578C22—44no—
579C22—48no—
580C22—44no—
581C22—48no—
582C22—44no—
583C22—44no—
584C22—42no—
585C22—44no—
586C22—42no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—48no—
591C22—44no—
592C22—48no—
593C22C22×#12111yescharacteristic
594C22C2×#53411yescharacteristic
595C22C2×#53411yescharacteristic
596C22C2×#53411yescharacteristic
597C22C2×#53511yescharacteristic
598C22C2×#53612yes—
599C22#344912yes—
600C22C2×#53811yescharacteristic
601C22C2×#53811yescharacteristic
602C22#348914yes—
603C22#349514yes—
604C22#353812yes—
605C22#353812yes—
606C22#353912yes—
607C22#354012yes—
608C22#354114yes—
609C22#354114yes—
610C4—24no—
611C4—44no—
612C4—44no—
613C4—44no—
614C4—84no—
615C4—84no—
616C4—84no—
617C4—84no—
618C4—84no—
619C4#261718yes—
620C2—48no—
621C2—44no—
622C2C2×#261711yescharacteristic
623C2#3530612yes—
624C2#3737412yes—
625C2#3737512yes—
626C2#3738012yes—
627C2#3738112yes—
628C2#3744912yes—
629C2#3745311yescharacteristic
630C2#3745511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2621C2semidirect
4. #16675C2semidirect
5. #16561C2semidirect
7. #15236C2semidirect
8. #15233C2semidirect
9. #15182C2semidirect
10. #15178C2semidirect
11. #14340C2semidirect
12. #14258C2semidirect
14. C22×C4⋊Q8C22semidirect
22. C2×#398C22semidirect
24. C2×#391C22semidirect
25. C2×#391C22semidirect
26. C2×#391C22semidirect
30. C2×#384C22semidirect
31. C2×#382C22semidirect
32. C2×#382C22semidirect
43. #1444C22semidirect
44. #1443C22semidirect
45. #1442C22semidirect
46. #1441C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
55. C4⋊#95C22semidirect
56. C4⋊#95C22semidirect
57. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C2×#26212
C2×#22802
#166752
#165612
#163912
#152362
#152332
#151822
#151782
#143402
#142582