Finite Group Databasebeta

← #626822

Subgroups — #626822

Order 512.

5106 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic, complemented
2C2×#2848C212yesmaximal
3C22×#385C211yesmaximal, characteristic
4#15996C218yesmaximal, complemented
5#15265C218yesmaximal, complemented
6#14720C212yesmaximal, complemented
7#14717C212yesmaximal
8#14339C211yesmaximal, characteristic, complemented
9#14281C212yesmaximal, complemented
10#14128C212yesmaximal, complemented
11#13868C212yesmaximal
12C25×C4C2211yescharacteristic
13C2×#544—216no—
14C2×#544—28no—
15C22×C4⋊Q8C2214yescomplemented
16C22×#125C2214yescomplemented
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2214yes—
20C22×#117C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25#2848—216no—
26C2×#395C2214yescomplemented
27C2×#393C2214yes—
28C2×#385—28no—
29C2×#385C2214yes—
30C2×#385C2218yes—
31C2×#385C2218yes—
32C2×#381C2212yes—
33C2×#380C2212yes—
34C2×#380C2214yes—
35C2×#379C2214yes—
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38#1446C2214yescomplemented
39#1445C2214yescomplemented
40#1444C2214yescomplemented
41#1444C2218yescomplemented
42#1442C2214yescomplemented
43#1441C2218yescomplemented
44#1441C2214yescomplemented
45#1441C2218yescomplemented
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2218yes—
50#1437C2218yes—
51#1436C2218yescomplemented
52C22⋊#95C2212yes—
53C22⋊#95C2214yes—
54C22⋊#95C2214yes—
55C26C2311yescharacteristic
56C23×Q8C2312yes—
57C23×Q8C2312yes—
58C24×C4—28no—
59C24×C4—24no—
60C24×C4—44no—
61C24×C4—44no—
62C24×C4C2311yescharacteristic
63C24×C4C2314yes—
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69#544—2128no—
70C2×C4⋊Q8—216no—
71C2×C4⋊Q8—216no—
72C2×C4⋊Q8—216no—
73C2×#125—216no—
74C2×#125—216no—
75C2×#125—216no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—28no—
78C2×C22⋊Q8—28no—
79C2×C22⋊Q8—216no—
80C2×C22⋊Q8—216no—
81C2×C22⋊Q8—48no—
82C2×C22⋊Q8—48no—
83C2×C22⋊Q8D4116yes—
84C2×#117—28no—
85C2×#117—216no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4D418yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2318yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4Q818yescomplemented
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2318yes—
108C22×C22⋊C4C2318yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C42C2314yes—
112#395—48no—
113#393—48no—
114#385—232no—
115#385—232no—
116#385—216no—
117#381—28no—
118#380—28no—
119#380—216no—
120#379—216no—
121C2×#95—24no—
122C2×#95—24no—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2318yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2318yes—
130C2×#95C2314yes—
131C2×#95C2318yes—
132C2×#95C2314yes—
133C2×#95C2318yes—
134C2×#95C2318yes—
135C25—22no—
136C25—22no—
137C25—24no—
138C25—24no—
139C25—24no—
140C25—24no—
141C25—42no—
142C25—42no—
143C25C2×Q814yes—
144C25C2412yes—
145C25C2411yescharacteristic
146C22×Q8—24no—
147C22×Q8—24no—
148C22×Q8—24no—
149C22×Q8—28no—
150C22×Q8—44no—
151C22×Q8—44no—
152C22×Q8—44no—
153C22×Q8—44no—
154C22×Q8C2×D418yes—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—28no—
161C23×C4—24no—
162C23×C4—28no—
163C23×C4—24no—
164C23×C4—28no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—28no—
169C23×C4—28no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—416no—
182C23×C4—464no—
183C23×C4—416no—
184C23×C4—416no—
185C23×C4C2×D414yes—
186C23×C4C2×D414yes—
187C23×C4C2×D418yes—
188C23×C4C2×D414yes—
189C23×C4C2×Q814yes—
190C23×C4C2×Q814yes—
191C23×C4#4114yes—
192C23×C4C2418yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C4⋊Q8—2128no—
202#125—2128no—
203C22⋊Q8—264no—
204C22⋊Q8—264no—
205C22⋊Q8—464no—
206#117—264no—
207C2×C4⋊C4—232no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—216no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—216no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—416no—
233C2×C4⋊C4C2×D4116yes—
234C2×C4⋊C4C2×D418yes—
235C2×C4⋊C4C2×D4116yes—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—232no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—216no—
245C2×C22⋊C4—216no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—216no—
249C2×C22⋊C4—216no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—48no—
252C2×C22⋊C4—48no—
253C2×C22⋊C4—416no—
254C2×C22⋊C4C2×D4116yes—
255C2×C22⋊C4C2×Q8116yes—
256C2×C42—28no—
257C2×C42—28no—
258C2×C42—28no—
259#95—28no—
260#95—28no—
261#95—216no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—48no—
266#95—48no—
267#95—44no—
268#95—48no—
269#95—48no—
270#95—44no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—28no—
276C24—28no—
277C24—216no—
278C24—28no—
279C24—216no—
280C24—22no—
281C24—24no—
282C24—22no—
283C24—24no—
284C24—22no—
285C24—22no—
286C24—24no—
287C24—24no—
288C24—216no—
289C24—48no—
290C24—416no—
291C24—416no—
292C24—416no—
293C24—416no—
294C24—44no—
295C24—48no—
296C24—44no—
297C24—44no—
298C24—44no—
299C24—48no—
300C24C22×D412yes—
301C24C22×D414yes—
302C24C22×Q814yes—
303C24C22×Q812yes—
304C24C2×#4112yes—
305C24H(5,𝔽2)14yes—
306C242−1+414yes—
307C242−1+414yes—
308C24C2511yescharacteristic
309C2×Q8—216no—
310C2×Q8—216no—
311C2×Q8—416no—
312C2×Q8—88no—
313C2×Q8—84no—
314C2×Q8—84no—
315C2×Q8—84no—
316C2×Q8—816no—
317C2×Q8—84no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—216no—
321C22×C4—216no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—28no—
327C22×C4—232no—
328C22×C4—28no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4—22no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—24no—
358C22×C4—28no—
359C22×C4—24no—
360C22×C4—416no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—48no—
364C22×C4—416no—
365C22×C4—416no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—416no—
369C22×C4—416no—
370C22×C4—416no—
371C22×C4—416no—
372C22×C4—44no—
373C22×C4—48no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—4128no—
377C22×C4—44no—
378C22×C4—416no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—88no—
383C22×C4—88no—
384C22×C4#120116yes—
385C22×C4C22⋊Q8116yescomplemented
386C22×C4C22×D418yes—
387C22×C4C22×D418yes—
388C22×C4C22×D414yes—
389C22×C4C22×D414yes—
390C22×C4C22×D414yes—
391C22×C4C22×D414yes—
392C22×C4C22×D414yes—
393C22×C4C22×Q814yes—
394C22×C4C22×Q814yes—
395C22×C4C2×#4114yes—
396C22×C4H(5,𝔽2)14yes—
397C4⋊C4—216no—
398C4⋊C4—216no—
399C4⋊C4—216no—
400C4⋊C4—232no—
401C4⋊C4—232no—
402C4⋊C4—216no—
403C4⋊C4—232no—
404C4⋊C4—416no—
405C4⋊C4—432no—
406C4⋊C4—416no—
407C22⋊C4—232no—
408C22⋊C4—232no—
409C22⋊C4—432no—
410C22⋊C4—416no—
411C22⋊C4—432no—
412C22⋊C4—416no—
413C42—232no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—22no—
418C23—28no—
419C23—22no—
420C23—22no—
421C23—28no—
422C23—24no—
423C23—24no—
424C23—28no—
425C23—28no—
426C23—232no—
427C23—24no—
428C23—24no—
429C23—28no—
430C23—28no—
431C23—24no—
432C23—24no—
433C23—28no—
434C23—24no—
435C23—24no—
436C23—28no—
437C23—28no—
438C23—28no—
439C23—464no—
440C23—416no—
441C23—416no—
442C23—416no—
443C23—416no—
444C23—416no—
445C23—464no—
446C23—42no—
447C23—416no—
448C23—42no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23C2×#12014yes—
458C23C2×C22⋊Q814yes—
459C23#53614yes—
460C23#54418yescomplemented
461C23#54818yes—
462C23D4×Q818yes—
463C23C23×D411yescharacteristic
464C23C23×D412yes—
465C23C23×Q811yescharacteristic
466C23C22×#4111yescharacteristic
467C23C2×H(5,𝔽2)12yes—
468C23C2×H(5,𝔽2)12yes—
469C23C2×2−1+412yes—
470C23C2×2−1+412yes—
471C23#58512yes—
472Q8—816no—
473Q8—816no—
474C2×C4—24no—
475C2×C4—232no—
476C2×C4—24no—
477C2×C4—28no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—28no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—216no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—216no—
493C2×C4—28no—
494C2×C4—416no—
495C2×C4—464no—
496C2×C4—416no—
497C2×C4—416no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—48no—
516C2×C4—42no—
517C2×C4—48no—
518C2×C4—48no—
519C2×C4—48no—
520C2×C4—48no—
521C2×C4—48no—
522C2×C4—88no—
523C2×C4—88no—
524C2×C4C2×#12018yes—
525C2×C4C2×C22⋊Q818yes—
526C22—24no—
527C22—216no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—216no—
533C22—24no—
534C22—216no—
535C22—28no—
536C22—464no—
537C22—44no—
538C22—48no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—48no—
544C22—48no—
545C22—44no—
546C22#240918yes—
547C22C22×#12011yescharacteristic
548C22C22×C22⋊Q811yescharacteristic
549C22C2×#53611yescharacteristic
550C22C2×#54412yes—
551C22#346012yes—
552C22C2×#54812yes—
553C22C2×D4×Q812yes—
554C22#348414yes—
555C22#348411yescharacteristic
556C22#348411yescharacteristic
557C22#348611yescharacteristic
558C22#348711yescharacteristic
559C22#348914yes—
560C22#349212yes—
561C22#350112yes—
562C22#353514yes—
563C22#353814yes—
564C4—28no—
565C4—28no—
566C4—44no—
567C4—44no—
568C4—48no—
569C4—48no—
570C4—44no—
571C4—82no—
572C4—88no—
573C2—28no—
574C2—48no—
575C2C2×#240911yescharacteristic
576C2#3517411yescharacteristic
577C2#3522411yescharacteristic
578C2#3534611yescharacteristic
579C2#3565911yescharacteristic
580C2#3670811yescharacteristic
581C2#3671111yescharacteristic
582C2#3768514yes—
583C2#3775314yes—
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
1. C22×#544C2semidirect
4. #15996C2semidirect
5. #15265C2semidirect
6. #14720C2semidirect
8. #14339C2semidirect
9. #14281C2semidirect
10. #14128C2semidirect
15. C22×C4⋊Q8C22semidirect
16. C22×#125C22semidirect
26. C2×#395C22semidirect
38. #1446C22semidirect
39. #1445C22semidirect
40. #1444C22semidirect
41. #1444C22semidirect
42. #1442C22semidirect
43. #1441C22semidirect
44. #1441C22semidirect
45. #1441C22semidirect
51. #1436C22semidirect
104. C22×C22⋊C4Q8semidirect
385. C22×C4C22⋊Q8semidirect
460. C23#544semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#28482
C22×#3852
#159962
#152652
#147202
#147172
#143392
#142812
#141282
#138682