Finite Group Databasebeta

← #5025143

Subgroups — #5025143

Order 512.

2662 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2660C211yesmaximal, characteristic, complemented
3C2×#2633C211yesmaximal, characteristic
4C2×#2381C211yesmaximal, characteristic, complemented
5#15886C214yesmaximal, complemented
6#15765C214yesmaximal, complemented
7#15661C214yesmaximal
8#15629C214yesmaximal, complemented
9#15279C214yesmaximal, complemented
10#15276C214yesmaximal, complemented
11#14289C211yesmaximal, characteristic, complemented
12#14164C211yesmaximal, characteristic, complemented
13#13959C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×C22⋊Q8C2214yes—
17C22×C4×D4C2211yescharacteristic
18#2660—28no—
19#2633—28no—
20#2542—28no—
21C2×#399C2214yes—
22C2×#399C2212yes—
23C2×#398C2212yes—
24C2×#398C2214yes—
25C2×#398C2214yes—
26C2×#397C2212yes—
27C2×#396C2214yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#394C2212yes—
31#2381—28no—
32C2×#391C2214yes—
33C2×#390C2212yes—
34C2×#389C2214yes—
35C2×#388C2214yes—
36C2×#387C2214yes—
37C2×#387C2214yes—
38C2×#385C2214yes—
39C2×#384C2212yes—
40C2×#384C2212yes—
41C2×#383C2211yescharacteristic
42C2×#383C2211yescharacteristic
43C2×#382C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×#376C2211yescharacteristic
46#1445C2212yes—
47#1445C2212yes—
48#1445C2212yes—
49#1444C2214yes—
50#1444C2214yes—
51#1444C2212yes—
52#1442C2214yes—
53#1442C2212yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2212yes—
63#1437C2214yes—
64#1437C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C23×Q8C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4C2312yes—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8D418yes—
77C2×#125—24no—
78C2×#125—24no—
79C2×#125—24no—
80C2×C22⋊Q8—216no—
81C2×C22⋊Q8—48no—
82C2×C22⋊Q8—48no—
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4—24no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2314yes—
111C22×C42C2311yescharacteristic
112#399—216no—
113#399—28no—
114#398—216no—
115#398—28no—
116#398—48no—
117#397—28no—
118#396—48no—
119#395—28no—
120#395—28no—
121#394—28no—
122#391—216no—
123#390—28no—
124#389—48no—
125#388—48no—
126#387—216no—
127#387—216no—
128#385—216no—
129#384—28no—
130#384—28no—
131#383—24no—
132#383D418yes—
133#382—28no—
134C4×C4⋊C4—24no—
135#376—24no—
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C25C2412yes—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—44no—
164C22×Q8—44no—
165C22×Q8C2×D418yes—
166C22×D4—24no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—42no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C4⋊Q8—232no—
189#125—416no—
190C22⋊Q8—464no—
191C4×D4—416no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C2×D418yes—
235C2×C4⋊C4C2×D418yes—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
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—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—28no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42C2×D414yes—
267C2×C42C2×D414yes—
268C2×C42#4114yes—
269#95—22no—
270#95—22no—
271#95—24no—
272#95—24no—
273#95—28no—
274#95—28no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95C2×D418yes—
291#95C2×D418yes—
292#95C2×D418yes—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—22no—
301C24—22no—
302C24—22no—
303C242−1+414yes—
304C24C2511yescharacteristic
305C2×Q8—216no—
306C2×Q8—416no—
307C2×Q8—84no—
308C2×Q8—84no—
309C2×Q8—88no—
310C2×D4—28no—
311C2×D4—48no—
312C2×D4—48no—
313C2×D4—44no—
314C2×D4—44no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
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—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—416no—
369C22×C4—48no—
370C22×C4—88no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4C22×D418yes—
374C22×C4C22×D418yes—
375C22×C4C22×D412yes—
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C2×#4112yes—
382C22×C4C2×#4112yes—
383C22×C4C2×#4112yes—
384C22×C4H(5,𝔽2)18yes—
385C22×C42−1+414yes—
386C22×C42−1+414yes—
387C4⋊C4—216no—
388C4⋊C4—216no—
389C4⋊C4—216no—
390C4⋊C4—416no—
391C4⋊C4—416no—
392C4⋊C4—416no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—88no—
399C4⋊C4—88no—
400C4⋊C4—88no—
401C4⋊C4—88no—
402C22⋊C4—416no—
403C22⋊C4—416no—
404C22⋊C4—416no—
405C22⋊C4—416no—
406C22⋊C4—48no—
407C22⋊C4—88no—
408C22⋊C4—88no—
409C42—216no—
410C42—48no—
411C42—88no—
412C42#121116yes—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—28no—
417C23—28no—
418C23—42no—
419C23—44no—
420C23—44no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—44no—
425C23—44no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23C23×D411yescharacteristic
437C23C23×D411yescharacteristic
438C23C22×#4111yescharacteristic
439C23C2×H(5,𝔽2)14yes—
440C23C2×2−1+412yes—
441C23C2×2−1+412yes—
442C23C2×2−1+412yes—
443C23#58512yes—
444Q8—816no—
445D4—416no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—816no—
511C2×C4C2×#12114yes—
512C2×C4C2×#12114yes—
513C2×C4C2×#12114yes—
514C22—44no—
515C22—48no—
516C22—44no—
517C22—48no—
518C22—44no—
519C22—82no—
520C22—84no—
521C22—82no—
522C22—82no—
523C22—84no—
524C22—82no—
525C22C22×#12111yescharacteristic
526C22#348412yes—
527C22#348612yes—
528C22#348611yescharacteristic
529C22#348711yescharacteristic
530C22#348912yes—
531C22#349512yes—
532C22#349812yes—
533C22#349912yes—
534C22#350812yes—
535C22#350911yescharacteristic
536C22#350911yescharacteristic
537C22#353514yes—
538C22#353814yes—
539C22#355214yes—
540C22#355914yes—
541C4—24no—
542C4—24no—
543C4—24no—
544C4—48no—
545C4—44no—
546C4—44no—
547C4—48no—
548C4—48no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—84no—
553C2—84no—
554C2#3552811yescharacteristic
555C2#3566111yescharacteristic
556C2#3566711yescharacteristic
557C2#5065114yes—
558C2#5091412yes—
559C2#5122612yes—
560C2#5200612yes—
561C2#5200712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2660C2semidirect
4. C2×#2381C2semidirect
5. #15886C2semidirect
6. #15765C2semidirect
8. #15629C2semidirect
9. #15279C2semidirect
10. #15276C2semidirect
11. #14289C2semidirect
12. #14164C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#26602
C2×#26332
C2×#23812
#158862
#157652
#156612
#156292
#152792
#152762
#142892
#141642
#139592