Finite Group Databasebeta

← #5035178

Subgroups — #5035178

Order 512.

2576 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2620C211yesmaximal, characteristic
2C2×#2379C211yesmaximal, characteristic, complemented
3#16110C212yesmaximal, complemented
4#15709C214yesmaximal, complemented
5#15706C214yesmaximal, complemented
6#15466C212yesmaximal, complemented
7#15392C214yesmaximal, complemented
8#15384C214yesmaximal, complemented
9#15342C212yesmaximal, complemented
10#15260C212yesmaximal
11#14354C212yesmaximal, complemented
12#14169C211yesmaximal, characteristic, complemented
13#13957C212yesmaximal, complemented
14C22×#124C2211yescharacteristic
15C22×#123C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#2620—28no—
20C2×#399C2212yes—
21C2×#398C2212yes—
22C2×#397C2214yes—
23C2×#396C2214yes—
24C2×#394C2214yes—
25C2×#394C2214yes—
26#2379—28no—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#388C2214yes—
31C2×#386C2212yes—
32C2×#385C2214yes—
33C2×#385C2214yes—
34C2×#385C2212yes—
35C2×#384C2211yescharacteristic
36C2×#384C2214yes—
37C2×#382C2212yes—
38C2×#382C2214yes—
39C2×#376C2212yes—
40#1446C2212yes—
41#1445C2214yes—
42#1444C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1442C2214yes—
48#1441C2212yes—
49#1441C2214yes—
50#1441C2212yes—
51#1440C2212yes—
52#1440C2214yes—
53#1439C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2212yes—
58#1438C2212yes—
59#1438C2212yes—
60#1437C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1437C2212yes—
64#1436C2214yes—
65#1436C2212yes—
66#1434C2214yes—
67#1434C2214yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C2×#124—28no—
75C2×#124D418yes—
76C2×#123—48no—
77C2×#123—44no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—42no—
80C2×C22⋊Q8—42no—
81C2×#121—24no—
82C2×#121—42no—
83C2×#121—42no—
84C2×#117—24no—
85C2×#117—28no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2314yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#399—44no—
115#398—44no—
116#397—48no—
117#396—216no—
118#394—216no—
119#394—48no—
120#391—24no—
121#391—28no—
122#390—24no—
123#388—216no—
124#386—28no—
125#385—216no—
126#385—44no—
127#385—48no—
128#384—24no—
129#384—48no—
130#382—28no—
131#382—48no—
132#376—28no—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C25C2412yes—
154C25C2411yescharacteristic
155C22×Q8—24no—
156C22×Q8—41no—
157C22×Q8—41no—
158C22×Q8—42no—
159C22×Q8C2×D414yes—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—41no—
164C22×D4—41no—
165C22×D4—42no—
166C22×D4—44no—
167C22×D4—42no—
168C22×D4—44no—
169C22×D4C2×D414yes—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—42no—
176C23×C4—44no—
177C23×C4—42no—
178C23×C4—44no—
179C23×C4—42no—
180C23×C4C2412yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2411yescharacteristic
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191#124—232no—
192#123—432no—
193C22⋊Q8—416no—
194#121—416no—
195#117—416no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4C2×D4116yes—
263C2×C42—28no—
264C2×C42—28no—
265C2×C42—28no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—44no—
270C2×C42C2×D414yes—
271C2×C42#4118yes—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—24no—
278#95—24no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—42no—
291#95—42no—
292C24—21no—
293C24—22no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—22no—
299C24—21no—
300C24—22no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—22no—
305C24—24no—
306C24C22×D414yes—
307C24H(5,𝔽2)12yes—
308C24C2511yescharacteristic
309C2×Q8—28no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—84no—
313C2×Q8—84no—
314C2×D4—28no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—48no—
318C2×D4—48no—
319C2×D4—84no—
320C2×D4—84no—
321C2×D4—88no—
322C2×D4—84no—
323C2×D4—88no—
324C22×C4—24no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—28no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
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—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—48no—
384C22×C4—84no—
385C22×C4—88no—
386C22×C4—84no—
387C22×C4—88no—
388C22×C4C22×D414yes—
389C22×C4C22×D412yes—
390C22×C4C22×D414yes—
391C22×C4C2×#4114yes—
392C22×C4C2×#4114yes—
393C22×C4C2×#4114yes—
394C22×C4H(5,𝔽2)12yes—
395C22×C4H(5,𝔽2)18yes—
396C22×C42−1+418yes—
397C4⋊C4—48no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—416no—
401C4⋊C4—48no—
402C4⋊C4—84no—
403C4⋊C4—88no—
404C4⋊C4—84no—
405C4⋊C4—84no—
406C4⋊C4—88no—
407C4⋊C4—88no—
408C4⋊C4—88no—
409C22⋊C4—232no—
410C22⋊C4—416no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—416no—
416C22⋊C4—48no—
417C22⋊C4—88no—
418C22⋊C4—84no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C42—232no—
422C42—44no—
423C42—84no—
424C42#123116yescomplemented
425C23—24no—
426C23—22no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—28no—
431C23—28no—
432C23—24no—
433C23—24no—
434C23—41no—
435C23—41no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—42no—
440C23—44no—
441C23—44no—
442C23—42no—
443C23—42no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—42no—
461C23C23×D411yescharacteristic
462C23C22×#4112yes—
463C23C2×H(5,𝔽2)11yescharacteristic
464C23C2×H(5,𝔽2)11yescharacteristic
465C23C2×H(5,𝔽2)14yes—
466C23C2×2−1+414yes—
467C23#58512yes—
468Q8—88no—
469D4—88no—
470D4—816no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—28no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—416no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—816no—
534C2×C4C2×#12314yes—
535C2×C4C2×#12318yes—
536C22—42no—
537C22—44no—
538C22—48no—
539C22—48no—
540C22—44no—
541C22—42no—
542C22—48no—
543C22—42no—
544C22—44no—
545C22—82no—
546C22—84no—
547C22—84no—
548C22—84no—
549C22—84no—
550C22—84no—
551C22—82no—
552C22C22×#12311yescharacteristic
553C22#348411yescharacteristic
554C22#348511yescharacteristic
555C22#349214yes—
556C22#349412yes—
557C22#349412yes—
558C22#349812yes—
559C22#349912yes—
560C22#351114yes—
561C22#353212yes—
562C22#353514yes—
563C22#353812yes—
564C22#354412yes—
565C22#354714yes—
566C22#354912yes—
567C4—28no—
568C4—24no—
569C4—48no—
570C4—48no—
571C4—42no—
572C4—44no—
573C4—84no—
574C4—82no—
575C4—84no—
576C4—82no—
577C4—82no—
578C2—84no—
579C2—82no—
580C2#3553012yes—
581C2#3554011yescharacteristic
582C2#5154211yescharacteristic
583C2#5154612yes—
584C2#5157314yes—
585C2#5157414yes—
586C2#5190211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2379C2semidirect
3. #16110C2semidirect
4. #15709C2semidirect
5. #15706C2semidirect
6. #15466C2semidirect
7. #15392C2semidirect
8. #15384C2semidirect
9. #15342C2semidirect
11. #14354C2semidirect
12. #14169C2semidirect
13. #13957C2semidirect
424. C42#123semidirect

Maximal subgroups

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

subgroupindex
C2×#26202
C2×#23792
#161102
#157092
#157062
#154662
#153922
#153842
#153422
#152602
#143542
#141692
#139572