Finite Group Databasebeta

← #4466169

Subgroups — #4466169

Order 512.

2526 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2811C211yesmaximal, characteristic, complemented
2C2×#2620C211yesmaximal, characteristic
3#17104C214yesmaximal, complemented
4#16814C212yesmaximal, complemented
5#16510C214yesmaximal, complemented
6#16294C214yesmaximal, complemented
7#16094C212yesmaximal, complemented
8#15673C212yesmaximal
9#15633C214yesmaximal, complemented
10#15625C212yesmaximal, complemented
11#15466C212yesmaximal, complemented
12#14887C212yesmaximal, complemented
13#14489C211yesmaximal, characteristic, complemented
14C22×#124C2211yescharacteristic
15C22×#124C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#2811—28no—
20#2620—28no—
21C2×#399C2212yes—
22C2×#399C2214yes—
23C2×#398C2212yes—
24C2×#396C2214yes—
25C2×#395C2212yes—
26C2×#395C2214yes—
27C2×#395C2214yes—
28C2×#395C2212yes—
29C2×#394C2214yes—
30C2×#394C2214yes—
31C2×#393C2214yes—
32C2×#393C2212yes—
33C2×#393C2212yes—
34C2×#391C2214yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#389C2214yes—
40C2×#388C2214yes—
41C2×#384C2211yescharacteristic
42#1446C2212yes—
43#1444C2214yes—
44#1444C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2212yes—
50#1440C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2212yes—
60#1438C2214yes—
61#1437C2212yes—
62#1437C2212yes—
63#1436C2214yes—
64#1436C2212yes—
65#1436C2212yes—
66#1434C2214yes—
67#1434C2212yes—
68C22⋊#95C2214yes—
69C23×Q8C2312yes—
70C23×Q8C2311yescharacteristic
71C23×Q8C2312yes—
72C23×D4C2311yescharacteristic
73C23×D4C2312yes—
74C24×C4C2311yescharacteristic
75C2×#124—24no—
76C2×#124—44no—
77C2×#124—48no—
78C2×#124—44no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—42no—
81C2×C22⋊Q8—42no—
82C2×#121—24no—
83C2×#121—42no—
84C2×#121—42no—
85C2×#117—24no—
86C2×#117—44no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113#399—48no—
114#399—44no—
115#398—44no—
116#396—216no—
117#395—28no—
118#395—216no—
119#395—48no—
120#395—44no—
121#394—216no—
122#394—48no—
123#393—44no—
124#393—48no—
125#393—44no—
126#391—28no—
127#391—28no—
128#391—216no—
129#391D418yes—
130#390—24no—
131#389—48no—
132#388—48no—
133#384—24no—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C25C2412yes—
156C25C2411yescharacteristic
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—28no—
160C22×Q8—24no—
161C22×Q8—41no—
162C22×Q8—41no—
163C22×Q8—42no—
164C22×Q8—44no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8—44no—
168C22×Q8—42no—
169C22×Q8—44no—
170C22×Q8—42no—
171C22×Q8C2×D414yes—
172C22×D4—24no—
173C22×D4—22no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—44no—
181C22×D4—44no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—42no—
187C23×C4C2412yes—
188C23×C4C2414yes—
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2411yescharacteristic
196C23×C4C2414yes—
197C23×C4C2411yescharacteristic
198#124—416no—
199#124—432no—
200C22⋊Q8—416no—
201#121—416no—
202#117—416no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C2×D414yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
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—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
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⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—28no—
271C2×C42—24no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—44no—
275C2×C42C2×D414yes—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—44no—
288#95—42no—
289#95—44no—
290#95—42no—
291#95—44no—
292#95—42no—
293#95—44no—
294#95—42no—
295#95—42no—
296#95C2×D4116yes—
297C24—21no—
298C24—22no—
299C24—24no—
300C24—21no—
301C24—24no—
302C24—22no—
303C24—22no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—22no—
308C24—22no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24H(5,𝔽2)12yes—
313C24C2511yescharacteristic
314C2×Q8—216no—
315C2×Q8—28no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—48no—
319C2×Q8—84no—
320C2×Q8—84no—
321C2×Q8—84no—
322C2×Q8—84no—
323C2×Q8—84no—
324C2×Q8—88no—
325C2×Q8—88no—
326C2×Q8—88no—
327C2×Q8—84no—
328C2×Q8—88no—
329C2×D4—48no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—48no—
333C2×D4—44no—
334C2×D4—84no—
335C2×D4—84no—
336C2×D4—88no—
337C2×D4—84no—
338C2×D4—88no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
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—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4C22×D414yes—
399C22×C4C22×D414yes—
400C22×C4C22×D414yes—
401C22×C4C22×D412yes—
402C22×C4H(5,𝔽2)18yes—
403C22×C4H(5,𝔽2)12yes—
404C22×C4H(5,𝔽2)14yes—
405C22×C42−1+418yes—
406C22×C42−1+414yes—
407C4⋊C4—44no—
408C4⋊C4—416no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—44no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C4⋊C4—88no—
415C4⋊C4—88no—
416C4⋊C4—88no—
417C4⋊C4—88no—
418C22⋊C4—416no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—416no—
422C22⋊C4—48no—
423C22⋊C4—416no—
424C22⋊C4—416no—
425C22⋊C4—88no—
426C22⋊C4—88no—
427C22⋊C4—88no—
428C22⋊C4—88no—
429C42—28no—
430C42—44no—
431C42—48no—
432C42—84no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—42no—
441C23—44no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—44no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—44no—
452C23—44no—
453C23—42no—
454C23—42no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—42no—
468C23—42no—
469C23C23×D411yescharacteristic
470C23C2×H(5,𝔽2)11yescharacteristic
471C23C2×H(5,𝔽2)12yes—
472C23C2×H(5,𝔽2)11yescharacteristic
473C23C2×H(5,𝔽2)14yes—
474C23C2×2−1+412yes—
475C23C2×2−1+414yes—
476Q8—88no—
477Q8—816no—
478Q8—816no—
479D4—88no—
480D4—816no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—28no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—416no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
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—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4#53614yes—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—48no—
549C22—82no—
550C22—82no—
551C22—84no—
552C22—84no—
553C22—84no—
554C22—84no—
555C22—84no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—84no—
560C22—84no—
561C22C2×#53611yescharacteristic
562C22#348411yescharacteristic
563C22#348411yescharacteristic
564C22#348612yes—
565C22#349812yes—
566C22#353212yes—
567C22#353614yes—
568C22#353614yes—
569C22#353612yes—
570C22#353612yes—
571C22#354414yes—
572C22#354412yes—
573C22#354912yes—
574C22#355014yes—
575C22#355012yes—
576C4—24no—
577C4—44no—
578C4—44no—
579C4—48no—
580C4—42no—
581C4—44no—
582C4—44no—
583C4—48no—
584C4—84no—
585C4—82no—
586C4—84no—
587C2—82no—
588C2—162no—
589C2#3671811yescharacteristic
590C2#4278912yes—
591C2#4635612yes—
592C2#5190211yescharacteristic
593C2#5207211yescharacteristic
594C2#5500114yes—
595C2#5895914yes—
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
1. C2×#2811C2semidirect
3. #17104C2semidirect
4. #16814C2semidirect
5. #16510C2semidirect
6. #16294C2semidirect
7. #16094C2semidirect
9. #15633C2semidirect
10. #15625C2semidirect
11. #15466C2semidirect
12. #14887C2semidirect
13. #14489C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28112
C2×#26202
#171042
#168142
#165102
#162942
#160942
#156732
#156332
#156252
#154662
#148872
#144892