Finite Group Databasebeta

← #919291

Subgroups — #919291

Order 512.

2422 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic
2C22×#398C211yesmaximal, characteristic
3#17206C214yesmaximal, complemented
4#16883C214yesmaximal, complemented
5#16629C212yesmaximal, complemented
6#16563C212yesmaximal, complemented
7#15856C212yesmaximal, complemented
8#15790C212yesmaximal, complemented
9#15092C212yesmaximal
10#14920C212yesmaximal
11#14720C212yesmaximal
12#14265C212yesmaximal
13#14044C212yesmaximal
14#14039C212yesmaximal
15#13923C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2502—28no—
21C2×#398—28no—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#395C2214yes—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#387C2214yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#384C2211yescharacteristic
38C2×#384C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C22×#95C2212yes—
43C22×#95C2211yescharacteristic
44#1445C2214yes—
45#1445C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1443C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2212yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1436C2212yes—
66#1436C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—42no—
85C2×C22⋊Q8—42no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—48no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4—42no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4D418yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119#398—28no—
120#398—28no—
121#398—28no—
122#398—28no—
123#398—28no—
124#398—28no—
125#395—48no—
126#391—24no—
127#391—24no—
128#387—48no—
129#385—28no—
130#385—28no—
131#385—44no—
132#385—44no—
133#385Q8116yescomplemented
134#384—24no—
135#384—24no—
136#380—42no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—42no—
139C2×#95—24no—
140C2×#95—28no—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C25C2411yescharacteristic
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8—41no—
173C22×Q8—41no—
174C22×Q8—42no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—44no—
188C23×C4—42no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4C2×D414yes—
195C23×C4C2×D414yes—
196C23×C4C2×D414yes—
197C23×C4C2×Q818yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C22⋊Q8—416no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—416no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—416no—
247C2×C4⋊C4—432no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×Q8116yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4C2×Q818yes—
276C2×C22⋊C4C2×Q818yes—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—44no—
283#95—24no—
284#95—24no—
285#95—28no—
286#95—28no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—42no—
302#95—42no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95C2×Q8116yes—
311C24—22no—
312C24—22no—
313C24—24no—
314C24C22×D412yes—
315C24C22×Q814yes—
316C24H(5,𝔽2)12yes—
317C24H(5,𝔽2)12yes—
318C242−1+414yes—
319C24C2511yescharacteristic
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—84no—
324C2×Q8—84no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—216no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
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—28no—
368C22×C4—28no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—44no—
394C22×C4—48no—
395C22×C4—416no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—416no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×Q814yes—
418C22×C4C22×Q818yes—
419C22×C4C22×Q818yes—
420C22×C4C22×Q814yes—
421C22×C4H(5,𝔽2)14yes—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—416no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—416no—
430C4⋊C4—44no—
431C4⋊C4—48no—
432C4⋊C4—416no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C4⋊C4—88no—
439C4⋊C4—88no—
440C22⋊C4—216no—
441C22⋊C4—216no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C42—44no—
445C42—44no—
446C42—84no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—28no—
451C23—28no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—28no—
457C23—28no—
458C23#53414yes—
459C23#54418yes—
460C23D4×Q818yes—
461C23Q8214yes—
462C23#56314yes—
463C23C23×D411yescharacteristic
464C23C23×Q812yes—
465C23C2×H(5,𝔽2)11yescharacteristic
466C23C2×H(5,𝔽2)11yescharacteristic
467C23C2×H(5,𝔽2)12yes—
468C23C2×2−1+412yes—
469C23#58512yes—
470C23#58514yes—
471Q8—88no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—416no—
498C2×C4—416no—
499C2×C4—432no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
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—88no—
549C2×C4—816no—
550C2×C4#53514yes—
551C22—24no—
552C22—24no—
553C22—28no—
554C22—28no—
555C22—216no—
556C22—28no—
557C22—28no—
558C22#273418yes—
559C22C2×#53411yescharacteristic
560C22C2×#53511yescharacteristic
561C22#344911yescharacteristic
562C22C2×#54412yes—
563C22C2×D4×Q812yes—
564C22C2×Q8211yescharacteristic
565C22#348512yes—
566C22#350114yes—
567C22#350414yes—
568C22C2×#56311yescharacteristic
569C22#353312yes—
570C22#353312yes—
571C22#353412yes—
572C22#353412yes—
573C22#354114yes—
574C22#355314yes—
575C4—24no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—44no—
582C4—48no—
583C4—44no—
584C4—48no—
585C4—84no—
586C4—82no—
587C4—82no—
588C4—84no—
589C2—28no—
590C2#3695511yescharacteristic
591C2#3744011yescharacteristic
592C2#3958912yes—
593C2#3959412yes—
594C2#4061514yes—
595C2#4312612yes—
596C2C2×#273411yescharacteristic
597C2#5245312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #17206C2semidirect
4. #16883C2semidirect
5. #16629C2semidirect
6. #16563C2semidirect
7. #15856C2semidirect
8. #15790C2semidirect
133. #385Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
C22×#3982
#172062
#168832
#166292
#165632
#158562
#157902
#150922
#149202
#147202
#142652
#140442
#140392
#139232