Finite Group Databasebeta

← #847422

Subgroups — #847422

Order 512.

2970 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic
2C22×#391C211yesmaximal, characteristic
3#16630C212yesmaximal, complemented
4#16629C212yesmaximal, complemented
5#15860C214yesmaximal, complemented
6#15856C212yesmaximal, complemented
7#15854C212yesmaximal, complemented
8#14920C212yesmaximal
9#14914C212yesmaximal
10#14874C212yesmaximal
11#14695C212yesmaximal
12#14554C214yesmaximal, complemented
13#14044C212yesmaximal
14#14040C212yesmaximal
15#13923C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2583—28no—
21C2×#398C2211yescharacteristic
22C2×#398C2211yescharacteristic
23C2×#395C2211yescharacteristic
24C2×#395C2211yescharacteristic
25C2×#395C2214yes—
26C2×#395C2214yes—
27C2×#391—28no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#387C2214yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#380C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C22×#95C2212yes—
44C22×#95C2212yes—
45#1446C2214yes—
46#1446C2214yes—
47#1446C2214yes—
48#1445C2214yes—
49#1445C2214yes—
50#1444C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2214yes—
61#1436C2212yes—
62#1436C2212yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2214yes—
68#1434C2214yes—
69#1434C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8—28no—
78C23×Q8—24no—
79C23×Q8—42no—
80C23×Q8—42no—
81C23×Q8—44no—
82C23×Q8C2312yes—
83C23×Q8C2312yes—
84C23×Q8C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—42no—
92C2×C22⋊Q8—42no—
93C22×C4⋊C4—42no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4D418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C42—24no—
115C22×C42—44no—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120#398—24no—
121#398—24no—
122#395—216no—
123#395—24no—
124#395—24no—
125#395—48no—
126#391—28no—
127#391—28no—
128#391—28no—
129#391—28no—
130#389—28no—
131#389—28no—
132#387—48no—
133#385—28no—
134#385—28no—
135#385—44no—
136#385—44no—
137#385Q8116yescomplemented
138#380—42no—
139C4×C22⋊C4—24no—
140C4×C22⋊C4—42no—
141C2×#95—28no—
142C2×#95—28no—
143C2×#95C2311yescharacteristic
144C2×#95C2311yescharacteristic
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C25C2411yescharacteristic
173C22×Q8—232no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—22no—
177C22×Q8—28no—
178C22×Q8—24no—
179C22×Q8—28no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—42no—
186C22×Q8—41no—
187C22×Q8—41no—
188C22×Q8—44no—
189C22×Q8—432no—
190C22×Q8—432no—
191C22×Q8—416no—
192C22×Q8—416no—
193C22×Q8#4118yes—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×Q818yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C22⋊Q8—416no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—416no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×Q8116yescomplemented
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4C2×Q818yes—
281C2×C22⋊C4C2×Q818yes—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—28no—
285C2×C42—24no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—416no—
289C2×C42—42no—
290C2×C42—44no—
291C2×C42—44no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—42no—
312#95—42no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95C2×Q8116yescomplemented
321C24—22no—
322C24—22no—
323C24—24no—
324C24C22×D412yes—
325C24C22×Q814yes—
326C24H(5,𝔽2)12yes—
327C24H(5,𝔽2)12yes—
328C242−1+414yes—
329C24C2511yescharacteristic
330C2×Q8—216no—
331C2×Q8—216no—
332C2×Q8—48no—
333C2×Q8—48no—
334C2×Q8—416no—
335C2×Q8—416no—
336C2×Q8—4128no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—48no—
342C2×Q8—44no—
343C2×Q8—48no—
344C2×Q8—84no—
345C2×Q8—84no—
346C2×Q8—84no—
347C22×C4—216no—
348C22×C4—216no—
349C22×C4—28no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—216no—
359C22×C4—216no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—416no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—416no—
421C22×C4—416no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—416no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×Q814yes—
438C22×C4C22×Q818yes—
439C22×C4C22×Q818yes—
440C22×C4C22×Q814yes—
441C22×C4C2×#4118yes—
442C22×C4C2×#4114yes—
443C22×C4H(5,𝔽2)14yes—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—416no—
449C4⋊C4—48no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C4⋊C4—88no—
456C4⋊C4—88no—
457C4⋊C4—84no—
458C22⋊C4—216no—
459C22⋊C4—216no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C42—48no—
463C42—48no—
464C42—84no—
465C42—84no—
466C23—22no—
467C23—22no—
468C23—28no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—24no—
476C23—24no—
477C23#53414yes—
478C23#54418yes—
479C23D4×Q818yes—
480C23Q8214yes—
481C23#56314yes—
482C23C23×D411yescharacteristic
483C23C23×Q812yes—
484C23C22×#4112yes—
485C23C2×H(5,𝔽2)11yescharacteristic
486C23C2×H(5,𝔽2)11yescharacteristic
487C23C2×H(5,𝔽2)12yes—
488C23C2×2−1+412yes—
489C23#58514yes—
490Q8—432no—
491Q8—432no—
492Q8—88no—
493C2×C4—232no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—432no—
524C2×C4—432no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
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—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—88no—
569C2×C4—88no—
570C2×C4#52514yes—
571C2×C4#53514yes—
572C22—28no—
573C22—28no—
574C22—216no—
575C22—28no—
576C22—28no—
577C22—24no—
578C22—24no—
579C22#273418yes—
580C22C2×#52511yescharacteristic
581C22C2×#53411yescharacteristic
582C22C2×#53511yescharacteristic
583C22C2×#54412yes—
584C22C2×D4×Q812yes—
585C22C2×Q8211yescharacteristic
586C22Q8×#4114yes—
587C22#348512yes—
588C22#349012yes—
589C22#349012yes—
590C22#350114yes—
591C22#351614yes—
592C22C2×#56311yescharacteristic
593C22#354114yes—
594C22#354512yes—
595C22#354512yes—
596C4—24no—
597C4—24no—
598C4—48no—
599C4—44no—
600C4—48no—
601C4—44no—
602C4—48no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—84no—
608C4—82no—
609C4—82no—
610C2—28no—
611C2#3505011yescharacteristic
612C2#3654712yes—
613C2#3711911yescharacteristic
614C2#3844014yes—
615C2#3959412yes—
616C2#4634912yes—
617C2C2×#273411yescharacteristic
618C2#5992312yes—
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. #16630C2semidirect
4. #16629C2semidirect
5. #15860C2semidirect
6. #15856C2semidirect
7. #15854C2semidirect
12. #14554C2semidirect
137. #385Q8semidirect
269. C2×C4⋊C4C2×Q8semidirect
320. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C22×#3912
#166302
#166292
#158602
#158562
#158542
#149202
#149142
#148742
#146952
#145542
#140442
#140402
#139232