Finite Group Databasebeta

← #7500047

Subgroups — #7500047

Order 512.

1891 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59822C211yesmaximal, characteristic, complemented
2#39363C211yesmaximal, characteristic, complemented
3#32668C212yesmaximal, complemented
4#32628C212yesmaximal, complemented
5#30874C214yesmaximal, complemented
6#30733C214yesmaximal, complemented
7#30038C212yesmaximal, complemented
8#29364C212yesmaximal, complemented
9#27422C214yesmaximal, complemented
10#27190C214yesmaximal, complemented
11#25933C212yesmaximal, complemented
12#25631C212yesmaximal, complemented
13#23363C211yesmaximal, characteristic, complemented
14C2×#555—21no—
15C2×#550C2211yescharacteristic
16C2×D4×Q8C2211yescharacteristic
17C2×#536—21no—
18C2×#493C2212yes—
19C2×#486C2212yes—
20C2×#459C2212yes—
21C2×#459C2212yes—
22#2858C2211yescharacteristic
23#2733—21no—
24#2733—21no—
25#2681—24no—
26#2620—21no—
27#2548C2212yes—
28#2548C2212yes—
29#2548C2211yescharacteristic
30#2546C2211yescharacteristic
31#2541C2214yes—
32#2537—21no—
33#2452C2214yes—
34#2445C2214yes—
35#2436—21no—
36#2432—21no—
37#2414—24no—
38#2354C2214yes—
39C4×C4⋊Q8C2211yescharacteristic
40C4×#127C2212yes—
41#2293C2211yescharacteristic
42#2099C2212yes—
43#2093C2212yes—
44#2071C2214yes—
45#2067C2214yes—
46#2054C2214yes—
47#2041C2214yes—
48#2041C2212yes—
49#2023C2214yes—
50#2023C2214yes—
51#2011C2212yes—
52#2011C2214yes—
53#2011C2214yes—
54#2011C2214yes—
55#2011C2214yes—
56#1988C2212yes—
57#1984C2212yes—
58#1984C2212yes—
59#1982C2212yes—
60#1970C2214yes—
61#1934C2214yes—
62#1925C2214yes—
63#1925C2214yes—
64#1918C2214yes—
65#1916C2214yes—
66#1916C2214yes—
67#1879C2214yes—
68#1879C2214yes—
69#1879C2214yes—
70#1879C2214yes—
71#1879C2214yes—
72#1879C2214yes—
73#1862C2214yes—
74#1839C2212yes—
75C4×#102C2212yes—
76C4×#102C2212yes—
77C22×#41—21no—
78C23×Q8—21no—
79C22×D8C2314yes—
80C22×D8C2314yes—
81C22×D8C2314yes—
82#555—48no—
83#550—42no—
84#550—42no—
85D4×Q8—42no—
86D4×Q8—42no—
87#536—48no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8D411yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2312yes—
93C2×#127C2312yes—
94C2×#127C2311yescharacteristic
95C2×#127C2312yes—
96C2×#127C2312yes—
97C2×#124—21no—
98C2×#124—22no—
99C2×#124—22no—
100C2×#124—21no—
101C2×#124—21no—
102C2×#123—24no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—22no—
106C2×#121—21no—
107C2×#121—24no—
108C2×#121—22no—
109C2×#121—24no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2314yes—
118C2×#117—24no—
119C2×#117—21no—
120#493—44no—
121#486—44no—
122#459—44no—
123#459—44no—
124C2×#107C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C4×C8C2312yes—
136C2×C4×C8C2312yes—
137#398—22no—
138#396—24no—
139#394—24no—
140#394—22no—
141#394—22no—
142#394—22no—
143#394—22no—
144#394—22no—
145#394—24no—
146#393—24no—
147#392—24no—
148#392—24no—
149#391—21no—
150#391D411yescharacteristic
151#391D411yescharacteristic
152#390—21no—
153#390—22no—
154#390—22no—
155#390—22no—
156#390—22no—
157#390—22no—
158#390C2314yes—
159#390C2314yes—
160#390C2314yes—
161#390C2314yes—
162#390C2314yes—
163#390C2312yes—
164#389D414yes—
165#389C2314yes—
166#389C2312yes—
167#388—22no—
168#388—24no—
169#388—24no—
170#386—24no—
171#385—24no—
172#384D411yescharacteristic
173#384C2312yes—
174#380—24no—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2311yescharacteristic
178C4×C4⋊C4C2314yes—
179C4×C22⋊C4—21no—
180C43C2311yescharacteristic
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#336C2314yes—
186#336C2312yes—
187#336C2312yes—
188#336C2314yes—
189#336C2314yes—
190#336C2314yes—
191C2×#41—44no—
192C2×#41—44no—
193C2×#41—82no—
194C22×Q8—21no—
195C22×Q8—22no—
196C22×Q8—44no—
197C22×Q8—44no—
198C22×Q8—82no—
199C22×Q8C2×D411yescharacteristic
200C22×Q8C2×D411yescharacteristic
201C22×D4—22no—
202C22×D4—21no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—21no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4C2414yes—
212C22×D4C2412yes—
213C22×D4C2411yescharacteristic
214C23×C4—21no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C2×D8—28no—
219C2×D8—44no—
220C2×D8—44no—
221C2×D8—44no—
222C2×D8—44no—
223C2×D8—44no—
224C2×D8—44no—
225C2×D8—44no—
226C2×D8—44no—
227C22×C8C2414yes—
228C22×C8C2414yes—
229C4⋊Q8—24no—
230C4⋊Q8—42no—
231C4⋊Q8—41no—
232C4⋊Q8—41no—
233C4⋊Q8—44no—
234C4⋊Q8D818yescomplemented
235#127—24no—
236#127—44no—
237#127—42no—
238#127—41no—
239#127—41no—
240#127—42no—
241#127—42no—
242#127—42no—
243#127—42no—
244#124—44no—
245#124—48no—
246#124—48no—
247#124—44no—
248#124—82no—
249#123—416no—
250C22⋊Q8—48no—
251C22⋊Q8—44no—
252C22⋊Q8—416no—
253#121—48no—
254#121—44no—
255#121—416no—
256#121—88no—
257C4×Q8—41no—
258C4×Q8—41no—
259C4×Q8—41no—
260C4×Q8—41no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—44no—
266C4×D4—44no—
267C4×D4—42no—
268C4×D4—42no—
269C4×D4—48no—
270C4×D4—84no—
271#117—416no—
272#117—44no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4C2×D411yescharacteristic
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2411yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—21no—
308C2×C42C2×D412yes—
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2412yes—
312C2×C42C2411yescharacteristic
313C2×C42C2412yes—
314C2×C42C2412yes—
315#107—44no—
316C4⋊C8—42no—
317C4⋊C8—42no—
318#102—28no—
319#102—44no—
320#102—44no—
321#102—44no—
322#102—44no—
323#102—44no—
324#102—44no—
325#102—44no—
326C4×C8—42no—
327C4×C8—42no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—21no—
332#95—24no—
333#95C2×D414yes—
334#95C2×D414yes—
335C24—21no—
336C24—22no—
337C24—24no—
338#41—816no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—44no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—44no—
347C2×Q8—42no—
348C2×Q8—82no—
349C2×Q8—816no—
350C2×Q8C2×D814yes—
351C2×Q8C2×D814yes—
352C2×D4—21no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—44no—
365C2×D4—42no—
366C2×D4—44no—
367C2×D4—42no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—42no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—42no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4—82no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4—82no—
385C2×D4—82no—
386C2×D4—82no—
387C2×D4—82no—
388C2×D4—82no—
389C2×D4—84no—
390C2×D4H(5,𝔽2)12yes—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—42no—
402C22×C4—48no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C2511yescharacteristic
416D8—88no—
417D8—88no—
418D8—88no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—41no—
434C4⋊C4—42no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—48no—
441C4⋊C4—82no—
442C4⋊C4C2×D8116yes—
443C4⋊C4H(5,𝔽2)12yes—
444C22⋊C4—24no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—44no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—82no—
455C22⋊C4—84no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—21no—
465C42—21no—
466C42—24no—
467C42—42no—
468C42—41no—
469C42—44no—
470C42—44no—
471C42—42no—
472C42C2×D814yes—
473C23—22no—
474C23—42no—
475C23—41no—
476C23—42no—
477C23—41no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—44no—
490C23C23×D411yescharacteristic
491Q8—44no—
492Q8—44no—
493Q8—88no—
494Q8—84no—
495D4—24no—
496D4—44no—
497D4—44no—
498D4—44no—
499D4—42no—
500D4—42no—
501D4—44no—
502D4—44no—
503D4—42no—
504D4—88no—
505D4—88no—
506D4—84no—
507D4—88no—
508D4—164no—
509D4—164no—
510D4—164no—
511D4—162no—
512D4—164no—
513D4—164no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—48no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—88no—
545C2×C4—88no—
546C2×C4—816no—
547C2×C4C22×D812yes—
548C2×C4C22×D814yes—
549C2×C4C22×D814yes—
550C2×C4C22×D814yes—
551C2×C4C2×H(5,𝔽2)11yescharacteristic
552C2×C4C2×H(5,𝔽2)11yescharacteristic
553C2×C4#58514yes—
554C8—44no—
555C8—44no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—84no—
563C22—82no—
564C22—84no—
565C22—84no—
566C22—84no—
567C22—84no—
568C22—82no—
569C22—84no—
570C22—84no—
571C22—84no—
572C22—84no—
573C22#348411yescharacteristic
574C22#348712yes—
575C22C23×D811yescharacteristic
576C22#358511yescharacteristic
577C22#358512yes—
578C4—22no—
579C4—24no—
580C4—24no—
581C4—24no—
582C4—42no—
583C4—41no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—82no—
588C4—82no—
589C4—84no—
590C4#354512yes—
591C4#355112yes—
592C2—82no—
593C2—162no—
594C2—164no—
595C2#5208311yescharacteristic
596C2#6155911yescharacteristic
597C2#6167311yescharacteristic
598C2#6188012yes—
599C2#6193812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59822C2semidirect
2. #39363C2semidirect
3. #32668C2semidirect
4. #32628C2semidirect
5. #30874C2semidirect
6. #30733C2semidirect
7. #30038C2semidirect
8. #29364C2semidirect
9. #27422C2semidirect
10. #27190C2semidirect
11. #25933C2semidirect
12. #25631C2semidirect
13. #23363C2semidirect
234. C4⋊Q8D8semidirect

Maximal subgroups

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

subgroupindex
#598222
#393632
#326682
#326282
#308742
#307332
#300382
#293642
#274222
#271902
#259332
#256312
#233632