Finite Group Databasebeta

← #6482293

Subgroups — #6482293

Order 512.

1476 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic
3#33150C212yesmaximal
4#33149C211yesmaximal, characteristic
5#33147C211yesmaximal, characteristic
6#29973C212yesmaximal
7#29969C211yesmaximal, characteristic
8#29967C211yesmaximal, characteristic
9#26520C214yesmaximal, complemented
10#26519C214yesmaximal, complemented
11#26454C214yesmaximal, complemented
12#26453C214yesmaximal, complemented
13#21955C211yesmaximal, characteristic
14#21841C212yesmaximal
15#21363C212yesmaximal
16C22×C4⋊Q8C2211yescharacteristic
17C2×#432C2212yes—
18#2857C2211yescharacteristic
19#2857C2211yescharacteristic
20#2851C2212yes—
21#2627—21no—
22#2627—21no—
23#2606—22no—
24#2605—22no—
25#2570C2212yes—
26#2570C2212yes—
27#2550C2212yes—
28#2550C2212yes—
29#2369—22no—
30#2369—22no—
31#2369C2214yes—
32#2369C2214yes—
33#2329—21no—
34#2329—21no—
35C2×#391—21no—
36C2×#391—21no—
37C2×#389—21no—
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40#2283C2212yes—
41C2×C4×C4⋊C4—21no—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46#2101C2214yes—
47#2100C2212yes—
48#2099C2212yes—
49#2088C2214yes—
50#2088C2214yes—
51#2087C2214yes—
52#2086C2214yes—
53#2059C2214yes—
54#2058C2212yes—
55#2057C2212yes—
56#1993C2214yes—
57#1993C2214yes—
58#1989C2214yes—
59#1988C2214yes—
60#1988C2214yes—
61#1988C2212yes—
62#1987C2212yes—
63#1949C2214yes—
64#1949C2214yes—
65#1949C2214yes—
66#1948C2212yes—
67#1947C2212yes—
68#1940C2214yes—
69#1940C2214yes—
70#1921C2214yes—
71#1920C2214yes—
72#1919C2214yes—
73#1918C2214yes—
74#1853C2212yes—
75#1853C2212yes—
76#1852C2212yes—
77#1852C2212yes—
78#1824C2212yes—
79#1824C2212yes—
80#1824C2212yes—
81#1824C2212yes—
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84#1797C2212yes—
85#1776C2214yes—
86#1776C2214yes—
87C23×Q8—21no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8Q818yescomplemented
91C2×C4⋊Q8C2314yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×#125C2312yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4D411yescharacteristic
101C22×C4⋊C4D411yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C42—21no—
107C22×C42—21no—
108C22×C42—21no—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#432—28no—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2314yes—
118C2×C4⋊C8C2314yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2312yes—
121C2×C22⋊C8C2312yes—
122C2×C22⋊C8C2312yes—
123C2×#97C2314yes—
124#400—22no—
125#400—22no—
126#398—22no—
127#398—22no—
128#398—22no—
129#398—22no—
130#395—22no—
131#395—22no—
132#391—21no—
133#391—21no—
134#391—22no—
135#391—22no—
136#391—24no—
137#391—24no—
138#391—24no—
139#391—24no—
140#389—24no—
141#389—24no—
142#389—21no—
143#389—21no—
144#389—22no—
145#389—21no—
146#389—21no—
147#389—22no—
148#389—22no—
149#389C2312yes—
150#389C2312yes—
151#389C2312yes—
152#389C2312yes—
153#389C2312yes—
154#389C2312yes—
155#389C2314yes—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#385—22no—
163#385—22no—
164#384C2311yescharacteristic
165#384C2311yescharacteristic
166#383—22no—
167#382C2312yes—
168#382C2312yes—
169#382C2312yes—
170#381—22no—
171#380—21no—
172#380—21no—
173#380C2312yes—
174#380C2312yes—
175C4×C4⋊C4—24no—
176C4×C4⋊C4—24no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4D412yes—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2314yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—22no—
192#376—22no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95D411yescharacteristic
199C2×#95D411yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215C22×Q8—28no—
216C22×Q8—24no—
217C22×Q8—21no—
218C22×Q8—21no—
219C22×Q8—82no—
220C22×Q8—82no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2411yescharacteristic
232C22×C8C2414yes—
233C22×C8C2414yes—
234C4⋊Q8—216no—
235C4⋊Q8—28no—
236C4⋊Q8—42no—
237C4⋊Q8—42no—
238C4⋊Q8C2×Q8116yes—
239#125—44no—
240C22⋊Q8—44no—
241C22⋊Q8—44no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—82no—
270C2×C4⋊C4—82no—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×Q814yes—
276C2×C4⋊C4C2×Q814yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—44no—
296C2×C42—82no—
297C2×C42—82no—
298C2×C42C2×D412yes—
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D412yes—
302C2×C42C2×Q814yes—
303C2×C42#4114yes—
304C2×C42C2412yes—
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307C2×C42C2412yes—
308#107—28no—
309#107—44no—
310#106—28no—
311#106—44no—
312C4⋊C8—28no—
313C4⋊C8—44no—
314C4⋊C8—42no—
315C4⋊C8—42no—
316C22⋊C8—24no—
317C22⋊C8—24no—
318#97—28no—
319#95—24no—
320#95—22no—
321#95—22no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95C2×D412yes—
341#95C2×D412yes—
342#95C2×D412yes—
343#95C2×D412yes—
344C24C22×D411yescharacteristic
345C2×Q8—216no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—84no—
349C2×Q8—84no—
350C2×Q8—81no—
351C2×Q8—81no—
352C2×Q8—81no—
353C2×Q8—81no—
354C2×Q8—816no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4C22⋊Q814yes—
393C22×C4C22⋊Q814yes—
394C22×C4#13714yes—
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×Q812yes—
405C22×C4C2×#4112yes—
406C22×C4C2511yescharacteristic
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C4⋊C4—24no—
414C4⋊C4—216no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—81no—
435C4⋊C4—82no—
436C4⋊C4—81no—
437C4⋊C4—82no—
438C4⋊C4—84no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C4⋊C4C22×Q814yes—
446C4⋊C4C22×Q814yes—
447C4⋊C42−1+414yes—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C42—24no—
451C42—22no—
452C42—22no—
453C42—41no—
454C42—41no—
455C42—42no—
456C42—48no—
457C42—82no—
458C42—82no—
459C42—82no—
460C42—82no—
461C42—84no—
462C42C22⋊Q818yes—
463C42C22×Q814yes—
464C42C2×#4114yes—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23C2×C22⋊Q812yes—
470C23C2×#13712yes—
471C23#57512yes—
472C23C23×D411yescharacteristic
473Q8—816no—
474Q8—162no—
475Q8—162no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—28no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—84no—
534C2×C4C2×C22⋊Q812yes—
535C2×C4C2×C22⋊Q814yes—
536C2×C4C2×C22⋊Q814yes—
537C2×C4C2×#13712yes—
538C2×C4C2×#13712yes—
539C2×C4C23×Q811yescharacteristic
540C2×C4C22×#4111yescharacteristic
541C2×C4C2×2−1+412yes—
542C2×C4#58512yes—
543C8—44no—
544C8—44no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22#307814yes—
552C22C22×C22⋊Q811yescharacteristic
553C22#348511yescharacteristic
554C22#348611yescharacteristic
555C22C22×#13711yescharacteristic
556C22C2×#57511yescharacteristic
557C22#358511yescharacteristic
558C22#358511yescharacteristic
559C4—22no—
560C4—22no—
561C4—24no—
562C4—24no—
563C4—22no—
564C4—42no—
565C4—41no—
566C4—42no—
567C4—41no—
568C4—44no—
569C4—44no—
570C4—44no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4#346112yes—
577C4#350212yes—
578C2—24no—
579C2#3515411yescharacteristic
580C2C2×#307811yescharacteristic
581C2#6049011yescharacteristic
582C2#6170911yescharacteristic
583C2#6171011yescharacteristic
584C2#6183011yescharacteristic
585C2#6183111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #26520C2semidirect
10. #26519C2semidirect
11. #26454C2semidirect
12. #26453C2semidirect
90. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351042
C2×#23692
#331502
#331492
#331472
#299732
#299692
#299672
#265202
#265192
#264542
#264532
#219552
#218412
#213632