Finite Group Databasebeta

← #6482292

Subgroups — #6482292

Order 512.

1584 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#33151C212yesmaximal
4#33148C211yesmaximal, characteristic
5#33146C211yesmaximal, characteristic
6#29972C212yesmaximal
7#29968C211yesmaximal, characteristic
8#29966C211yesmaximal, characteristic
9#26521C214yesmaximal, complemented
10#26518C214yesmaximal, complemented
11#26455C214yesmaximal, complemented
12#26452C214yesmaximal, complemented
13#21955C211yesmaximal, characteristic
14#21840C212yesmaximal
15#21363C212yesmaximal
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8—21no—
18C2×#432C2212yes—
19#2857C2211yescharacteristic
20#2857C2211yescharacteristic
21#2851C2212yes—
22#2627—21no—
23#2627—21no—
24#2606—22no—
25#2605—22no—
26#2570C2212yes—
27#2570C2212yes—
28#2550C2212yes—
29#2550C2212yes—
30#2365—22no—
31#2365—22no—
32#2365C2214yes—
33#2365C2214yes—
34#2329—21no—
35#2329—21no—
36C2×#391—21no—
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#387—21no—
40C2×#387—21no—
41#2283C2212yes—
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#1988C2212yes—
60#1987C2214yes—
61#1987C2214yes—
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#1823C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1823C2212yes—
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—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8—48no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C42—21no—
110C22×C42—21no—
111C22×C42—21no—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#432—28no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2314yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2314yes—
123C2×C4⋊C8C2312yes—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2312yes—
126C2×#97C2314yes—
127#400—22no—
128#400—22no—
129#398—22no—
130#398—22no—
131#398—22no—
132#398—22no—
133#395—22no—
134#395—22no—
135#391—24no—
136#391—24no—
137#391—21no—
138#391—21no—
139#391—22no—
140#391—22no—
141#389—21no—
142#389—21no—
143#389—22no—
144#389—21no—
145#389—21no—
146#389—22no—
147#389—22no—
148#389C2312yes—
149#389C2312yes—
150#389C2312yes—
151#389C2312yes—
152#389C2312yes—
153#389C2312yes—
154#389C2314yes—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—24no—
162#387—24no—
163#387—24no—
164#387—24no—
165#385—22no—
166#385—22no—
167#384C2311yescharacteristic
168#384C2311yescharacteristic
169#383—22no—
170#382C2312yes—
171#382C2312yes—
172#382C2312yes—
173#381—22no—
174#380—21no—
175#380—21no—
176#380C2312yes—
177#380C2312yes—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4D412yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2314yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2314yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C22⋊C4—22no—
193#376—22no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95D411yescharacteristic
200C2×#95D411yescharacteristic
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2312yes—
214#336C2312yes—
215#336C2314yes—
216C22×Q8—28no—
217C22×Q8—24no—
218C22×Q8—21no—
219C22×Q8—21no—
220C22×Q8—44no—
221C22×Q8—44no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2411yescharacteristic
233C22×C8C2414yes—
234C22×C8C2414yes—
235C4⋊Q8—216no—
236C4⋊Q8—28no—
237C4⋊Q8—42no—
238C4⋊Q8—42no—
239C4⋊Q8C2×Q8116yes—
240#125—44no—
241C22⋊Q8—44no—
242C22⋊Q8—44no—
243C4×Q8—416no—
244C4×Q8—416no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×Q814yes—
279C2×C4⋊C4C2×Q814yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42C2×D412yes—
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D412yes—
305C2×C42C2×Q814yes—
306C2×C42#4114yes—
307C2×C42C2412yes—
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2412yes—
311#107—28no—
312#107—44no—
313#106—28no—
314#106—44no—
315C4⋊C8—28no—
316C4⋊C8—44no—
317C4⋊C8—42no—
318C4⋊C8—42no—
319C22⋊C8—24no—
320C22⋊C8—24no—
321#97—28no—
322#95—24no—
323#95—24no—
324#95—22no—
325#95—22no—
326#95—24no—
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#95—22no—
341#95—22no—
342#95—22no—
343#95C2×D412yes—
344#95C2×D412yes—
345#95C2×D412yes—
346#95C2×D412yes—
347C24C22×D411yescharacteristic
348C2×Q8—216no—
349C2×Q8—48no—
350C2×Q8—432no—
351C2×Q8—48no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—81no—
355C2×Q8—81no—
356C2×Q8—81no—
357C2×Q8—81no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
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—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4C22⋊Q814yes—
396C22×C4C22⋊Q814yes—
397C22×C4AGL(1,ℤ/8)14yes—
398C22×C4C22×D412yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×Q812yes—
408C22×C4C2×#4112yes—
409C22×C4C2511yescharacteristic
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C4⋊C4—24no—
417C4⋊C4—216no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—48no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—81no—
441C4⋊C4—81no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4C22×Q814yes—
449C4⋊C4C22×Q814yes—
450C4⋊C42−1+414yes—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C42—24no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—42no—
459C42—48no—
460C42—48no—
461C42—44no—
462C42—44no—
463C42—82no—
464C42—82no—
465C42C22⋊Q818yes—
466C42C22×Q814yes—
467C42C2×#4114yes—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23C2×C22⋊Q812yes—
473C23C2×AGL(1,ℤ/8)12yes—
474C23#57512yes—
475C23C23×D411yescharacteristic
476Q8—432no—
477Q8—162no—
478Q8—162no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—28no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—48no—
513C2×C4—48no—
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—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—84no—
537C2×C4C2×C22⋊Q812yes—
538C2×C4C2×C22⋊Q814yes—
539C2×C4C2×C22⋊Q814yes—
540C2×C4C2×AGL(1,ℤ/8)12yes—
541C2×C4C2×AGL(1,ℤ/8)12yes—
542C2×C4C23×Q811yescharacteristic
543C2×C4C22×#4111yescharacteristic
544C2×C4C2×2−1+412yes—
545C2×C4#58512yes—
546C8—44no—
547C8—44no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22#307714yes—
555C22C22×C22⋊Q811yescharacteristic
556C22#348511yescharacteristic
557C22#348611yescharacteristic
558C22C22×AGL(1,ℤ/8)11yescharacteristic
559C22C2×#57511yescharacteristic
560C22#358611yescharacteristic
561C22#358611yescharacteristic
562C4—22no—
563C4—22no—
564C4—24no—
565C4—24no—
566C4—22no—
567C4—42no—
568C4—41no—
569C4—42no—
570C4—41no—
571C4—44no—
572C4—44no—
573C4—44no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4#346112yes—
580C4#350212yes—
581C2—24no—
582C2#3515411yescharacteristic
583C2C2×#307711yescharacteristic
584C2#6049111yescharacteristic
585C2#6170611yescharacteristic
586C2#6171211yescharacteristic
587C2#6182711yescharacteristic
588C2#6183211yescharacteristic
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. #26521C2semidirect
10. #26518C2semidirect
11. #26455C2semidirect
12. #26452C2semidirect
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×#23652
#331512
#331482
#331462
#299722
#299682
#299662
#265212
#265182
#264552
#264522
#219552
#218402
#213632