Finite Group Databasebeta

← #6484988

Subgroups — #6484988

Order 512.

1506 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3#33100C212yesmaximal
4#33098C212yesmaximal
5#29913C212yesmaximal
6#29909C212yesmaximal
7#26333C214yesmaximal, complemented
8#26293C214yesmaximal, complemented
9#25075C214yesmaximal, complemented
10#25009C214yesmaximal, complemented
11#21986C211yesmaximal, characteristic
12#21866C211yesmaximal, characteristic
13#21822C211yesmaximal, characteristic
14#21394C211yesmaximal, characteristic
15#21365C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20#2855C2212yes—
21#2855C2212yes—
22#2627—21no—
23#2627—21no—
24#2582C2212yes—
25#2582C2212yes—
26#2581—22no—
27#2568C2212yes—
28#2567C2212yes—
29#2565—22no—
30#2355—22no—
31#2355—22no—
32#2355C2214yes—
33#2355C2214yes—
34#2317—21no—
35#2310—21no—
36C2×#391—21no—
37C2×#387—21no—
38C2×#387—21no—
39#2285C2211yescharacteristic
40C2×#384C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×C4×C4⋊C4—21no—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47#2101C2214yes—
48#2100C2214yes—
49#2097C2214yes—
50#2095C2214yes—
51#2085C2214yes—
52#2083C2214yes—
53#2074C2214yes—
54#2072C2214yes—
55#1993C2212yes—
56#1988C2212yes—
57#1985C2214yes—
58#1985C2214yes—
59#1981C2212yes—
60#1960C2212yes—
61#1949C2214yes—
62#1948C2212yes—
63#1947C2212yes—
64#1928C2214yes—
65#1927C2214yes—
66#1927C2214yes—
67#1927C2214yes—
68#1920C2214yes—
69#1920C2214yes—
70#1919C2214yes—
71#1918C2214yes—
72#1872C2212yes—
73#1872C2212yes—
74#1872C2212yes—
75#1872C2212yes—
76#1853C2212yes—
77#1853C2212yes—
78#1852C2212yes—
79#1852C2212yes—
80#1847C2212yes—
81#1836C2212yes—
82#1817C2212yes—
83#1817C2212yes—
84#1805C2214yes—
85#1776C2214yes—
86#1763C2214yes—
87C23×Q8D411yescharacteristic
88#533—24no—
89#533—24no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#125—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8C2312yes—
95C2×C4×Q8—21no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8D412yes—
99C2×C4×Q8D412yes—
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2312yes—
103C2×#117—21no—
104C2×#117—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C42—21no—
115C22×C42—21no—
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#432—24no—
121#432—24no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×#97C2312yes—
135C2×#97C2312yes—
136#400—22no—
137#400—22no—
138#398—22no—
139#398—22no—
140#398—22no—
141#398—22no—
142#395—22no—
143#395—22no—
144#391—22no—
145#391—22no—
146#391—22no—
147#391—22no—
148#391—24no—
149#391—24no—
150#391C2312yes—
151#391C2312yes—
152#391C2312yes—
153#391C2312yes—
154#389—22no—
155#389—22no—
156#389C2312yes—
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#387—24no—
161#387—24no—
162#387—24no—
163#387—24no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#385—22no—
169#385—22no—
170#384—22no—
171#384—22no—
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#384C2314yes—
175#382—22no—
176#382—21no—
177#382—22no—
178#382—21no—
179#382—21no—
180#382C2312yes—
181#382C2314yes—
182#382C2312yes—
183#381—22no—
184#381—22no—
185#380—21no—
186#380C2312yes—
187C4×C4⋊C4—24no—
188C4×C4⋊C4—24no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2314yes—
196C4×C4⋊C4C2314yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2314yes—
216#336C2314yes—
217#336C2314yes—
218C22×Q8—24no—
219C22×Q8—24no—
220C22×Q8—21no—
221C22×Q8—42no—
222C22×Q8C2×D412yes—
223C22×Q8C2×D412yes—
224C22×Q8C2412yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2411yescharacteristic
236C22×C8C2414yes—
237C22×C8C2414yes—
238C4⋊Q8—22no—
239C4⋊Q8—22no—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242#125—44no—
243C22⋊Q8—24no—
244C22⋊Q8—24no—
245C22⋊Q8—44no—
246C4×Q8—28no—
247C4×Q8—28no—
248C4×Q8—24no—
249C4×Q8—24no—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×Q8—24no—
253C4×Q8—24no—
254C4×Q8—44no—
255#117—44no—
256#117—44no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—82no—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—44no—
311C2×C42—82no—
312C2×C42—82no—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42#4114yes—
318C2×C42#4114yes—
319C2×C42C2412yes—
320C2×C42C2412yes—
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323#107—44no—
324#107—44no—
325#106—28no—
326#106—44no—
327C4⋊C8—24no—
328C4⋊C8—24no—
329#103—28no—
330#103—44no—
331#103—44no—
332#103—44no—
333C22⋊C8—24no—
334C22⋊C8—24no—
335#97—24no—
336#97—24no—
337#95—24no—
338#95—22no—
339#95—22no—
340#95—24no—
341#95—24no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361C24C22×D411yescharacteristic
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—416no—
373C2×Q8—42no—
374C2×Q8—81no—
375C2×Q8—81no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4#12314yes—
414C22×C4#12314yes—
415C22×C4#13714yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2511yescharacteristic
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—48no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—84no—
468C4⋊C42−1+414yes—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C42—24no—
472C42—24no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—44no—
478C42—44no—
479C42—48no—
480C42—82no—
481C42—82no—
482C42—84no—
483C42#12314yes—
484C42#12314yes—
485C42C2×#4112yes—
486C42C2×#4112yes—
487C42C2×#4112yes—
488C42C2×#4112yes—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23C2×#12312yes—
494C23C2×#13712yes—
495C23#57512yes—
496C23C23×D411yescharacteristic
497Q8—48no—
498Q8—48no—
499Q8—44no—
500Q8—44no—
501Q8—84no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
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—84no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4C2×#12312yes—
560C2×C4C2×#12312yes—
561C2×C4C2×#12312yes—
562C2×C4C2×#12312yes—
563C2×C4C2×#12312yes—
564C2×C4C2×#13714yes—
565C2×C4C22×#4111yescharacteristic
566C2×C4C22×#4111yescharacteristic
567C2×C4C2×2−1+412yes—
568C2×C4#58512yes—
569C8—44no—
570C8—44no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22#309514yes—
578C22C22×#12311yescharacteristic
579C22#348611yescharacteristic
580C22#348711yescharacteristic
581C22C22×#13711yescharacteristic
582C22C2×#57511yescharacteristic
583C22#358511yescharacteristic
584C22#358611yescharacteristic
585C4—22no—
586C4—22no—
587C4—24no—
588C4—22no—
589C4—22no—
590C4—22no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—44no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4#346112yes—
603C4#350712yes—
604C2—24no—
605C2#3518211yescharacteristic
606C2C2×#309511yescharacteristic
607C2#6051711yescharacteristic
608C2#6182611yescharacteristic
609C2#6183211yescharacteristic
610C2#6188911yescharacteristic
611C2#6189511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #26333C2semidirect
8. #26293C2semidirect
9. #25075C2semidirect
10. #25009C2semidirect

Maximal subgroups

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

subgroupindex
#351792
C2×#23552
#331002
#330982
#299132
#299092
#263332
#262932
#250752
#250092
#219862
#218662
#218222
#213942
#213652