Finite Group Databasebeta

← #6854390

Subgroups — #6854390

Order 512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic
2Q8×#125C211yesmaximal, characteristic
3#33316C211yesmaximal, characteristic
4#30976C214yesmaximal
5#30968C214yesmaximal
6#30312C212yesmaximal
7#29213C211yesmaximal, characteristic
8#29184C212yesmaximal
9#29150C211yesmaximal, characteristic
10#28518C212yesmaximal
11#26047C212yesmaximal
12#26043C212yesmaximal
13#25062C212yesmaximal
14#25060C212yesmaximal
15#22225C212yesmaximal
16C4×#491C211yesmaximal, characteristic
17#21345C211yesmaximal, characteristic
18C2×#501C2214yes—
19C2×#491C2211yescharacteristic
20C2×#479C2214yes—
21C2×#479C2214yes—
22#2868C2211yescharacteristic
23#2859C2212yes—
24#2859C2211yescharacteristic
25#2859C2212yes—
26#2582C2212yes—
27#2567—21no—
28#2552—21no—
29#2552C2212yes—
30#2552C2211yescharacteristic
31#2552C2211yescharacteristic
32#2552C2212yes—
33#2550—22no—
34#2507—21no—
35#2507—22no—
36#2507—22no—
37#2507—21no—
38#2507—22no—
39#2451C2214yes—
40#2369—21no—
41#2355—21no—
42Q8×C4⋊C4—21no—
43Q8×C4⋊C4—21no—
44Q8×C4⋊C4C2211yescharacteristic
45Q8×C4⋊C4C2211yescharacteristic
46Q8×C4⋊C4C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×#125C2212yes—
50C4×#125C2211yescharacteristic
51C42×Q8C2211yescharacteristic
52#2096C2214yes—
53#2095C2214yes—
54#2088C2214yes—
55#2086C2214yes—
56#2084C2214yes—
57#2083C2214yes—
58#2059C2214yes—
59#2059C2214yes—
60#2058C2212yes—
61#2057C2212yes—
62#2026C2214yes—
63#1992C2211yescharacteristic
64#1992C2212yes—
65#1984C2212yes—
66#1960C2211yescharacteristic
67#1960C2212yes—
68#1948C2212yes—
69#1947C2212yes—
70#1939C2212yes—
71#1935C2212yes—
72#1935C2212yes—
73#1928C2214yes—
74#1927C2214yes—
75#1921C2214yes—
76#1920C2214yes—
77#1917C2214yes—
78#1865C2214yes—
79#1863C2214yes—
80#1847C2212yes—
81#1847C2212yes—
82#1847C2212yes—
83#1845C2212yes—
84#1844C2212yes—
85#1836C2212yes—
86#1836C2211yescharacteristic
87#1803C2212yes—
88#1803C2212yes—
89#1767C2212yes—
90#1767C2211yescharacteristic
91C4×#103C2212yes—
92C4×#103C2212yes—
93#1745C2212yes—
94C4×#97C2211yescharacteristic
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×#125—22no—
100C2×#125C2311yescharacteristic
101C2×#125C2312yes—
102C2×#125C2312yes—
103C2×#125C2312yes—
104C2×C4×Q8—21no—
105C2×C4×Q8—21no—
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2314yes—
110C2×C4×Q8C2311yescharacteristic
111#501—48no—
112#491Q818yescomplemented
113#479—48no—
114#479—48no—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×C4⋊C8C2314yes—
122C2×C4⋊C8C2312yes—
123C2×#103C2312yes—
124C2×#103C2312yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×#97C2312yes—
130C2×#97C2311yescharacteristic
131C2×#97C2312yes—
132C2×#97C2311yescharacteristic
133C2×C4×C8C2312yes—
134#398—24no—
135#398—24no—
136#398—22no—
137#398—22no—
138#398—22no—
139#398—22no—
140#398—22no—
141#398—22no—
142#395—24no—
143#391—22no—
144#391—24no—
145#391C2314yes—
146#391C2312yes—
147#389—22no—
148#389—21no—
149#389—21no—
150#389—22no—
151#389—22no—
152#389—21no—
153#389—21no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—21no—
158#389—21no—
159#389—22no—
160#389—22no—
161#389—21no—
162#389—22no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—21no—
168#389—22no—
169#389—21no—
170#389—24no—
171#389C2311yescharacteristic
172#389C2311yescharacteristic
173#389C2312yes—
174#389C2312yes—
175#389C2314yes—
176#389C2314yes—
177#387—24no—
178#387—22no—
179#387—21no—
180#387—22no—
181#387—21no—
182#384—22no—
183#384—21no—
184#384—21no—
185#384—22no—
186#384—22no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#384C2312yes—
196#382—22no—
197#382C2312yes—
198#382C2311yescharacteristic
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C43C2312yes—
219C43C2311yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#336C2312yes—
229#336C2314yes—
230#336C2314yes—
231#336C2314yes—
232C22×Q8—22no—
233C22×Q8C2×D411yescharacteristic
234C22×Q8C2412yes—
235C22×C8C2414yes—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C4⋊Q8—22no—
239C4⋊Q8—44no—
240C4⋊Q8—44no—
241C4⋊Q8—82no—
242C4⋊Q8C2×Q814yes—
243#125—22no—
244#125—44no—
245#125—44no—
246#125—44no—
247#125—84no—
248#125C2×Q814yes—
249C4×Q8—24no—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×Q8—24no—
253C4×Q8—42no—
254C4×Q8—44no—
255C4×Q8—44no—
256C4×Q8—44no—
257C4×Q8—44no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D412yes—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D412yes—
318C2×C42C2×D411yescharacteristic
319C2×C42C2412yes—
320C2×C42C2412yes—
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324#107—28no—
325#107—44no—
326#107—44no—
327#106—28no—
328#106—44no—
329#106—44no—
330C4⋊C8—28no—
331C4⋊C8—24no—
332#103—24no—
333#103—24no—
334#103—44no—
335#103—44no—
336#103C2×Q818yes—
337#103C2×Q818yes—
338#97—24no—
339#97—22no—
340#97—24no—
341#97C2×Q814yes—
342C4×C8—24no—
343#95—24no—
344#95—22no—
345#95—21no—
346#95—21no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—21no—
351#95—21no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—21no—
359#95—21no—
360#95—24no—
361#95C2×D412yes—
362#95C2×D412yes—
363#95C2×D412yes—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—44no—
372C2×Q8—82no—
373C2×Q8—82no—
374C2×Q8#13714yes—
375C2×Q8C22×Q814yes—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D412yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C2511yescharacteristic
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8C22×Q814yes—
404C2×C8H(5,𝔽2)14yes—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4—82no—
438C4⋊C4—82no—
439C4⋊C4—81no—
440C4⋊C4—82no—
441C4⋊C4—81no—
442C4⋊C4—82no—
443C4⋊C4—84no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—84no—
451C4⋊C4—84no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4C22×Q812yes—
455C4⋊C4C22×Q812yes—
456C42—24no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—24no—
461C42—24no—
462C42—41no—
463C42—42no—
464C42—42no—
465C42—42no—
466C42—41no—
467C42—44no—
468C42—44no—
469C42—44no—
470C42—42no—
471C42—42no—
472C42—84no—
473C42C22×Q812yes—
474C422−1+414yes—
475C422−1+412yes—
476C23C23×D411yescharacteristic
477Q8—28no—
478Q8—24no—
479Q8—24no—
480Q8—42no—
481Q8—42no—
482Q8—164no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—44no—
533C2×C4#53612yes—
534C2×C4#53612yes—
535C2×C4D4×Q812yes—
536C2×C4D4×Q812yes—
537C2×C4C2×#13714yes—
538C2×C4C2×#13712yes—
539C2×C4#57512yes—
540C2×C4C23×Q811yescharacteristic
541C2×C4C2×H(5,𝔽2)12yes—
542C2×C4C2×2−1+412yes—
543C2×C4C2×2−1+411yescharacteristic
544C8—24no—
545C8—24no—
546C8—44no—
547C22C2×#53611yescharacteristic
548C22C2×D4×Q811yescharacteristic
549C22#349811yescharacteristic
550C22C22×#13711yescharacteristic
551C22C2×#57511yescharacteristic
552C22#358511yescharacteristic
553C22#358611yescharacteristic
554C4—24no—
555C4—22no—
556C4—22no—
557C4—22no—
558C4—42no—
559C4—42no—
560C4—44no—
561C4—42no—
562C4—42no—
563C4—41no—
564C4—42no—
565C4—41no—
566C4—82no—
567C4—81no—
568C4—82no—
569C4—84no—
570C4—81no—
571C4—82no—
572C4#323614yes—
573C4#350312yes—
574C4#354412yes—
575C2#3961711yescharacteristic
576C2C2×#323611yescharacteristic
577C2#6088611yescharacteristic
578C2Q8×#13711yescharacteristic
579C2#6136911yescharacteristic
580C2#6245111yescharacteristic
581C2#6245411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
112. #491Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
Q8×#1252
#333162
#309762
#309682
#303122
#292132
#291842
#291502
#285182
#260472
#260432
#250622
#250602
#222252
C4×#4912
#213452