Finite Group Databasebeta

← #6461797

Subgroups — #6461797

Order 512.

1558 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35313C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#29526C214yesmaximal, complemented
4#29525C214yesmaximal, complemented
5#29520C212yesmaximal, complemented
6#29519C212yesmaximal, complemented
7#29515C212yesmaximal, complemented
8#29512C212yesmaximal, complemented
9#23744C211yesmaximal, characteristic
10#23732C211yesmaximal, characteristic
11#23552C211yesmaximal, characteristic
12C2×#2024C211yesmaximal, characteristic
13#21160C212yesmaximal
14#21088C212yesmaximal
15#21064C212yesmaximal
16#21036C212yesmaximal
17#20368C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#419C2212yes—
22C22×#103C2212yes—
23C2×#405C2212yes—
24#2863C2212yes—
25#2863C2212yes—
26#2645C2212yes—
27#2645C2212yes—
28#2514—21no—
29#2514—21no—
30#2511—21no—
31#2511—21no—
32#2493—22no—
33#2493—22no—
34#2369—21no—
35#2369—21no—
36#2365—21no—
37#2365—21no—
38#2345—22no—
39Q8×C22⋊C4—22no—
40#2331C2214yes—
41#2317C2211yescharacteristic
42#2316C2211yescharacteristic
43#2316C2212yes—
44#2310C2211yescharacteristic
45#2310C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48#2296C2211yescharacteristic
49C2×#391C2211yescharacteristic
50#2285C2211yescharacteristic
51#2274C2212yes—
52C4×#117C2211yescharacteristic
53#2101C2212yes—
54#2101C2212yes—
55#2100C2211yescharacteristic
56#2100C2211yescharacteristic
57#2099C2211yescharacteristic
58#2099C2211yescharacteristic
59#2097C2214yes—
60#2096C2212yes—
61#2095C2212yes—
62#2058C2212yes—
63#2058C2212yes—
64#2057C2212yes—
65#2057C2212yes—
66#2024C2212yes—
67#2024C2212yes—
68#2024C2214yes—
69#1944C2214yes—
70#1943C2214yes—
71#1942C2214yes—
72#1941C2214yes—
73#1935C2214yes—
74#1935C2214yes—
75#1933C2214yes—
76#1933C2214yes—
77#1928C2214yes—
78#1928C2214yes—
79#1927C2214yes—
80#1927C2214yes—
81#1921C2214yes—
82#1920C2214yes—
83#1919C2214yes—
84#1918C2214yes—
85#1746C2214yes—
86#1745C2212yes—
87#1744C2212yes—
88#1736C2212yes—
89#1735C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C2×#117C2312yes—
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C42C2311yescharacteristic
115#428—28no—
116#428—28no—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121#419—28no—
122C2×#103—28no—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127#405—28no—
128C2×#97C2314yes—
129C2×C4×C8C2314yes—
130#401—22no—
131#401—22no—
132#400—22no—
133#400—22no—
134#400—22no—
135#400—22no—
136#400—22no—
137#400—22no—
138#391—21no—
139#391—21no—
140#391—21no—
141#391—21no—
142#391—22no—
143#391—22no—
144#391—21no—
145#391—21no—
146#391—24no—
147#391—24no—
148#391C2312yes—
149#391C2312yes—
150#391C2312yes—
151#391C2312yes—
152#391C2312yes—
153#391C2312yes—
154#389—22no—
155#389—22no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389C2314yes—
161#389C2314yes—
162#389C2311yescharacteristic
163#389C2311yescharacteristic
164#389C2311yescharacteristic
165#389C2311yescharacteristic
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—24no—
177#387—24no—
178#385—21no—
179#385—21no—
180#385—22no—
181#384—21no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#383—22no—
185#383—22no—
186#382—21no—
187#382C2312yes—
188#381—22no—
189#381—22no—
190#380—21no—
191#380—21no—
192#380—24no—
193#380—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2314yes—
208C4×C22⋊C4—24no—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4D412yes—
211#376C2312yes—
212C2×#95—21no—
213C2×#95—21no—
214C43C2312yes—
215#340C2314yes—
216#340C2314yes—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#336C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2314yes—
233#336C2314yes—
234C22×Q8—28no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8—24no—
239C22×Q8—24no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—82no—
243C22×Q8—82no—
244C22×Q8C2412yes—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×M16—24no—
252C2×M16—24no—
253C2×M16#4118yes—
254C22×C8—24no—
255C22×C8—24no—
256C22×C8—24no—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262C4⋊Q8—28no—
263C4⋊Q8—28no—
264C4×Q8—84no—
265C4×Q8—84no—
266#117—24no—
267#117—22no—
268#117—22no—
269#117—24no—
270#117—24no—
271#117—24no—
272#117—22no—
273#117—22no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4#4118yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2414yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D412yes—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42C2×D412yes—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2412yes—
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329#107—28no—
330#107—28no—
331#106—28no—
332#106—28no—
333#103—28no—
334#103—28no—
335#103—28no—
336#103—28no—
337#97—28no—
338C4×C8—28no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—24no—
344#95—24no—
345#95—22no—
346#95—22no—
347#95—24no—
348#95—24no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369C24C22×D411yescharacteristic
370C2×Q8—28no—
371C2×Q8—28no—
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—44no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—44no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—82no—
384C2×Q8—82no—
385C2×Q8—816no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C2×#4114yes—
420C22×C4C2×#4114yes—
421C22×C42−1+412yes—
422C22×C42−1+412yes—
423C22×C4C2511yescharacteristic
424M16—216no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—48no—
434C2×C8C2×#4118yes—
435C4⋊C4—28no—
436C4⋊C4—28no—
437C4⋊C4—28no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—81no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—81no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4C2×#4118yes—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—42no—
473C22⋊C4—44no—
474C22⋊C4—42no—
475C42—22no—
476C42—22no—
477C42—24no—
478C42—24no—
479C42—24no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—44no—
488C42—42no—
489C42—84no—
490C42—84no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23#52512yes—
496C23#53212yes—
497C23#53612yes—
498C23C23×D411yescharacteristic
499Q8—416no—
500Q8—44no—
501Q8—44no—
502Q8—88no—
503Q8—84no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—88no—
556C2×C4—84no—
557C2×C4—162no—
558C2×C4#52512yes—
559C2×C4#53212yes—
560C2×C4#53612yes—
561C2×C4#55218yes—
562C2×C4#55618yes—
563C2×C4#57712yes—
564C2×C4#57812yes—
565C2×C4C22×#4112yes—
566C2×C4C22×#4112yes—
567C2×C4C2×2−1+411yescharacteristic
568C2×C4C2×2−1+411yescharacteristic
569C8—28no—
570C8—42no—
571C8—42no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22#260414yes—
579C22C2×#52511yescharacteristic
580C22C2×#53211yescharacteristic
581C22C2×#53611yescharacteristic
582C22C2×#57711yescharacteristic
583C22C2×#57811yescharacteristic
584C22#358511yescharacteristic
585C22#358611yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—41no—
591C4—48no—
592C4—41no—
593C4—42no—
594C4—42no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4#260414yes—
604C4C2×#55212yes—
605C4C2×#55612yes—
606C2—24no—
607C2C2×#260411yescharacteristic
608C2#6057911yescharacteristic
609C2#6058311yescharacteristic
610C2#6072311yescharacteristic
611C2#6072711yescharacteristic
612C2#6088911yescharacteristic
613C2#6089311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29526C2semidirect
4. #29525C2semidirect
5. #29520C2semidirect
6. #29519C2semidirect
7. #29515C2semidirect
8. #29512C2semidirect

Maximal subgroups

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

subgroupindex
#353132
#342162
#295262
#295252
#295202
#295192
#295152
#295122
#237442
#237322
#235522
C2×#20242
#211602
#210882
#210642
#210362
#203682