Finite Group Databasebeta

← #6434409

Subgroups — #6434409

Order 512.

1376 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34638C211yesmaximal, characteristic
2C2×#2645C211yesmaximal, characteristic
3#30394C214yesmaximal, complemented
4#30393C214yesmaximal, complemented
5#30391C214yesmaximal, complemented
6#30388C214yesmaximal, complemented
7#23747C211yesmaximal, characteristic
8#23732C211yesmaximal, characteristic
9#23670C212yesmaximal
10#23657C211yesmaximal, characteristic
11#23629C211yesmaximal, characteristic
12#23619C212yesmaximal
13#21159C212yesmaximal
14#21094C212yesmaximal
15#20429C211yesmaximal, characteristic
16C2×#428C2212yes—
17C2×#428C2212yes—
18C2×#423C2212yes—
19C2×#419C2212yes—
20C2×#419C2212yes—
21C2×C4×M16C2212yes—
22#2647C2212yes—
23#2647C2212yes—
24#2647C2212yes—
25#2647C2212yes—
26#2646—21no—
27#2646—21no—
28#2646—21no—
29#2646—21no—
30#2645—22no—
31#2645—22no—
32#2645C2214yes—
33#2645C2214yes—
34#2402—22no—
35#2402—22no—
36C2×#401—21no—
37C2×#400—21no—
38C2×#400—21no—
39C2×#400—21no—
40#2296C2212yes—
41#2296C2212yes—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48#2285C2211yescharacteristic
49#2285C2211yescharacteristic
50C2×#376C2211yescharacteristic
51#2101C2214yes—
52#2100C2212yes—
53#2099C2212yes—
54#2097C2214yes—
55#2096C2212yes—
56#2095C2212yes—
57#2088C2214yes—
58#2088C2214yes—
59#2087C2214yes—
60#2086C2214yes—
61#2085C2214yes—
62#2084C2212yes—
63#2083C2212yes—
64#2074C2214yes—
65#2072C2212yes—
66#2072C2212yes—
67#2068C2214yes—
68#2068C2214yes—
69#2066C2214yes—
70#2066C2214yes—
71#1944C2214yes—
72#1943C2214yes—
73#1942C2214yes—
74#1941C2214yes—
75#1935C2214yes—
76#1935C2214yes—
77#1933C2214yes—
78#1933C2214yes—
79#1768C2214yes—
80#1768C2214yes—
81#1736C2212yes—
82#1736C2212yes—
83#1736C2212yes—
84#1736C2212yes—
85#1728C2211yescharacteristic
86#1728C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C22×M16C2312yes—
89C22×M16C2312yes—
90C2×#117C2312yes—
91C2×#117C2312yes—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101#428—28no—
102#428—44no—
103C2×#107C2314yes—
104C2×#107C2314yes—
105C2×#106C2314yes—
106C2×#106C2314yes—
107#423—28no—
108C2×C4⋊C8C2314yes—
109C2×C4⋊C8C2314yes—
110#419—28no—
111#419—44no—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C4×M16—44no—
117C2×#97C2314yes—
118C2×C4×C8C2314yes—
119#401—24no—
120#401—24no—
121#400—24no—
122#400—24no—
123#400—24no—
124#400—24no—
125#400—24no—
126#400—24no—
127#400—22no—
128#400—22no—
129#400—22no—
130#400—22no—
131#400—22no—
132#400—22no—
133#400—22no—
134#400—22no—
135#398—22no—
136#398—22no—
137#398—22no—
138#398—22no—
139#398—22no—
140#398—22no—
141#398—22no—
142#398—22no—
143#391—22no—
144#391—21no—
145#391—21no—
146#391—22no—
147#391C2314yes—
148#391C2312yes—
149#391C2312yes—
150#391C2314yes—
151#389—21no—
152#389—21no—
153#389—22no—
154#389—22no—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#389C2314yes—
162#389C2314yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#385—22no—
168#385—22no—
169#385—22no—
170#385—22no—
171#384C2311yescharacteristic
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#382C2312yes—
176#382C2312yes—
177#380—21no—
178#380—21no—
179#380—21no—
180#380—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189#376D412yes—
190#376D412yes—
191#376C2312yes—
192#376C2312yes—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95D411yescharacteristic
203C2×#95D411yescharacteristic
204C2×#95D411yescharacteristic
205C2×#95D411yescharacteristic
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2312yes—
217#336C2312yes—
218C22×Q8—21no—
219C22×Q8—24no—
220C22×Q8—22no—
221C22×Q8—24no—
222C22×Q8—22no—
223C22×Q8C2412yes—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2411yescharacteristic
235C2×M16—24no—
236C2×M16—24no—
237C2×M16—24no—
238C2×M16—24no—
239C2×M16—24no—
240C2×M16#4118yes—
241C22×C8C2414yes—
242C22×C8C2414yes—
243#117—24no—
244#117—24no—
245#117—24no—
246#117—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
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—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—82no—
275C2×C4⋊C4—82no—
276C2×C4⋊C4—82no—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42C2×D412yes—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D412yes—
296C2×C42C2411yescharacteristic
297C2×C42C2411yescharacteristic
298C2×C42C2412yes—
299C2×C42C2412yes—
300#107—28no—
301#107—44no—
302#106—28no—
303#106—44no—
304C4⋊C8—28no—
305C4⋊C8—28no—
306#103—28no—
307#103—28no—
308#103—44no—
309#103—44no—
310#97—44no—
311C4×C8—44no—
312#95—22no—
313#95—22no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—22no—
325#95—22no—
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#95C2×D412yes—
337#95C2×D412yes—
338#95C2×D412yes—
339#95C2×D412yes—
340#95C2×D412yes—
341#95C2×D412yes—
342#95C2×D412yes—
343#95C2×D412yes—
344C24C22×D411yescharacteristic
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—48no—
353C2×Q8—48no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
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×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C2×#4114yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+414yes—
404C22×C42−1+412yes—
405C22×C4C2511yescharacteristic
406M16—216no—
407M16—48no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8C2×#4118yes—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4—81no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4—84no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4—84no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C42—24no—
452C42—22no—
453C42—22no—
454C42—24no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—44no—
462C42—44no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23#53212yes—
468C23#53612yes—
469C23#53712yes—
470C23C23×D411yescharacteristic
471Q8—44no—
472Q8—44no—
473Q8—84no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
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—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4—84no—
532C2×C4#53212yes—
533C2×C4#53612yes—
534C2×C4#53712yes—
535C2×C4#53918yes—
536C2×C4#54118yes—
537C2×C4C22×#4112yes—
538C2×C4C2×H(5,𝔽2)11yescharacteristic
539C2×C4C2×2−1+412yes—
540C2×C4C2×2−1+411yescharacteristic
541C8—28no—
542C8—44no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22#266414yes—
550C22C2×#53211yescharacteristic
551C22C2×#53611yescharacteristic
552C22C2×#53711yescharacteristic
553C22#358511yescharacteristic
554C22#358611yescharacteristic
555C22#358611yescharacteristic
556C22#358611yescharacteristic
557C4—42no—
558C4—42no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—42no—
564C4—42no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4#266414yes—
574C4C2×#53912yes—
575C4C2×#54112yes—
576C2—24no—
577C2C2×#266411yescharacteristic
578C2#6070311yescharacteristic
579C2#6070511yescharacteristic
580C2#6086911yescharacteristic
581C2#6087011yescharacteristic
582C2#6091911yescharacteristic
583C2#6092111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30394C2semidirect
4. #30393C2semidirect
5. #30391C2semidirect
6. #30388C2semidirect

Maximal subgroups

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

subgroupindex
#346382
C2×#26452
#303942
#303932
#303912
#303882
#237472
#237322
#236702
#236572
#236292
#236192
#211592
#210942
#204292