Finite Group Databasebeta

← #6434848

Subgroups — #6434848

Order 512.

1452 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2645C211yesmaximal, characteristic
2#34074C211yesmaximal, characteristic
3#30164C214yesmaximal, complemented
4#30162C214yesmaximal, complemented
5#30160C214yesmaximal, complemented
6#30159C214yesmaximal, complemented
7#23670C212yesmaximal
8#23657C211yesmaximal, characteristic
9#23629C211yesmaximal, characteristic
10#23619C212yesmaximal
11#23554C211yesmaximal, characteristic
12#23455C211yesmaximal, characteristic
13#21061C212yesmaximal
14#21034C212yesmaximal
15#20753C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#428C2212yes—
18C2×#428C2212yes—
19C2×#423C2212yes—
20C2×#419C2212yes—
21C2×#419C2212yes—
22#2645—22no—
23#2645—22no—
24#2645C2214yes—
25#2645C2214yes—
26C2×#401—21no—
27C2×#400—21no—
28C2×#400—21no—
29C2×#400—21no—
30#2331C2212yes—
31#2331C2212yes—
32#2330C2212yes—
33#2330C2212yes—
34#2329—21no—
35#2329—21no—
36#2329—21no—
37#2329—21no—
38#2296C2212yes—
39#2295—22no—
40#2295—22no—
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47#2285C2211yescharacteristic
48C4×#117C2212yes—
49C2×#376C2211yescharacteristic
50#2088C2214yes—
51#2088C2214yes—
52#2087C2214yes—
53#2086C2214yes—
54#2085C2214yes—
55#2084C2212yes—
56#2083C2212yes—
57#2074C2214yes—
58#2072C2212yes—
59#2072C2212yes—
60#2068C2214yes—
61#2068C2214yes—
62#2066C2214yes—
63#2066C2214yes—
64#2059C2214yes—
65#2058C2212yes—
66#2057C2212yes—
67#2026C2214yes—
68#2024C2212yes—
69#2024C2212yes—
70#1928C2214yes—
71#1928C2214yes—
72#1927C2214yes—
73#1927C2214yes—
74#1921C2214yes—
75#1920C2214yes—
76#1919C2214yes—
77#1918C2214yes—
78#1832C2214yes—
79#1832C2214yes—
80#1736C2212yes—
81#1736C2212yes—
82#1736C2212yes—
83#1736C2212yes—
84#1728C2211yescharacteristic
85#1728C2212yes—
86#1728C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C22×M16C2312yes—
89C22×M16C2312yes—
90#533—28no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×#125C2312yes—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C4×Q8—21no—
97C2×C4×Q8—21no—
98C2×#117C2312yes—
99C2×#117C2312yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109#428—28no—
110#428—44no—
111C2×#107C2314yes—
112C2×#107C2314yes—
113C2×#106C2314yes—
114C2×#106C2314yes—
115#423—44no—
116C2×C4⋊C8C2314yes—
117C2×C4⋊C8C2314yes—
118#419—28no—
119#419—44no—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124#401—24no—
125#401—24no—
126#400—24no—
127#400—24no—
128#400—24no—
129#400—24no—
130#400—24no—
131#400—24no—
132#391—21no—
133#391—21no—
134#391—22no—
135#391—22no—
136#391—22no—
137#391—22no—
138#391—22no—
139#391—22no—
140#391C2314yes—
141#391C2312yes—
142#391C2312yes—
143#391C2314yes—
144#389—21no—
145#389—21no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389C2312yes—
153#389C2312yes—
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#389C2314yes—
159#389C2314yes—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#385—22no—
169#385—22no—
170#384C2311yescharacteristic
171#384C2311yescharacteristic
172#382C2312yes—
173#380—21no—
174#380—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2312yes—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—22no—
185#376—22no—
186#376D412yes—
187#376D412yes—
188#376C2312yes—
189#376C2312yes—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95D411yescharacteristic
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202C2×#95D411yescharacteristic
203C43—22no—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2312yes—
215#336C2312yes—
216#336C2314yes—
217#336C2314yes—
218C22×Q8—21no—
219C22×Q8—21no—
220C22×Q8—21no—
221C22×Q8—24no—
222C22×Q8—22no—
223C22×Q8—24no—
224C22×Q8—22no—
225C22×Q8C2412yes—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2411yescharacteristic
237C2×M16—24no—
238C2×M16—24no—
239C2×M16—24no—
240C2×M16—24no—
241C2×M16—24no—
242C2×M16—24no—
243C22×C8C2414yes—
244C22×C8C2414yes—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247#125—24no—
248#125—24no—
249C22⋊Q8—28no—
250C22⋊Q8—28no—
251C4×Q8—28no—
252C4×Q8—28no—
253#117—24no—
254#117—24no—
255#117—24no—
256#117#4118yes—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
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—82no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4—82no—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308C2×C42C2412yes—
309C2×C42C2412yes—
310#107—28no—
311#107—44no—
312#106—28no—
313#106—44no—
314C4⋊C8—44no—
315C4⋊C8—44no—
316#103—28no—
317#103—28no—
318#103—44no—
319#103—44no—
320#95—24no—
321#95—24no—
322#95—22no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
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#95C2×D412yes—
343#95C2×D412yes—
344#95C2×D412yes—
345#95C2×D412yes—
346#95C2×D412yes—
347#95C2×D412yes—
348#95C2×D412yes—
349#95C2×D412yes—
350C24C22×D411yescharacteristic
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—22no—
355C2×Q8—24no—
356C2×Q8—22no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—81no—
363C2×Q8—81no—
364C2×Q8—81no—
365C2×Q8—81no—
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—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2×#4114yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C42−1+414yes—
416C22×C42−1+412yes—
417C22×C4C2511yescharacteristic
418M16—48no—
419M16—48no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—82no—
444C4⋊C4—81no—
445C4⋊C4—82no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4—84no—
452C4⋊C4—82no—
453C4⋊C4—84no—
454C4⋊C4—82no—
455C4⋊C4C2×#4114yes—
456C4⋊C4C2×#4114yes—
457C22⋊C4—28no—
458C22⋊C4—44no—
459C42—22no—
460C42—22no—
461C42—28no—
462C42—24no—
463C42—24no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—24no—
468C42—22no—
469C42—44no—
470C42—82no—
471C42—82no—
472C42—82no—
473C42—82no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23#52512yes—
479C23#53612yes—
480C23#53712yes—
481C23C23×D411yescharacteristic
482Q8—44no—
483Q8—44no—
484Q8—84no—
485Q8—84no—
486Q8—84no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
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—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—42no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4#52512yes—
546C2×C4#53612yes—
547C2×C4#53712yes—
548C2×C4#54118yes—
549C2×C4#54218yes—
550C2×C4C22×#4112yes—
551C2×C4C2×H(5,𝔽2)11yescharacteristic
552C2×C4C2×2−1+411yescharacteristic
553C2×C4C2×2−1+412yes—
554C8—44no—
555C8—44no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22#263714yes—
563C22C2×#52511yescharacteristic
564C22C2×#53611yescharacteristic
565C22C2×#53711yescharacteristic
566C22#358511yescharacteristic
567C22#358611yescharacteristic
568C22#358611yescharacteristic
569C22#358611yescharacteristic
570C4—41no—
571C4—41no—
572C4—44no—
573C4—41no—
574C4—41no—
575C4—44no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4#263714yes—
587C4C2×#54112yes—
588C4C2×#54212yes—
589C2—24no—
590C2C2×#263711yescharacteristic
591C2#6056711yescharacteristic
592C2#6056911yescharacteristic
593C2#6086911yescharacteristic
594C2#6087011yescharacteristic
595C2#6091911yescharacteristic
596C2#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. #30164C2semidirect
4. #30162C2semidirect
5. #30160C2semidirect
6. #30159C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26452
#340742
#301642
#301622
#301602
#301592
#236702
#236572
#236292
#236192
#235542
#234552
#210612
#210342
#207532