Finite Group Databasebeta

← #6434503

Subgroups — #6434503

Order 512.

1698 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35296C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic, complemented
3#30139C214yesmaximal, complemented
4#30080C212yesmaximal, complemented
5#30041C212yesmaximal, complemented
6#29960C212yesmaximal, complemented
7#29910C212yesmaximal, complemented
8#29722C212yesmaximal
9#29624C212yesmaximal, complemented
10#23829C212yesmaximal, complemented
11#23811C212yesmaximal, complemented
12#23683C212yesmaximal, complemented
13#23677C212yesmaximal, complemented
14#21462C214yesmaximal, complemented
15#20781C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#519C2212yes—
18C2×#448C2214yes—
19C2×#440C2214yes—
20C2×C4×SD16C2214yes—
21#2864C2211yescharacteristic
22#2858C2211yescharacteristic
23#2854C2212yes—
24#2644C2212yes—
25#2628C2211yescharacteristic
26#2619C2211yescharacteristic
27#2595—22no—
28#2557—22no—
29#2474—22no—
30#2471—22no—
31#2365—22no—
32#2365—22no—
33#2344—22no—
34#2344—22no—
35#2331C2211yescharacteristic
36#2328C2211yescharacteristic
37#2327C2212yes—
38#2327C2212yes—
39#2324C2212yes—
40#2310C2212yes—
41C4×C4⋊Q8C2212yes—
42C4×C4⋊Q8C2212yes—
43#2293C2212yes—
44#2285C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2101C2214yes—
47#2097C2214yes—
48#2092C2214yes—
49#2084C2212yes—
50#2083C2212yes—
51#2074C2214yes—
52#2071C2212yes—
53#2071C2212yes—
54#2068C2214yes—
55#2066C2214yes—
56#2055C2214yes—
57#2055C2214yes—
58#2043C2214yes—
59#2043C2212yes—
60#2043C2212yes—
61#2024C2214yes—
62#2014C2214yes—
63#2013C2214yes—
64#2013C2212yes—
65#2013C2212yes—
66#1880C2214yes—
67#1880C2214yes—
68#1875C2214yes—
69#1865C2214yes—
70#1863C2214yes—
71#1854C2214yes—
72#1839C2212yes—
73#1832C2212yes—
74#1817C2212yes—
75#1798C2212yes—
76#1787C2214yes—
77#1767C2211yescharacteristic
78C4×C4⋊C8C2211yescharacteristic
79#1763C2214yes—
80C4×#103C2214yes—
81#1750C2212yes—
82#1746C2212yes—
83C23×Q8C2312yes—
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#124—22no—
93C2×#123—22no—
94C2×#121C2312yes—
95#519—44no—
96#519—44no—
97C2×C4×Q8—24no—
98C2×C4×Q8C2314yes—
99C2×C4×Q8C2314yes—
100C2×C4×D4C2312yes—
101C2×#117—24no—
102C2×#117—22no—
103C2×#117C2312yes—
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C22×C4⋊C4—22no—
107C22×C42C2311yescharacteristic
108#448—216no—
109#440—48no—
110C4×SD16—48no—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×C4⋊C8C2312yes—
114C2×C4⋊C8C2312yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×C22⋊C8C2314yes—
124C2×#97C2312yes—
125C2×C4×C8C2312yes—
126#400—21no—
127#399—21no—
128#397—22no—
129#397—22no—
130#397—22no—
131#396—22no—
132#395—21no—
133#394—21no—
134#394—21no—
135#393—21no—
136#392—21no—
137#392—21no—
138#391—22no—
139#391—22no—
140#391—22no—
141#391C2314yes—
142#391C2314yes—
143#390—22no—
144#390—22no—
145#390C2312yes—
146#390C2312yes—
147#390C2312yes—
148#390C2312yes—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389C2312yes—
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#387—22no—
163#387—22no—
164#387—24no—
165#387—24no—
166#387—24no—
167#387—24no—
168#387—22no—
169#386—22no—
170#386—22no—
171#386—22no—
172#386—22no—
173#385—22no—
174#384C2312yes—
175#382C2312yes—
176#382C2311yescharacteristic
177#382C2311yescharacteristic
178#381—22no—
179#381—22no—
180#380—22no—
181#380C2312yes—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4D412yes—
188C4×C4⋊C4D412yes—
189C4×C4⋊C4C2312yes—
190C4×C22⋊C4—22no—
191C4×C22⋊C4D412yes—
192C4×C22⋊C4D412yes—
193#376C2311yescharacteristic
194#376C2311yescharacteristic
195C43C2311yescharacteristic
196C43C2311yescharacteristic
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2312yes—
203#336C2312yes—
204#336C2314yes—
205#336C2314yes—
206C22×Q8—22no—
207C22×Q8—21no—
208C22×Q8—24no—
209C22×Q8—42no—
210C22×Q8—42no—
211C22×Q8—42no—
212C22×Q8—42no—
213C22×Q8#4118yes—
214C22×Q8C2414yes—
215C22×D4—22no—
216C22×D4—21no—
217C22×D4—21no—
218C22×D4C2412yes—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4C2×D412yes—
222C23×C4C2411yescharacteristic
223C2×SD16—28no—
224C2×SD16—28no—
225C2×SD16—44no—
226C2×SD16—44no—
227C2×SD16—44no—
228C2×SD16—44no—
229C22×C8C2414yes—
230C22×C8C2414yes—
231C4⋊Q8—216no—
232C4⋊Q8—28no—
233C4⋊Q8—28no—
234C4⋊Q8—42no—
235#127—42no—
236#124—84no—
237#123—84no—
238#121—28no—
239C4×Q8—28no—
240C4×Q8—28no—
241C4×Q8—44no—
242C4×Q8—44no—
243C4×Q8—416no—
244C4×D4—24no—
245C4×D4—24no—
246#117—22no—
247#117—22no—
248#117—42no—
249#117—416no—
250#117—42no—
251#117—42no—
252#117—84no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2×D411yescharacteristic
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4C2×D412yes—
286C2×C22⋊C4C2×D411yescharacteristic
287C2×C22⋊C4C2×D411yescharacteristic
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304#107—44no—
305#106—44no—
306C4⋊C8—42no—
307C4⋊C8—42no—
308#103—28no—
309#103—28no—
310#103—44no—
311#103—44no—
312#103—44no—
313#103—44no—
314#102—44no—
315#102—44no—
316C22⋊C8—28no—
317#97—24no—
318C4×C8—24no—
319#95—22no—
320#95—22no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—22no—
327#95—22no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—22no—
333#95—22no—
334#95C2×D412yes—
335#95C2×D412yes—
336#95C2×D412yes—
337#95C2×D412yes—
338C24—22no—
339C24C22×D411yescharacteristic
340C2×Q8—28no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—42no—
353C2×Q8—82no—
354C2×Q8C2×#4118yes—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—44no—
362C2×D4—82no—
363C2×D4—82no—
364C2×D4—82no—
365C2×D4—82no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—28no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C2×#4114yes—
393C22×C4C2511yescharacteristic
394SD16—416no—
395SD16—416no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C4⋊C4—28no—
403C4⋊C4—28no—
404C4⋊C4—216no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—44no—
415C4⋊C4—48no—
416C4⋊C4—42no—
417C4⋊C4—48no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C22⋊C4—24no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C22⋊C4—82no—
434C22⋊C4—82no—
435C22⋊C4—82no—
436C22⋊C4—82no—
437C22⋊C4—82no—
438C42—24no—
439C42—24no—
440C42—24no—
441C42—22no—
442C42—22no—
443C42—21no—
444C42—21no—
445C42—22no—
446C42—22no—
447C42—42no—
448C42—42no—
449C42—44no—
450C42—48no—
451C42—48no—
452C42—82no—
453C42H(5,𝔽2)12yes—
454C42H(5,𝔽2)12yes—
455C23—22no—
456C23—21no—
457C23—21no—
458C23—22no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23#52512yes—
467C23C23×D411yescharacteristic
468Q8—28no—
469Q8—28no—
470Q8—416no—
471Q8—44no—
472Q8—44no—
473Q8—162no—
474D4—44no—
475D4—44no—
476D4—162no—
477D4—162no—
478D4—164no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—28no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—48no—
503C2×C4—416no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—88no—
519C2×C4#52512yes—
520C2×C4#53514yes—
521C2×C4#57714yes—
522C2×C4C22×#4112yes—
523C2×C4C2×H(5,𝔽2)11yescharacteristic
524C2×C4C2×H(5,𝔽2)11yescharacteristic
525C2×C4#58512yes—
526C8—44no—
527C8—44no—
528C22—22no—
529C22—24no—
530C22—41no—
531C22—41no—
532C22—42no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22C2×#52511yescharacteristic
541C22C2×#53511yescharacteristic
542C22#344911yescharacteristic
543C22C2×#57712yes—
544C22#358612yes—
545C4—22no—
546C4—24no—
547C4—24no—
548C4—48no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—84no—
557C4—82no—
558C4—82no—
559C4#345912yes—
560C4#345912yes—
561C2—42no—
562C2—162no—
563C2#3459311yescharacteristic
564C2#6058212yes—
565C2#6083612yes—
566C2#6101211yescharacteristic
567C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2619C22
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
2. #34209C2semidirect
3. #30139C2semidirect
4. #30080C2semidirect
5. #30041C2semidirect
6. #29960C2semidirect
7. #29910C2semidirect
9. #29624C2semidirect
10. #23829C2semidirect
11. #23811C2semidirect
12. #23683C2semidirect
13. #23677C2semidirect
14. #21462C2semidirect
15. #20781C2semidirect

Maximal subgroups

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

subgroupindex
#352962
#342092
#301392
#300802
#300412
#299602
#299102
#297222
#296242
#238292
#238112
#236832
#236772
#214622
#207812