Finite Group Databasebeta

← #6633295

Subgroups — #6633295

Order 512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37421C211yesmaximal, characteristic
2#35674C211yesmaximal, characteristic, complemented
3#32915C211yesmaximal, characteristic, complemented
4#32912C211yesmaximal, characteristic
5#31663C214yesmaximal, complemented
6#31659C214yesmaximal, complemented
7#30764C214yesmaximal, complemented
8#30756C214yesmaximal, complemented
9#29724C211yesmaximal, characteristic, complemented
10#29722C211yesmaximal, characteristic
11#29339C212yesmaximal, complemented
12#25726C212yesmaximal, complemented
13#23946C212yesmaximal, complemented
14#23946C212yesmaximal, complemented
15#21917C211yesmaximal, characteristic, complemented
16C2×#542C2212yes—
17C22×C4⋊Q8C2211yescharacteristic
18C2×#519C2211yescharacteristic
19C2×#497C2214yes—
20C2×#462C2212yes—
21C2×#460C2212yes—
22C2×#451C2212yes—
23C2×#451C2212yes—
24C2×#448C2212yes—
25C2×#448C2212yes—
26#2850C2212yes—
27#2712—22no—
28#2702—24no—
29#2693—22no—
30#2621C2211yescharacteristic
31#2619C2211yescharacteristic
32#2568C2212yes—
33#2563C2212yes—
34#2545C2212yes—
35#2544C2212yes—
36#2484—22no—
37#2484—22no—
38#2446—24no—
39#2446C2214yes—
40#2446C2214yes—
41#2310C2211yescharacteristic
42#2310C2211yescharacteristic
43#2310C2211yescharacteristic
44#2296C2211yescharacteristic
45#2296C2211yescharacteristic
46#2101C2214yes—
47#2074C2212yes—
48#2074C2212yes—
49#2056C2214yes—
50#2055C2214yes—
51#2042C2214yes—
52#2024C2214yes—
53#2024C2214yes—
54#2016C2212yes—
55#2016C2214yes—
56#2015C2214yes—
57#2015C2214yes—
58#2014C2212yes—
59#2013C2214yes—
60#1960C2214yes—
61#1936C2214yes—
62#1935C2214yes—
63#1917C2214yes—
64#1917C2214yes—
65#1880C2214yes—
66#1878C2214yes—
67#1878C2214yes—
68#1878C2214yes—
69#1877C2214yes—
70#1877C2214yes—
71#1875C2212yes—
72#1875C2212yes—
73#1873C2212yes—
74#1873C2212yes—
75#1818C2212yes—
76#1805C2212yes—
77#1805C2212yes—
78#1798C2211yescharacteristic
79#1798C2211yescharacteristic
80#1749C2212yes—
81C23×Q8C2311yescharacteristic
82C23×Q8C2311yescharacteristic
83C22×Q16C2314yes—
84C22×Q16C2314yes—
85C22×SD16C2314yes—
86C22×SD16C2314yes—
87#542—44no—
88#542—44no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—44no—
91C2×C4⋊Q8C2314yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2312yes—
94C2×#127C2311yescharacteristic
95C2×#126—22no—
96C2×#124—22no—
97C2×#124C2312yes—
98C2×#124C2312yes—
99C2×#123—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8C2312yes—
102C2×#121C2312yes—
103#519—44no—
104C2×C4×Q8—22no—
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2312yes—
107C2×C4×Q8C2312yes—
108C2×C4×D4C2312yes—
109C2×#117—22no—
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C22×C4⋊C4—22no—
115C22×C42C2311yescharacteristic
116#497—48no—
117#462—44no—
118#460—44no—
119#451—44no—
120#451—44no—
121#448—44no—
122#448—44no—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2312yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×#97C2314yes—
136#400—24no—
137#400—24no—
138#400—24no—
139#398—22no—
140#398—24no—
141#398—24no—
142#397—22no—
143#396—24no—
144#394—22no—
145#394—24no—
146#393—24no—
147#392—22no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—24no—
152#391C2312yes—
153#391C2312yes—
154#391C2314yes—
155#391C2314yes—
156#391C2312yes—
157#391C2312yes—
158#390C2312yes—
159#390C2312yes—
160#389D414yes—
161#389C2314yes—
162#388—24no—
163#388—24no—
164#387—24no—
165#387—24no—
166#387—24no—
167#387—24no—
168#386—24no—
169#386—24no—
170#385—22no—
171#385D412yes—
172#385D412yes—
173#382C2311yescharacteristic
174#382C2312yes—
175#382C2311yescharacteristic
176#381—21no—
177#381—21no—
178#381—21no—
179#381—21no—
180#381—22no—
181#380—24no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2312yes—
190#336C2312yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194C22×Q8—24no—
195C22×Q8—21no—
196C22×Q8—21no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8C2412yes—
206C22×Q8C2412yes—
207C22×Q8C2412yes—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4C2412yes—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—22no—
214C23×C4C2×D412yes—
215C23×C4C2411yescharacteristic
216C2×Q16—28no—
217C2×Q16—44no—
218C2×Q16—44no—
219C2×Q16—44no—
220C2×Q16—44no—
221C2×Q16—44no—
222C2×SD16—28no—
223C2×SD16—44no—
224C2×SD16—44no—
225C2×SD16—44no—
226C2×SD16—44no—
227C2×SD16—44no—
228C22×C8C2414yes—
229C22×C8C2414yes—
230C4⋊Q8—28no—
231C4⋊Q8—42no—
232C4⋊Q8—42no—
233C4⋊Q8—44no—
234C4⋊Q8—42no—
235C4⋊Q8—48no—
236#127—42no—
237#126—48no—
238#124—42no—
239#124—42no—
240#124—44no—
241#124—48no—
242#123—48no—
243C22⋊Q8—24no—
244C22⋊Q8—24no—
245C22⋊Q8—48no—
246#121—24no—
247#121—24no—
248C4×Q8—24no—
249C4×Q8—24no—
250C4×Q8—42no—
251C4×Q8—42no—
252C4×Q8—42no—
253C4×Q8—42no—
254C4×Q8—48no—
255C4×D4—42no—
256C4×D4—42no—
257#117—22no—
258#117—22no—
259#117—42no—
260#117—42no—
261#117—42no—
262#117—48no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4C2×D411yescharacteristic
292C2×C22⋊C4C2×D411yescharacteristic
293C2×C22⋊C4C2×D412yes—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—22no—
299C2×C42—42no—
300C2×C42C2×D412yes—
301C2×C42C2412yes—
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2412yes—
305C4⋊C8—42no—
306C4⋊C8—42no—
307#103—44no—
308#103—44no—
309#103—44no—
310#103—44no—
311#103—44no—
312#103—44no—
313#102—44no—
314#102—44no—
315C22⋊C8—24no—
316C22⋊C8—24no—
317#97—44no—
318#95—24no—
319#95—22no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95C2×D414yes—
331#95C2×D414yes—
332C24—22no—
333C24C22×D411yescharacteristic
334C2×Q8—24no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—24no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—24no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—44no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—84no—
357C2×Q8—84no—
358C2×Q8—84no—
359C2×Q8H(5,𝔽2)14yes—
360C2×D4—24no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—44no—
365C2×D4—82no—
366C2×D4—84no—
367C2×D4—84no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—84no—
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C22×D414yes—
392C22×C4C22×D411yescharacteristic
393C22×C4H(5,𝔽2)12yes—
394C22×C4H(5,𝔽2)12yes—
395C22×C4C2511yescharacteristic
396Q16—88no—
397Q16—88no—
398SD16—88no—
399SD16—88no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—48no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—82no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—42no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—84no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C42—24no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—42no—
445C42—44no—
446C42—44no—
447C42—42no—
448C42—42no—
449C42—44no—
450C23—24no—
451C23—21no—
452C23—21no—
453C23—44no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23#53512yes—
459C23C23×D411yescharacteristic
460Q8—24no—
461Q8—24no—
462Q8—48no—
463Q8—42no—
464Q8—42no—
465Q8—44no—
466Q8—44no—
467Q8—44no—
468Q8—44no—
469Q8—84no—
470Q8—164no—
471D4—44no—
472D4—44no—
473D4—164no—
474D4—164no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—48no—
492C2×C4—48no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
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—84no—
510C2×C4—164no—
511C2×C4#53512yes—
512C2×C4C2×H(5,𝔽2)11yescharacteristic
513C2×C4C2×H(5,𝔽2)11yescharacteristic
514C2×C4C2×H(5,𝔽2)12yes—
515C2×C4#58512yes—
516C8—44no—
517C8—44no—
518C22—22no—
519C22—24no—
520C22—42no—
521C22—42no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—84no—
526C22—84no—
527C22C2×#53511yescharacteristic
528C22#348411yescharacteristic
529C22#348711yescharacteristic
530C22#358614yes—
531C4—22no—
532C4—44no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—44no—
537C4—42no—
538C4—82no—
539C4—82no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4#353412yes—
545C4#353412yes—
546C2—42no—
547C2—162no—
548C2#3671011yescharacteristic
549C2#6082612yes—
550C2#6164612yes—
551C2#6194612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2619C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35674C2semidirect
3. #32915C2semidirect
5. #31663C2semidirect
6. #31659C2semidirect
7. #30764C2semidirect
8. #30756C2semidirect
9. #29724C2semidirect
11. #29339C2semidirect
12. #25726C2semidirect
13. #23946C2semidirect
14. #23946C2semidirect
15. #21917C2semidirect

Maximal subgroups

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

subgroupindex
#374212
#356742
#329152
#329122
#316632
#316592
#307642
#307562
#297242
#297222
#293392
#257262
#239462
#219172