Finite Group Databasebeta

← #6654080

Subgroups — #6654080

Order 512.

1460 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic
2#34938C211yesmaximal, characteristic
3#33109C212yesmaximal
4#31576C214yesmaximal, complemented
5#31122C214yesmaximal
6#30816C214yesmaximal, complemented
7#30635C214yesmaximal
8#29923C212yesmaximal
9#29312C212yesmaximal, complemented
10#29259C212yesmaximal, complemented
11#28748C212yesmaximal, complemented
12#28284C212yesmaximal, complemented
13#21922C211yesmaximal, characteristic
14C2×#533—21no—
15C2×#487C2214yes—
16C2×#484C2214yes—
17C2×#477C2214yes—
18#2868C2211yescharacteristic
19#2859C2211yescharacteristic
20#2857C2212yes—
21#2647—21no—
22#2629—21no—
23#2552C2212yes—
24#2551C2212yes—
25#2550—22no—
26#2549—22no—
27#2507—22no—
28#2497—22no—
29#2446C2214yes—
30#2440C2212yes—
31#2440C2212yes—
32#2429C2214yes—
33#2399—21no—
34#2399—21no—
35#2330—21no—
36#2314—21no—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40#2296—21no—
41#2285C2211yescharacteristic
42C4×#117C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44#2095C2214yes—
45#2095C2214yes—
46#2083C2214yes—
47#2072C2214yes—
48#2068C2214yes—
49#2068C2214yes—
50#2057C2214yes—
51#2043C2214yes—
52#2043C2214yes—
53#2036C2214yes—
54#2036C2214yes—
55#2024C2214yes—
56#1981C2212yes—
57#1981C2212yes—
58#1981C2212yes—
59#1981C2212yes—
60#1956C2214yes—
61#1947C2212yes—
62#1947C2212yes—
63#1947C2212yes—
64#1947C2212yes—
65#1942C2214yes—
66#1942C2214yes—
67#1928C2214yes—
68#1928C2214yes—
69#1923C2214yes—
70#1916C2214yes—
71#1872C2214yes—
72#1863C2214yes—
73#1863C2214yes—
74#1842C2211yescharacteristic
75#1831C2211yescharacteristic
76#1803C2214yes—
77#1797C2212yes—
78C4×C4⋊C8C2212yes—
79#533—48no—
80C2×C4⋊Q8—22no—
81C2×C4⋊Q8C2312yes—
82C2×C4⋊Q8C2314yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×#125—22no—
86C2×#125—21no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8C2314yes—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8C2314yes—
93C2×#117—21no—
94C2×#117D411yescharacteristic
95C2×#117D411yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C42—21no—
98C22×C42C2311yescharacteristic
99#487—48no—
100#484—48no—
101#477—48no—
102C2×#107C2314yes—
103C2×#107C2314yes—
104C2×#107C2314yes—
105C2×#107C2314yes—
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2312yes—
108C2×C4⋊C8C2314yes—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×C22⋊C8C2314yes—
118C2×C4×C8C2314yes—
119#400—24no—
120#398—24no—
121#398—24no—
122#398—22no—
123#398—22no—
124#398—22no—
125#398—22no—
126#396—24no—
127#395—22no—
128#395—22no—
129#395—22no—
130#395—22no—
131#391—22no—
132#391—22no—
133#391C2314yes—
134#391C2314yes—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—24no—
145#389C2312yes—
146#389C2312yes—
147#389C2312yes—
148#389C2312yes—
149#389C2314yes—
150#388—24no—
151#387—24no—
152#387—24no—
153#385—22no—
154#385—22no—
155#384—21no—
156#384—21no—
157#384—22no—
158#384C2312yes—
159#384C2312yes—
160#384C2312yes—
161#384C2312yes—
162#382—21no—
163#382—21no—
164#380—22no—
165#380—22no—
166#380—21no—
167#380C2311yescharacteristic
168#380C2311yescharacteristic
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4D412yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2311yescharacteristic
180C4×C4⋊C4C2311yescharacteristic
181C4×C4⋊C4C2312yes—
182C4×C22⋊C4D412yes—
183C4×C22⋊C4C2312yes—
184#376D411yescharacteristic
185#376D411yescharacteristic
186C2×#95—21no—
187C43C2311yescharacteristic
188C43C2311yescharacteristic
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2314yes—
195#336C2314yes—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8C2414yes—
199C23×C4—22no—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2411yescharacteristic
203C22×C8C2414yes—
204C22×C8C2414yes—
205C4⋊Q8—24no—
206C4⋊Q8—24no—
207C4⋊Q8—24no—
208C4⋊Q8—24no—
209C4⋊Q8—42no—
210C4⋊Q8—42no—
211C4⋊Q8—44no—
212C4⋊Q8—44no—
213C4⋊Q8—48no—
214#125—48no—
215#125—44no—
216C22⋊Q8—48no—
217C22⋊Q8—48no—
218C22⋊Q8—44no—
219C22⋊Q8—44no—
220C4×Q8—48no—
221C4×Q8—44no—
222C4×Q8—44no—
223C4×Q8—48no—
224#117—24no—
225#117—44no—
226#117Q1618yes—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2412yes—
254C2×C4⋊C4C2414yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2412yes—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4C2×D412yes—
262C2×C22⋊C4C2×D412yes—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—24no—
270C2×C42—44no—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2412yes—
280C2×C42C2412yes—
281C2×C42C2411yescharacteristic
282C2×C42C2411yescharacteristic
283#107—28no—
284#107—44no—
285#107—44no—
286#107—44no—
287C4⋊C8—24no—
288C4⋊C8—24no—
289C4⋊C8—44no—
290#103—28no—
291#103—28no—
292#103—44no—
293#103—44no—
294#103—44no—
295#103—44no—
296#103—44no—
297#103—44no—
298C22⋊C8—44no—
299C4×C8—44no—
300#95—22no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95C2×D412yes—
316#95C2×D412yes—
317#95C2×D412yes—
318#95C2×D412yes—
319C24C22×D411yescharacteristic
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—82no—
331C2×Q8—82no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—82no—
349C22×C4C2×Q1614yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D412yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D412yes—
355C22×C4C22×D412yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D412yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C2511yescharacteristic
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—48no—
390C4⋊C4—44no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C4⋊C4—84no—
398C4⋊C4C2×Q1618yes—
399C4⋊C4H(5,𝔽2)14yes—
400C22⋊C4—24no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—84no—
404C22⋊C4C2×Q1618yes—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—21no—
411C42—21no—
412C42—22no—
413C42—22no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42—42no—
418C42—44no—
419C42—44no—
420C42—84no—
421C42C2×Q1614yes—
422C42C2×Q1614yes—
423C42AGL(1,ℤ/8)14yes—
424C422−1+412yes—
425C422−1+412yes—
426C23—22no—
427C23—21no—
428C23—22no—
429C23—21no—
430C23C22×Q1612yes—
431C23C23×D411yescharacteristic
432Q8—44no—
433Q8—44no—
434Q8—44no—
435Q8—44no—
436Q8—88no—
437Q8—88no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—28no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—82no—
477C2×C4—88no—
478C2×C4#53612yes—
479C2×C4#53612yes—
480C2×C4C22×Q1614yes—
481C2×C4C22×Q1614yes—
482C2×C4C22×Q1612yes—
483C2×C4C22×Q1612yes—
484C2×C4C2×AGL(1,ℤ/8)14yes—
485C2×C4C2×AGL(1,ℤ/8)12yes—
486C2×C4C2×H(5,𝔽2)12yes—
487C2×C4C2×2−1+411yescharacteristic
488C2×C4C2×2−1+411yescharacteristic
489C2×C4#58512yes—
490C8—44no—
491C8—44no—
492C22—24no—
493C22—22no—
494C22—42no—
495C22—41no—
496C22—41no—
497C22C2×#53611yescharacteristic
498C22#348812yes—
499C22C23×Q1611yescharacteristic
500C22C22×AGL(1,ℤ/8)11yescharacteristic
501C22#358612yes—
502C4—24no—
503C4—24no—
504C4—24no—
505C4—22no—
506C4—22no—
507C4—42no—
508C4—44no—
509C4—41no—
510C4—41no—
511C4—84no—
512C4—82no—
513C4—82no—
514C4—82no—
515C4—84no—
516C4#323014yes—
517C4#354811yescharacteristic
518C4#355412yes—
519C4#355711yescharacteristic
520C2—42no—
521C2#3727411yescharacteristic
522C2C2×#323011yescharacteristic
523C2#6086911yescharacteristic
524C2#6195212yes—
525C2#6196212yes—
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
4. #31576C2semidirect
6. #30816C2semidirect
9. #29312C2semidirect
10. #29259C2semidirect
11. #28748C2semidirect
12. #28284C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#349382
#331092
#315762
#311222
#308162
#306352
#299232
#293122
#292592
#287482
#282842
#219222