Finite Group Databasebeta

← #6495476

Subgroups — #6495476

Order 512.

1401 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic
2C4×#541C211yesmaximal, characteristic
3#33200C212yesmaximal, complemented
4#33196C212yesmaximal, complemented
5#29949C212yesmaximal
6#29701C212yesmaximal
7#29327C212yesmaximal, complemented
8#29274C212yesmaximal, complemented
9#29138C212yesmaximal, complemented
10#28205C212yesmaximal, complemented
11#27642C214yesmaximal
12#24710C212yesmaximal
13#24703C212yesmaximal
14#22128C214yesmaximal, complemented
15#20815C211yesmaximal, characteristic
16C2×#541C2211yescharacteristic
17C2×#479C2214yes—
18C2×#475C2212yes—
19C2×#475C2212yes—
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2647C2212yes—
26#2629C2212yes—
27#2507—22no—
28#2497—22no—
29#2446—22no—
30#2429—22no—
31#2336—22no—
32#2334—22no—
33#2330C2211yescharacteristic
34#2330C2212yes—
35#2327C2211yescharacteristic
36#2323C2212yes—
37#2320C2212yes—
38#2314C2211yescharacteristic
39#2314C2212yes—
40C4×C4⋊Q8C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×#125—22no—
43C4×#125C2211yescharacteristic
44C4×#126—22no—
45C4×C22⋊Q8C2212yes—
46C42×Q8C2212yes—
47C4×#117C2211yescharacteristic
48#2101C2214yes—
49#2084C2214yes—
50#2077C2214yes—
51#2077C2214yes—
52#2059C2214yes—
53#2043C2214yes—
54#2036C2214yes—
55#2035C2214yes—
56#1989C2214yes—
57#1988C2212yes—
58#1988C2212yes—
59#1949C2214yes—
60#1948C2212yes—
61#1948C2214yes—
62#1948C2214yes—
63#1947C2212yes—
64#1943C2214yes—
65#1941C2214yes—
66#1935C2214yes—
67#1933C2214yes—
68#1927C2214yes—
69#1927C2214yes—
70#1921C2214yes—
71#1919C2214yes—
72#1842C2212yes—
73#1831C2212yes—
74#1805C2214yes—
75#1803C2214yes—
76#1803C2212yes—
77#1803C2212yes—
78#1803C2214yes—
79#1798C2212yes—
80#1798C2212yes—
81#1768C2212yes—
82#1768C2212yes—
83#541—28no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2312yes—
87C2×#126—22no—
88C2×#125—22no—
89C2×#125C2311yescharacteristic
90C2×#125C2312yes—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8C2312yes—
93C2×C4×Q8—22no—
94C2×C4×Q8C2312yes—
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2312yes—
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#479—216no—
101#475—44no—
102#475—44no—
103#432—42no—
104#431—42no—
105C2×#107C2314yes—
106C2×#106C2314yes—
107C2×#106C2314yes—
108C2×#106C2314yes—
109C2×C4⋊C8C2312yes—
110C2×C4⋊C8C2312yes—
111C2×C4⋊C8C2314yes—
112C2×C4⋊C8C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×C22⋊C8C2312yes—
118C2×C22⋊C8C2312yes—
119C2×#97C2312yes—
120C2×C4×C8C2312yes—
121#400—24no—
122#398—24no—
123#398—24no—
124#396—24no—
125#391—22no—
126#391—22no—
127#391C2312yes—
128#391C2312yes—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—24no—
137#389C2312yes—
138#389C2314yes—
139#389C2312yes—
140#389C2314yes—
141#388—22no—
142#388—22no—
143#388—22no—
144#388—24no—
145#387—22no—
146#387—22no—
147#387—24no—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—24no—
156#385—22no—
157#384—22no—
158#384C2311yescharacteristic
159#384C2312yes—
160#384C2312yes—
161#384C2312yes—
162#383—22no—
163#382—22no—
164#382C2311yescharacteristic
165#382C2312yes—
166#380—22no—
167#380C2312yes—
168#380C2314yes—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4D412yes—
177C4×C4⋊C4D412yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2314yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2312yes—
182C4×C22⋊C4—22no—
183C4×C22⋊C4D412yes—
184C4×C22⋊C4D412yes—
185#376C2311yescharacteristic
186#376C2311yescharacteristic
187C43—22no—
188C43C2311yescharacteristic
189C43C2311yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C22×Q8—22no—
201C22×Q8C2412yes—
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2411yescharacteristic
205C22×C8C2414yes—
206C22×C8C2414yes—
207C4⋊Q8—24no—
208C4⋊Q8—24no—
209C4⋊Q8—24no—
210C4⋊Q8—84no—
211#126—216no—
212#125—216no—
213#125—24no—
214#125—28no—
215C22⋊Q8—24no—
216C22⋊Q8—24no—
217C22⋊Q8—84no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—24no—
222C4×Q8—24no—
223C4×Q8—84no—
224C4×Q8#4118yes—
225#117—24no—
226#117—24no—
227C2×C4⋊C4—22no—
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—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2414yes—
250C2×C4⋊C4C2412yes—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4C2×D412yes—
255C2×C22⋊C4C2×D412yes—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2411yescharacteristic
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277#107—28no—
278#106—28no—
279#106—44no—
280#106—44no—
281C4⋊C8—28no—
282C4⋊C8—24no—
283C4⋊C8—24no—
284C4⋊C8—28no—
285#103—28no—
286#103—28no—
287#103—44no—
288#103—44no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291#97—42no—
292C4×C8—42no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95C2×D412yes—
310#95C2×D412yes—
311#95C2×D412yes—
312#95C2×D412yes—
313C24C22×D411yescharacteristic
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8C2×#4114yes—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—42no—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D412yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D412yes—
342C22×C4C22×D412yes—
343C22×C4C22×D412yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C2511yescharacteristic
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C4⋊C4—28no—
354C4⋊C4—28no—
355C4⋊C4—28no—
356C4⋊C4—28no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—44no—
369C4⋊C4—48no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C4⋊C4—84no—
381C4⋊C4C2×#4114yes—
382C22⋊C4—28no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—82no—
386C22⋊C4—84no—
387C42—28no—
388C42—22no—
389C42—22no—
390C42—24no—
391C42—24no—
392C42—22no—
393C42—21no—
394C42—21no—
395C42—22no—
396C42—42no—
397C42—42no—
398C42—44no—
399C42—44no—
400C42—44no—
401C42—82no—
402C42—82no—
403C42—84no—
404C42C2×#4114yes—
405C42H(5,𝔽2)12yes—
406C422−1+414yes—
407C422−1+412yes—
408C23—22no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—22no—
413C23C23×D411yescharacteristic
414Q8—24no—
415Q8—24no—
416Q8—42no—
417Q8—42no—
418Q8—88no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—82no—
459C2×C4#52514yes—
460C2×C4#53614yes—
461C2×C4#53712yes—
462C2×C4#53712yes—
463C2×C4#57714yes—
464C2×C4C22×#4112yes—
465C2×C4C2×H(5,𝔽2)11yescharacteristic
466C2×C4C2×2−1+411yescharacteristic
467C2×C4C2×2−1+412yes—
468C8—44no—
469C8—44no—
470C22—42no—
471C22—42no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22C2×#52511yescharacteristic
476C22C2×#53611yescharacteristic
477C22C2×#53711yescharacteristic
478C22C2×#57712yes—
479C22#358612yes—
480C4—24no—
481C4—24no—
482C4—22no—
483C4—42no—
484C4—44no—
485C4—42no—
486C4—41no—
487C4—44no—
488C4—41no—
489C4—41no—
490C4—84no—
491C4—82no—
492C4—84no—
493C4—82no—
494C4#326014yes—
495C4#350912yes—
496C4#351112yes—
497C2—81no—
498C2#3529711yescharacteristic
499C2#6059912yes—
500C2#6088912yes—
501C2C2×#326011yescharacteristic
502C2#6092111yescharacteristic
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
3. #33200C2semidirect
4. #33196C2semidirect
7. #29327C2semidirect
8. #29274C2semidirect
9. #29138C2semidirect
10. #28205C2semidirect
14. #22128C2semidirect

Maximal subgroups

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

subgroupindex
#352842
C4×#5412
#332002
#331962
#299492
#297012
#293272
#292742
#291382
#282052
#276422
#247102
#247032
#221282
#208152