Finite Group Databasebeta

← #6495052

Subgroups — #6495052

Order 512.

1424 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35335C211yesmaximal, characteristic
2#35283C211yesmaximal, characteristic
3#33317C211yesmaximal, characteristic, complemented
4#33317C211yesmaximal, characteristic, complemented
5#33297C211yesmaximal, characteristic, complemented
6#33294C211yesmaximal, characteristic, complemented
7#30315C214yesmaximal, complemented
8#29217C214yesmaximal, complemented
9#28693C212yesmaximal, complemented
10#28692C212yesmaximal, complemented
11#27959C214yesmaximal
12#25819C212yesmaximal
13#25818C212yesmaximal
14#21992C214yesmaximal
15#20465C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C2×#501C2212yes—
19C2×#501C2212yes—
20C2×#501C2212yes—
21C2×#501C2212yes—
22C2×#433C2212yes—
23#2868C2211yescharacteristic
24#2862C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2857C2212yes—
30#2660C2212yes—
31#2660C2212yes—
32#2647C2212yes—
33#2628C2212yes—
34#2584—22no—
35#2584—22no—
36#2583—22no—
37#2583—22no—
38#2511—22no—
39#2492—22no—
40#2451—22no—
41#2423—22no—
42#2331C2212yes—
43#2317C2212yes—
44#2317C2212yes—
45#2314C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×#117C2211yescharacteristic
48#2085C2214yes—
49#2085C2214yes—
50#2084C2214yes—
51#2083C2214yes—
52#2059C2214yes—
53#2059C2214yes—
54#2058C2214yes—
55#2057C2214yes—
56#1993C2214yes—
57#1992C2212yes—
58#1992C2212yes—
59#1992C2212yes—
60#1992C2212yes—
61#1990C2214yes—
62#1949C2214yes—
63#1949C2214yes—
64#1948C2214yes—
65#1947C2214yes—
66#1854C2214yes—
67#1854C2214yes—
68#1853C2214yes—
69#1852C2214yes—
70#1847C2214yes—
71#1847C2214yes—
72#1846C2214yes—
73#1846C2214yes—
74#1836C2214yes—
75#1834C2214yes—
76#1768C2212yes—
77#1767C2212yes—
78#1750C2212yes—
79#1749C2212yes—
80#1746C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83#533—44no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×#125—22no—
89C2×#125—44no—
90C2×#125C2311yescharacteristic
91C2×#125C2311yescharacteristic
92C2×#125C2311yescharacteristic
93C2×#125C2311yescharacteristic
94C2×#125C2312yes—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C4×Q8—22no—
98C2×C4×Q8—22no—
99C2×#117C2312yes—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4—22no—
103C22×C4⋊C4C2312yes—
104C22×C42C2311yescharacteristic
105#501—44no—
106#501—44no—
107#501—44no—
108#501—44no—
109#433Q8116yescomplemented
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#107C2314yes—
113C2×#107C2314yes—
114C2×#106C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×C4⋊C8C2314yes—
119C2×C22⋊C8C2314yes—
120C2×#97C2312yes—
121C2×#97C2312yes—
122C2×#97C2312yes—
123C2×#97C2312yes—
124C2×C4×C8C2312yes—
125C2×C4×C8C2312yes—
126#400—22no—
127#400—24no—
128#399—22no—
129#398—22no—
130#398—22no—
131#398—22no—
132#398—22no—
133#398—24no—
134#395—22no—
135#395—22no—
136#395—22no—
137#395—22no—
138#395—22no—
139#393—22no—
140#391—22no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391—24no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389C2314yes—
160#389C2314yes—
161#388—24no—
162#387—24no—
163#387—24no—
164#385—22no—
165#385—22no—
166#385—22no—
167#385—22no—
168#384—22no—
169#384C2312yes—
170#384C2312yes—
171#384C2312yes—
172#384C2312yes—
173#384C2312yes—
174#384C2312yes—
175#382C2312yes—
176#382C2312yes—
177#382C2312yes—
178#382C2312yes—
179#382C2312yes—
180#382C2312yes—
181#380—22no—
182#380C2314yes—
183C4×C4⋊C4—22no—
184C4×C4⋊C4D412yes—
185C4×C4⋊C4D412yes—
186C4×C4⋊C4C2314yes—
187C4×C22⋊C4D412yes—
188C4×C22⋊C4D412yes—
189#376C2311yescharacteristic
190#376C2311yescharacteristic
191C43C2311yescharacteristic
192C43C2311yescharacteristic
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2314yes—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C23×C4—22no—
207C23×C4C2×D412yes—
208C23×C4C2411yescharacteristic
209C22×C8C2414yes—
210C22×C8C2414yes—
211C4⋊Q8—24no—
212C4⋊Q8—24no—
213C4⋊Q8—28no—
214C4⋊Q8—42no—
215C4⋊Q8—42no—
216C4⋊Q8—84no—
217#125—24no—
218#125—24no—
219#125—24no—
220#125—24no—
221#125—42no—
222#125—42no—
223#125—48no—
224C22⋊Q8—48no—
225C22⋊Q8—84no—
226C4×Q8—48no—
227C4×Q8—84no—
228#117—24no—
229#117—42no—
230#117—42no—
231#117C2×Q818yes—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
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—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2414yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4C2×D411yescharacteristic
270C2×C22⋊C4C2×D411yescharacteristic
271C2×C22⋊C4C2×D412yes—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2411yescharacteristic
281C2×C42C2411yescharacteristic
282C2×C42C2411yescharacteristic
283C2×C42C2411yescharacteristic
284C2×C42C2411yescharacteristic
285C2×C42C2411yescharacteristic
286#107—44no—
287#107—44no—
288#107—44no—
289#107—44no—
290#106—44no—
291#106—44no—
292#106—44no—
293#106—44no—
294C4⋊C8C2×Q8116yes—
295C22⋊C8C2×Q8116yes—
296#97—24no—
297#97—24no—
298#97—24no—
299#97C2×Q818yes—
300C4×C8—24no—
301C4×C8C2×Q818yes—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—24no—
316#95—22no—
317#95C2×D412yes—
318#95C2×D412yes—
319#95C2×D412yes—
320#95C2×D412yes—
321C24C22×D411yescharacteristic
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8—82no—
328C2×Q8—82no—
329C2×Q8—84no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D412yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D412yes—
348C22×C4C22×D412yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D412yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×Q814yes—
354C22×C4C2511yescharacteristic
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8C22×Q818yes—
360C2×C8C22×Q818yes—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—28no—
374C4⋊C4—44no—
375C4⋊C4—44no—
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—82no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C4⋊C4C4⋊Q818yes—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—44no—
401C22⋊C4—84no—
402C22⋊C4C4⋊Q818yes—
403C42—24no—
404C42—22no—
405C42—22no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—44no—
411C42—84no—
412C42C22×Q814yes—
413C42C22×Q814yes—
414C42H(5,𝔽2)14yes—
415C422−1+414yes—
416C23—22no—
417C23—22no—
418C23—21no—
419C23—21no—
420C23C2×C4⋊Q812yes—
421C23C23×D411yescharacteristic
422Q8—88no—
423Q8—88no—
424Q8—164no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—28no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—84no—
462C2×C4—88no—
463C2×C4C2×C4⋊Q814yes—
464C2×C4C2×C4⋊Q814yes—
465C2×C4C2×C4⋊Q812yes—
466C2×C4#53514yes—
467C2×C4#53612yes—
468C2×C4#53612yes—
469C2×C4C23×Q812yes—
470C2×C4C2×H(5,𝔽2)12yes—
471C2×C4C2×2−1+412yes—
472C8—28no—
473C8—28no—
474C22—22no—
475C22—24no—
476C22—42no—
477C22—41no—
478C22—41no—
479C22C22×C4⋊Q811yescharacteristic
480C22C2×#53511yescharacteristic
481C22C2×#53611yescharacteristic
482C22#358512yes—
483C22#358612yes—
484C4—24no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—24no—
489C4—24no—
490C4—42no—
491C4—41no—
492C4—41no—
493C4—82no—
494C4—82no—
495C4—84no—
496C4—82no—
497C4—82no—
498C4—84no—
499C4#350314yes—
500C2—42no—
501C2#3528111yescharacteristic
502C2#6066812yes—
503C2#6081012yes—
504C2#6088311yescharacteristic
505C2#6088411yescharacteristic
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. #33317C2semidirect
4. #33317C2semidirect
5. #33297C2semidirect
6. #33294C2semidirect
7. #30315C2semidirect
8. #29217C2semidirect
9. #28693C2semidirect
10. #28692C2semidirect
109. #433Q8semidirect

Maximal subgroups

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

subgroupindex
#353352
#352832
#333172
#332972
#332942
#303152
#292172
#286932
#286922
#279592
#258192
#258182
#219922
#204652