Finite Group Databasebeta

← #6475414

Subgroups — #6475414

Order 512.

1790 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×#2851C211yesmaximal, characteristic
3#33218C214yesmaximal, complemented
4#30297C214yesmaximal, complemented
5#29300C214yesmaximal, complemented
6#29298C214yesmaximal, complemented
7#27956C212yesmaximal, complemented
8#27954C212yesmaximal, complemented
9#25819C212yesmaximal, complemented
10#25817C212yesmaximal, complemented
11#21944C214yesmaximal
12C4×#407C211yesmaximal, characteristic
13C2×#533C2211yescharacteristic
14C22×#125C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C2×#501C2214yescomplemented
17C2×#407C2212yes—
18C2×C4×M16C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2855—22no—
22#2851—22no—
23#2851—22no—
24#2851C2212yes—
25#2851C2212yes—
26#2851C2212yes—
27#2851C2212yes—
28#2851C2212yes—
29#2851C2212yes—
30#2848—22no—
31C4×C22⋊Q8—22no—
32C4×C22⋊Q8—22no—
33#2296C2212yes—
34C2×#389C2212yes—
35C2×#387—22no—
36C2×#385—22no—
37#2285C2211yescharacteristic
38C2×#382C2211yescharacteristic
39#2280C2212yes—
40C2×#380C2212yes—
41#2085C2218yescomplemented
42#2085C2218yescomplemented
43#2059C2218yescomplemented
44#2059C2218yescomplemented
45#1993C2214yescomplemented
46#1992C2214yescomplemented
47#1942C2218yescomplemented
48#1941C2218yescomplemented
49#1854C2218yes—
50#1854C2218yes—
51#1853C2214yes—
52#1853C2214yes—
53#1852C2214yes—
54#1852C2214yes—
55#1847C2214yescomplemented
56#1809C2218yes—
57#1809C2218yes—
58#1749C2212yes—
59#1749C2212yes—
60C4×C22⋊C8C2212yes—
61C4×C22⋊C8C2212yes—
62#1728C2212yes—
63C24×C4D412yes—
64C24×C4C2311yescharacteristic
65C22×M16C2314yes—
66#533—44no—
67C2×C4⋊Q8C2312yes—
68C2×#125—22no—
69C2×#125—44no—
70C2×#125C2312yes—
71C2×#125C2312yes—
72C2×#125C2312yes—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C4×Q8—22no—
77C2×#117C2312yes—
78C2×#117C2312yes—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—22no—
84C22×C42—44no—
85C22×C42D412yes—
86C22×C42D412yes—
87C22×C42D412yes—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#501—48no—
95C2×#107C2318yes—
96C2×#107C2318yes—
97C2×#106C2318yes—
98C2×#106C2318yes—
99#407Q8116yescomplemented
100C2×C22⋊C8C2314yes—
101C2×C22⋊C8C2314yes—
102C4×M16—44no—
103C2×#97C2314yes—
104C2×C4×C8C2314yes—
105#391—22no—
106#391—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—22no—
111#389C2314yes—
112#389C2314yes—
113#389C2314yes—
114#389C2314yes—
115#387—24no—
116#387—24no—
117#387—24no—
118#387—24no—
119#387—24no—
120#387—24no—
121#385—24no—
122#385—24no—
123#385—24no—
124#385—24no—
125#385—24no—
126#385—24no—
127#385—24no—
128#384C2312yes—
129#382—22no—
130#382C2312yes—
131#382C2312yes—
132#382C2312yes—
133#381—22no—
134#380—22no—
135#380—22no—
136#380—22no—
137#380—22no—
138#380—22no—
139#380C2314yes—
140#380C2314yes—
141#380C2314yes—
142#380C2314yes—
143#379—22no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—24no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4—22no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95—22no—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#336C2314yes—
156#336C2314yes—
157C25C2×D411yescharacteristic
158C22×Q8—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—44no—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×Q818yes—
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178C23×C4C2411yescharacteristic
179C2×M16—44no—
180C2×M16—44no—
181C2×M16C2×Q8116yes—
182C22×C8C2418yes—
183C4⋊Q8—28no—
184#125—28no—
185#125—42no—
186#125—42no—
187#125—44no—
188#125—48no—
189C22⋊Q8—48no—
190C22⋊Q8—48no—
191C22⋊Q8—48no—
192C4×Q8—48no—
193#117—44no—
194#117—44no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—48no—
211C2×C4⋊C4C2414yes—
212C2×C4⋊C4C2414yes—
213C2×C4⋊C4C2414yes—
214C2×C4⋊C4C2414yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—48no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2411yescharacteristic
240C2×C42C2411yescharacteristic
241C2×C42C2411yescharacteristic
242C2×C42C2411yescharacteristic
243#107—48no—
244#107—48no—
245#106—48no—
246#106—48no—
247C22⋊C8C2×Q8116yes—
248C22⋊C8C2×Q8116yes—
249#97—28no—
250C4×C8—44no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262C24—24no—
263C24—41no—
264C24—41no—
265C24—42no—
266C24C4⋊Q814yes—
267C24C22×D411yescharacteristic
268C24C22×D411yescharacteristic
269C24C22×D411yescharacteristic
270C2×Q8—44no—
271C2×Q8—44no—
272C2×Q8—44no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—416no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—432no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4C4⋊Q814yes—
316C22×C4C4⋊Q814yes—
317C22×C4C4⋊Q814yes—
318C22×C4C22×D412yes—
319C22×C4C22×D412yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D412yes—
324C22×C4C22×D412yes—
325C22×C4C22×Q814yes—
326C22×C4C22×Q814yes—
327C22×C4C22×Q814yes—
328C22×C4C2511yescharacteristic
329M16—416no—
330C2×C8—28no—
331C2×C8—28no—
332C2×C8C22×Q8116yes—
333C4⋊C4—24no—
334C4⋊C4—28no—
335C4⋊C4—24no—
336C4⋊C4—28no—
337C4⋊C4—28no—
338C4⋊C4—44no—
339C4⋊C4—48no—
340C4⋊C4—44no—
341C4⋊C4—48no—
342C4⋊C4—48no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—48no—
346C4⋊C4—48no—
347C4⋊C4—44no—
348C4⋊C4—48no—
349C4⋊C4—48no—
350C22⋊C4—48no—
351C22⋊C4—48no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—48no—
355C22⋊C4—48no—
356C42—24no—
357C42—24no—
358C42—24no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—44no—
366C42—44no—
367C42H(5,𝔽2)14yes—
368C23—21no—
369C23—21no—
370C23—22no—
371C23—28no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—44no—
381C23—44no—
382C23—48no—
383C23—48no—
384C23C2×C4⋊Q812yes—
385C23C2×C4⋊Q812yes—
386C23C2×C4⋊Q812yes—
387C23#57514yes—
388C23C23×D411yescharacteristic
389Q8—88no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—48no—
413C2×C4—48no—
414C2×C4—48no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—88no—
431C2×C4—88no—
432C2×C4C2×C4⋊Q812yes—
433C2×C4C2×C4⋊Q812yes—
434C2×C4C2×C4⋊Q812yes—
435C2×C4#53414yes—
436C2×C4#57514yes—
437C2×C4C23×Q812yes—
438C2×C4C2×H(5,𝔽2)12yes—
439C2×C4#58512yes—
440C8—216no—
441C22—24no—
442C22—24no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—416no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22#316614yes—
457C22C22×C4⋊Q811yescharacteristic
458C22C2×#53411yescharacteristic
459C22#344911yescharacteristic
460C22C2×#57512yes—
461C22C2×#57512yes—
462C4—24no—
463C4—24no—
464C4—22no—
465C4—22no—
466C4—22no—
467C4—42no—
468C4—42no—
469C4—84no—
470C4—84no—
471C4—84no—
472C4—84no—
473C4#350114yes—
474C2—24no—
475C2—42no—
476C2—42no—
477C2#3501111yescharacteristic
478C2C2×#316611yescharacteristic
479C2#6066311yescharacteristic
480C2#6075212yes—
481C2#6095412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33218C2semidirect
4. #30297C2semidirect
5. #29300C2semidirect
6. #29298C2semidirect
7. #27956C2semidirect
8. #27954C2semidirect
9. #25819C2semidirect
10. #25817C2semidirect
16. C2×#501C22semidirect
41. #2085C22semidirect
42. #2085C22semidirect
43. #2059C22semidirect
44. #2059C22semidirect
45. #1993C22semidirect
46. #1992C22semidirect
47. #1942C22semidirect
48. #1941C22semidirect
55. #1847C22semidirect
99. #407Q8semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#28512
#332182
#302972
#293002
#292982
#279562
#279542
#258192
#258172
#219442
C4×#4072