Finite Group Databasebeta

← #6436928

Subgroups — #6436928

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic
3#29941C214yesmaximal, complemented
4#29926C214yesmaximal, complemented
5#29913C214yesmaximal, complemented
6#29895C212yesmaximal
7#29893C212yesmaximal
8#23765C218yesmaximal, complemented
9#21394C212yesmaximal, complemented
10#21365C212yesmaximal, complemented
11#20715C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C2×#432C2212yes—
15#2857C2212yes—
16#2857C2212yes—
17#2855—21no—
18#2855C2212yes—
19#2851—22no—
20#2848—21no—
21#2848C2212yes—
22C4×C22⋊Q8—22no—
23C4×C22⋊Q8C2214yes—
24#2296—21no—
25#2285C2211yescharacteristic
26#2283—22no—
27#2283—22no—
28#2283C2214yes—
29#2283C2214yes—
30C2×#383—21no—
31C2×#382C2212yes—
32#2280—21no—
33C2×#381—21no—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4—21no—
38C2×#376—21no—
39#2101C2218yescomplemented
40#2097C2218yescomplemented
41#2085C2218yescomplemented
42#2077C2218yes—
43#2076C2218yes—
44#2074C2218yescomplemented
45#2036C2218yes—
46#2035C2218yes—
47#1988C2214yescomplemented
48#1981C2214yescomplemented
49#1948C2214yescomplemented
50#1947C2214yescomplemented
51#1872C2214yes—
52#1872C2214yes—
53#1853C2214yes—
54#1852C2214yes—
55#1817C2212yes—
56#1797C2214yes—
57#1779C2212yes—
58#1776C2218yescomplemented
59#1763C2218yescomplemented
60#1749C2214yes—
61C24×C4D411yescharacteristic
62C24×C4D411yescharacteristic
63C24×C4C2311yescharacteristic
64#533—28no—
65C2×C4⋊Q8C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×#125—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8C2312yes—
71C2×C22⋊Q8C2314yes—
72C2×C4×Q8—21no—
73C2×C4×Q8C2312yes—
74C2×#117—21no—
75C2×#117—21no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C42—28no—
81C22×C42—22no—
82C22×C42D412yes—
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D412yes—
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90#432—28no—
91C2×#107C2318yes—
92C2×#106C2318yes—
93C2×C4⋊C8C2314yes—
94C2×#103C2318yes—
95C2×#103C2318yes—
96C2×C22⋊C8C2314yes—
97C2×C22⋊C8C2318yes—
98C2×#97C2314yes—
99#391—22no—
100#391C2314yes—
101#389—22no—
102#389—22no—
103#389—22no—
104#389C2314yes—
105#387—24no—
106#387—24no—
107#385—24no—
108#385—22no—
109#385—24no—
110#385—22no—
111#384C2312yes—
112#383—24no—
113#383—24no—
114#382—21no—
115#382—22no—
116#382—22no—
117#382C2314yes—
118#382C2314yes—
119#381—22no—
120#381—24no—
121#381—24no—
122#380—21no—
123#380—22no—
124#380—22no—
125#380—22no—
126#380—22no—
127#380—22no—
128#380C2314yes—
129#380C2314yes—
130#380C2314yes—
131#380C2312yes—
132#379—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2314yes—
139C4×C4⋊C4C2312yes—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—24no—
142C4×C22⋊C4—24no—
143#376—24no—
144#376—24no—
145C2×#95—21no—
146C2×#95—21no—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95—22no—
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#336C2318yes—
155C25C2×D411yescharacteristic
156C22×Q8—21no—
157C22×Q8C2412yes—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—42no—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C22×C8C2418yes—
180C4⋊Q8—24no—
181C4⋊Q8—24no—
182#125—44no—
183C22⋊Q8—28no—
184C22⋊Q8—48no—
185C22⋊Q8—48no—
186C22⋊Q8—44no—
187C4×Q8—28no—
188C4×Q8—44no—
189#117—44no—
190#117—44no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4C2414yes—
205C2×C4⋊C4C2414yes—
206C2×C4⋊C4C2412yes—
207C2×C4⋊C4C2414yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—28no—
220C2×C42—28no—
221C2×C42—28no—
222C2×C42—28no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—48no—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42#4118yes—
236C2×C42C2412yes—
237C2×C42C2411yescharacteristic
238C2×C42C2411yescharacteristic
239#107—48no—
240#106—48no—
241C4⋊C8—28no—
242#103—48no—
243#103—48no—
244C22⋊C8—28no—
245C22⋊C8—48no—
246#97—28no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—22no—
259#95—24no—
260#95—22no—
261C24—22no—
262C24—22no—
263C24—42no—
264C24—42no—
265C24#13714yes—
266C24C22×D412yes—
267C24C22×D411yescharacteristic
268C2×Q8—24no—
269C2×Q8—28no—
270C2×Q8—42no—
271C2×Q8—44no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
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—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—416no—
307C22×C4—416no—
308C22×C4—416no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4#12314yes—
316C22×C4#12314yes—
317C22×C4C22×D412yes—
318C22×C4C22×D412yes—
319C22×C4C22×D412yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C2×#4114yes—
326C22×C4C2511yescharacteristic
327C2×C8—28no—
328C2×C8—28no—
329C2×C8—28no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—28no—
335C4⋊C4—24no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—48no—
340C4⋊C4—48no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C22⋊C4—48no—
348C22⋊C4—48no—
349C22⋊C4—48no—
350C22⋊C4—44no—
351C22⋊C4—48no—
352C22⋊C4—44no—
353C42—24no—
354C42—24no—
355C42—24no—
356C42—24no—
357C42—24no—
358C42—24no—
359C42—48no—
360C42—44no—
361C42—44no—
362C42—48no—
363C42#12314yes—
364C42#12314yes—
365C42C2×#4114yes—
366C42C2×#4114yes—
367C23—22no—
368C23—22no—
369C23—28no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—48no—
377C23—44no—
378C23—44no—
379C23—48no—
380C23C2×#12312yes—
381C23C2×#13714yes—
382C23C2×#13712yes—
383C23#57512yes—
384C23C23×D411yescharacteristic
385Q8—48no—
386Q8—84no—
387C2×C4—28no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—28no—
395C2×C4—28no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—48no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—48no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
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—88no—
428C2×C4—88no—
429C2×C4C2×#12312yes—
430C2×C4C2×#12312yes—
431C2×C4C2×#12312yes—
432C2×C4C2×#12312yes—
433C2×C4C2×#12312yes—
434C2×C4#57514yes—
435C2×C4C22×#4112yes—
436C2×C4#58514yes—
437C8—48no—
438C22—24no—
439C22—24no—
440C22—22no—
441C22—22no—
442C22—24no—
443C22—24no—
444C22—24no—
445C22—416no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22#309514yes—
451C22C22×#12311yescharacteristic
452C22#344912yes—
453C22C22×#13711yescharacteristic
454C22C2×#57512yes—
455C22C2×#57511yescharacteristic
456C4—22no—
457C4—22no—
458C4—22no—
459C4—22no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—84no—
464C4—84no—
465C4—84no—
466C4—82no—
467C4—82no—
468C4#309314yes—
469C4#345514yes—
470C2—24no—
471C2—44no—
472C2#3463511yescharacteristic
473C2C2×#309311yescharacteristic
474C2C2×#309511yescharacteristic
475C2#6096012yes—
476C2#6097012yes—
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. #29941C2semidirect
4. #29926C2semidirect
5. #29913C2semidirect
8. #23765C2semidirect
9. #21394C2semidirect
10. #21365C2semidirect
39. #2101C22semidirect
40. #2097C22semidirect
41. #2085C22semidirect
44. #2074C22semidirect
47. #1988C22semidirect
48. #1981C22semidirect
49. #1948C22semidirect
50. #1947C22semidirect
58. #1776C22semidirect
59. #1763C22semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#22832
#299412
#299262
#299132
#298952
#298932
#237652
#213942
#213652
#207152