Finite Group Databasebeta

← #6475960

Subgroups — #6475960

Order 512.

1970 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C212yesmaximal
2#32937C214yesmaximal
3#30296C211yesmaximal, characteristic, complemented
4#30250C211yesmaximal, characteristic, complemented
5#30199C212yesmaximal, complemented
6#29165C214yesmaximal, complemented
7#28581C214yesmaximal, complemented
8#27855C214yesmaximal, complemented
9#25498C214yesmaximal, complemented
10#22188C214yesmaximal, complemented
11C2×#1728C211yesmaximal, characteristic
12C23×M16C2212yes—
13C2×#533C2212yes—
14C23×C42C2211yescharacteristic
15C2×#470C2214yes—
16C2×#470C2214yes—
17C2×#467C2214yes—
18#2857C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2857C2212yes—
22#2855—22no—
23#2855C2214yes—
24#2851—22no—
25#2851—22no—
26#2848—22no—
27#2848C2214yes—
28C4×C22⋊Q8—22no—
29C4×C22⋊Q8—22no—
30C4×C22⋊Q8C2214yes—
31C4×C22⋊Q8C2214yes—
32#2296—22no—
33#2285C2212yes—
34#2280—22no—
35#2099C2214yescomplemented
36#2096C2214yescomplemented
37#2084C2214yescomplemented
38#2072C2214yescomplemented
39#2057C2214yescomplemented
40#2043C2214yescomplemented
41#2024C2214yescomplemented
42#2016C2214yescomplemented
43#1927C2218yescomplemented
44#1920C2218yescomplemented
45#1894C2218yes—
46#1894C2218yes—
47#1892C2218yes—
48#1878C2218yescomplemented
49#1863C2218yescomplemented
50#1809C2214yes—
51#1809C2214yes—
52#1807C2214yes—
53#1728C2211yescharacteristic
54#1728C2214yes—
55#1728C2212yes—
56#1728C2211yescharacteristic
57C2×#336C2214yes—
58C24×C4D411yescharacteristic
59C24×C4D411yescharacteristic
60C24×C4C2311yescharacteristic
61C22×Q16C2314yes—
62C22×M16—28no—
63C22×M16—44no—
64C22×M16—44no—
65C22×M16C2314yes—
66C22×M16C2312yes—
67C22×M16C2312yes—
68C23×C8C2314yes—
69#533—48no—
70C2×C4⋊Q8C2312yes—
71C2×C4⋊Q8C2312yes—
72C2×#125—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8C2314yes—
77C2×C22⋊Q8C2314yes—
78C2×C22⋊Q8C2314yes—
79C2×C4×Q8—22no—
80C2×C4×Q8C2314yes—
81C2×#117—22no—
82C2×#117—22no—
83C22×C42—44no—
84C22×C42D411yescharacteristic
85C22×C42D412yes—
86C22×C42D412yes—
87C22×C42D411yescharacteristic
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92#470—216no—
93#470—48no—
94#467—48no—
95C2×#107C2314yes—
96C2×#106C2314yes—
97C2×#106C2314yes—
98C2×#103C2318yes—
99C2×#103C2318yes—
100C2×#103C2318yes—
101#391—22no—
102#391—22no—
103#391C2314yes—
104#391C2314yes—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—22no—
111#389C2314yes—
112#389C2314yes—
113#387—24no—
114#387—24no—
115#387—24no—
116#387—24no—
117#385—24no—
118#385—24no—
119#385—24no—
120#385—24no—
121#385—24no—
122#385—24no—
123#384C2312yes—
124#384C2312yes—
125#382—22no—
126#381—24no—
127#380—22no—
128#380—22no—
129#380—22no—
130#380—22no—
131#380—22no—
132#380—22no—
133#380—22no—
134#380C2314yes—
135#380C2314yes—
136#380C2314yes—
137#379—24no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4C2314yes—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—24no—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#336C2314yes—
148#336C2314yes—
149#336C2318yes—
150C25C2×D411yescharacteristic
151C22×Q8—22no—
152C22×Q8C2414yes—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—28no—
158C23×C4—44no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C23×C4C2411yescharacteristic
174C23×C4C2411yescharacteristic
175C2×Q16—28no—
176C2×Q16—44no—
177C2×Q16—44no—
178C2×M16—24no—
179C2×M16—28no—
180C2×M16—28no—
181C2×M16—42no—
182C2×M16—42no—
183C2×M16—44no—
184C2×M16—42no—
185C2×M16—42no—
186C2×M16—432no—
187C2×M16#4118yes—
188C22×C8—28no—
189C22×C8—28no—
190C22×C8—44no—
191C22×C8C2414yes—
192C22×C8C2414yes—
193C4⋊Q8—44no—
194C4⋊Q8—44no—
195#125—48no—
196C22⋊Q8—216no—
197C22⋊Q8—216no—
198C22⋊Q8—28no—
199C22⋊Q8—28no—
200C22⋊Q8—48no—
201C22⋊Q8—48no—
202C22⋊Q8—48no—
203C4×Q8—48no—
204C4×Q8—48no—
205#117—48no—
206#117—48no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4C2414yes—
221C2×C4⋊C4C2414yes—
222C2×C4⋊C4C2414yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C42—28no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—42no—
233C2×C42—44no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—44no—
237C2×C42—42no—
238C2×C42—44no—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2412yes—
246C2×C42C2412yes—
247#107—44no—
248#106—28no—
249#106—44no—
250#103—216no—
251#103—48no—
252#103—48no—
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—
262#95—24no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—24no—
267C24—42no—
268C24C22×D411yescharacteristic
269C24C22×D411yescharacteristic
270C24C22×D411yescharacteristic
271C2×Q8—28no—
272C2×Q8—24no—
273C2×Q8—28no—
274C2×Q8—24no—
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8—44no—
278C22×C4—216no—
279C22×C4—216no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—416no—
321C22×C4—416no—
322C22×C4#13718yes—
323C22×C4C22×D412yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D412yes—
328C22×C4C22×D412yes—
329C22×C4C22×D412yes—
330C22×C4C2×#4114yes—
331C22×C4H(5,𝔽2)14yes—
332C22×C4H(5,𝔽2)12yes—
333C22×C42−1+412yes—
334C22×C4C2511yescharacteristic
335Q16—88no—
336M16—48no—
337M16—416no—
338M16—48no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—416no—
345C2×C8C2×#4118yes—
346C4⋊C4—28no—
347C4⋊C4—24no—
348C4⋊C4—28no—
349C4⋊C4—24no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—48no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—48no—
356C4⋊C4—48no—
357C4⋊C4—48no—
358C4⋊C4—48no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—48no—
363C22⋊C4—48no—
364C22⋊C4—48no—
365C22⋊C4—48no—
366C22⋊C4—48no—
367C22⋊C4—48no—
368C22⋊C4—48no—
369C42—24no—
370C42—24no—
371C42—24no—
372C42—24no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—48no—
378C42—48no—
379C23—21no—
380C23—21no—
381C23—22no—
382C23—28no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—48no—
392C23—48no—
393C23—44no—
394C23—44no—
395C23#53212yes—
396C23#53412yes—
397C23#53712yes—
398C23C2×#13714yes—
399C23C23×D411yescharacteristic
400Q8—48no—
401Q8—48no—
402Q8—88no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—216no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—44no—
429C2×C4—48no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—48no—
433C2×C4—48no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4#53212yes—
448C2×C4#53412yes—
449C2×C4#53712yes—
450C2×C4C2×#13714yes—
451C2×C4C2×#13714yes—
452C2×C4#57514yes—
453C2×C4C22×#4112yes—
454C2×C4C2×H(5,𝔽2)11yescharacteristic
455C2×C4C2×H(5,𝔽2)12yes—
456C2×C4C2×2−1+411yescharacteristic
457C8—28no—
458C8—44no—
459C22—24no—
460C22—22no—
461C22—22no—
462C22—24no—
463C22—24no—
464C22—22no—
465C22—22no—
466C22—24no—
467C22—416no—
468C22—42no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—42no—
474C22#319514yes—
475C22C2×#53211yescharacteristic
476C22C2×#53411yescharacteristic
477C22C2×#53711yescharacteristic
478C22C22×#13712yes—
479C22C2×#57512yes—
480C4—24no—
481C4—24no—
482C4—24no—
483C4—22no—
484C4—22no—
485C4—22no—
486C4—42no—
487C4—84no—
488C4—84no—
489C4—84no—
490C4—84no—
491C4#349012yes—
492C4#349212yes—
493C2—24no—
494C2—42no—
495C2—42no—
496C2#3505111yescharacteristic
497C2#6070112yes—
498C2C2×#319511yescharacteristic
499C2#6074111yescharacteristic
500C2#6091812yes—
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. #30296C2semidirect
4. #30250C2semidirect
5. #30199C2semidirect
6. #29165C2semidirect
7. #28581C2semidirect
8. #27855C2semidirect
9. #25498C2semidirect
10. #22188C2semidirect
35. #2099C22semidirect
36. #2096C22semidirect
37. #2084C22semidirect
38. #2072C22semidirect
39. #2057C22semidirect
40. #2043C22semidirect
41. #2024C22semidirect
42. #2016C22semidirect
43. #1927C22semidirect
44. #1920C22semidirect
48. #1878C22semidirect
49. #1863C22semidirect

Maximal subgroups

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

subgroupindex
#366252
#329372
#302962
#302502
#301992
#291652
#285812
#278552
#254982
#221882
C2×#17282