Finite Group Databasebeta

← #6510747

Subgroups — #6510747

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#2499C211yesmaximal, characteristic
3#29111C212yesmaximal, complemented
4#29105C212yesmaximal, complemented
5#27969C212yesmaximal, complemented
6#27960C212yesmaximal, complemented
7#27722C218yesmaximal, complemented
8#27706C218yesmaximal, complemented
9#21765C212yesmaximal
10#20702C212yesmaximal
11#20524C211yesmaximal, characteristic
12C2×#543—21no—
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C2×#407C2212yes—
16#2857C2212yes—
17#2857C2212yes—
18#2659—21no—
19#2659—21no—
20#2549—24no—
21#2499—24no—
22#2499C2218yes—
23#2440C2214yes—
24#2440C2214yes—
25C2×#397—21no—
26C2×#397—21no—
27#2315—21no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#387—21no—
31C2×#387—21no—
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36#2101C2218yescomplemented
37#2099C2218yescomplemented
38#2072C2214yescomplemented
39#2072C2214yescomplemented
40#2024C2214yescomplemented
41#2024C2214yescomplemented
42#1993C2214yescomplemented
43#1990C2214yescomplemented
44#1935C2218yescomplemented
45#1933C2218yescomplemented
46#1894C2218yes—
47#1872C2214yes—
48#1872C2214yes—
49#1836C2214yescomplemented
50#1834C2214yescomplemented
51#1825C2218yes—
52#1824C2214yes—
53#1823C2214yes—
54#1823C2214yes—
55#1787C2214yes—
56#1753C2218yes—
57#1749C2212yes—
58C4×C22⋊C8C2212yes—
59#1736C2214yes—
60#1736C2214yes—
61C23×Q8C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C22×M16C2314yes—
64#543—48no—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—24no—
72C2×C22⋊Q8—28no—
73C2×C22⋊Q8C2314yes—
74C2×#117—22no—
75C2×#117—28no—
76C2×#117C2314yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4D411yescharacteristic
89C22×C22⋊C4D411yescharacteristic
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C2×#103C2318yes—
94C2×#103C2318yes—
95#407—28no—
96C2×C22⋊C8C2314yes—
97C2×C22⋊C8C2314yes—
98C2×#97C2314yes—
99C2×C4×C8C2314yes—
100#398—24no—
101#398—24no—
102#397—28no—
103#397—28no—
104#395—24no—
105#395—24no—
106#391—22no—
107#391—22no—
108#391—22no—
109#391—22no—
110#391C2314yes—
111#391C2314yes—
112#389—22no—
113#389—22no—
114#389C2314yes—
115#389C2314yes—
116#387—28no—
117#387—28no—
118#385—24no—
119#385—24no—
120#384—22no—
121#384C2312yes—
122#384C2312yes—
123#384C2314yes—
124#382—22no—
125#382C2314yes—
126#380—22no—
127#380—22no—
128#380—22no—
129#380—22no—
130#380C2314yes—
131#380C2314yes—
132C4×C22⋊C4D414yes—
133C4×C22⋊C4C2314yes—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—22no—
141C2×#95D411yescharacteristic
142C2×#95D411yescharacteristic
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#336C2314yes—
150#336C2314yes—
151C25C2×D411yescharacteristic
152C22×Q8—21no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—24no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8C2412yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4#4118yes—
174C23×C4C2411yescharacteristic
175C23×C4C2412yes—
176C2×M16—28no—
177C2×M16—44no—
178C2×M16—44no—
179C22×C8C2418yes—
180C4⋊Q8—44no—
181C4⋊Q8—44no—
182C22⋊Q8—28no—
183C22⋊Q8—28no—
184C22⋊Q8—416no—
185C22⋊Q8—416no—
186C22⋊Q8—48no—
187C22⋊Q8—48no—
188C22⋊Q8—48no—
189#117—28no—
190#117—28no—
191#117—48no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4C2412yes—
217C2×C4⋊C4C2412yes—
218C2×C4⋊C4C2412yes—
219C2×C4⋊C4C2414yes—
220C2×C4⋊C4C2414yes—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—48no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4C2×D414yes—
231C2×C22⋊C4C2×D414yes—
232C2×C42—24no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—44no—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2412yes—
239C2×C42C2412yes—
240#103—48no—
241#103—48no—
242C22⋊C8—28no—
243C22⋊C8—28no—
244#97—44no—
245C4×C8—44no—
246#95—24no—
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#95C2×D414yes—
257#95C2×D414yes—
258C24—22no—
259C24—22no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24—41no—
264C24#12414yes—
265C24C22×D412yes—
266C24C22×D411yescharacteristic
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—44no—
273C2×Q8—42no—
274C2×Q8—44no—
275C2×Q8—82no—
276C2×Q8—82no—
277C2×Q8—82no—
278C2×Q8—82no—
279C2×Q8—88no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—416no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4#12414yes—
324C22×C4#12418yes—
325C22×C4AGL(1,ℤ/8)14yes—
326C22×C4#13714yes—
327C22×C4C22×D412yes—
328C22×C4C22×D412yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D412yes—
331C22×C4C22×D412yes—
332C22×C4C22×D412yes—
333C22×C4C2×#4114yes—
334C22×C4C2×#4114yes—
335C22×C4C2×#4114yes—
336C22×C4C2511yescharacteristic
337M16—88no—
338C2×C8—28no—
339C2×C8—28no—
340C2×C8—28no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—28no—
345C4⋊C4—28no—
346C4⋊C4—44no—
347C4⋊C4—48no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—48no—
352C4⋊C4—48no—
353C4⋊C4—44no—
354C4⋊C4—48no—
355C4⋊C4—48no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C22⋊C4—28no—
363C22⋊C4—28no—
364C22⋊C4—48no—
365C22⋊C4—48no—
366C22⋊C4—48no—
367C22⋊C4—48no—
368C42—24no—
369C42—24no—
370C42—42no—
371C42—42no—
372C42—48no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—28no—
384C23—44no—
385C23—44no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23C2×#12414yes—
391C23C2×#12412yes—
392C23C2×AGL(1,ℤ/8)12yes—
393C23C2×#13712yes—
394C23C23×D411yescharacteristic
395Q8—48no—
396Q8—84no—
397Q8—164no—
398Q8—164no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—28no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—44no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4—164no—
445C2×C4—164no—
446C2×C4C2×#12414yes—
447C2×C4C2×#12412yes—
448C2×C4C2×AGL(1,ℤ/8)14yes—
449C2×C4C2×#13714yes—
450C2×C4C22×#4112yes—
451C2×C4#58512yes—
452C2×C4#58512yes—
453C8—48no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—24no—
461C22—24no—
462C22—24no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—88no—
468C22#314114yes—
469C22C22×#12411yescharacteristic
470C22#348511yescharacteristic
471C22#348511yescharacteristic
472C22C22×AGL(1,ℤ/8)11yescharacteristic
473C22C22×#13711yescharacteristic
474C22#358611yescharacteristic
475C22#358611yescharacteristic
476C4—24no—
477C4—24no—
478C4—24no—
479C4—22no—
480C4—42no—
481C4—42no—
482C4—84no—
483C4—84no—
484C4—82no—
485C4—82no—
486C4—82no—
487C4—82no—
488C4#349114yes—
489C2—24no—
490C2—44no—
491C2#3545511yescharacteristic
492C2C2×#314111yescharacteristic
493C2#6062011yescharacteristic
494C2#6170411yescharacteristic
495C2#6170511yescharacteristic
496C2#6172211yescharacteristic
497C2#6172311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29111C2semidirect
4. #29105C2semidirect
5. #27969C2semidirect
6. #27960C2semidirect
7. #27722C2semidirect
8. #27706C2semidirect
36. #2101C22semidirect
37. #2099C22semidirect
38. #2072C22semidirect
39. #2072C22semidirect
40. #2024C22semidirect
41. #2024C22semidirect
42. #1993C22semidirect
43. #1990C22semidirect
44. #1935C22semidirect
45. #1933C22semidirect
49. #1836C22semidirect
50. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#24992
#291112
#291052
#279692
#279602
#277222
#277062
#217652
#207022
#205242