Finite Group Databasebeta

← #6506779

Subgroups — #6506779

Order 512.

1742 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#2441C211yesmaximal, characteristic
3#29302C212yesmaximal, complemented
4#29297C212yesmaximal, complemented
5#28783C218yesmaximal, complemented
6#27950C212yesmaximal, complemented
7#26077C218yesmaximal, complemented
8#25819C212yesmaximal, complemented
9#21942C212yesmaximal
10#20846C212yesmaximal
11#20524C211yesmaximal, characteristic
12C2×#543—21no—
13C22×#125C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C2×#501C2214yescomplemented
16C2×#428C2214yes—
17C2×#407C2212yes—
18#2857C2212yes—
19#2857C2212yes—
20#2659—21no—
21#2659—21no—
22#2549—24no—
23#2441—24no—
24#2441C2218yes—
25#2440C2214yes—
26#2440C2214yes—
27C2×#399—21no—
28C2×#396—21no—
29#2315—21no—
30C2×#389—21no—
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#387—21no—
34C2×#384C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#2084C2214yescomplemented
39#2084C2214yescomplemented
40#2083C2218yescomplemented
41#2059C2218yescomplemented
42#2058C2214yescomplemented
43#2057C2214yescomplemented
44#1992C2214yescomplemented
45#1990C2214yescomplemented
46#1944C2218yescomplemented
47#1941C2218yescomplemented
48#1854C2218yes—
49#1853C2214yes—
50#1853C2214yes—
51#1853C2214yes—
52#1852C2214yes—
53#1852C2214yes—
54#1852C2214yes—
55#1845C2214yescomplemented
56#1809C2218yes—
57#1753C2218yes—
58#1749C2212yes—
59C4×C22⋊C8C2212yes—
60#1736C2214yes—
61C24×C4C2311yescharacteristic
62C22×M16C2314yes—
63#543—48no—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×#125—22no—
67C2×#125—44no—
68C2×#125C2314yes—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—24no—
72C2×C22⋊Q8—24no—
73C2×#117—22no—
74C2×#117—44no—
75C2×#117C2314yes—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—22no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4D411yescharacteristic
92C22×C22⋊C4D411yescharacteristic
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#501—48no—
97#428—48no—
98C2×#107C2318yes—
99C2×#107C2318yes—
100C2×#106C2318yes—
101C2×#106C2318yes—
102#407Q8116yescomplemented
103C2×C22⋊C8C2314yes—
104C2×C22⋊C8C2314yes—
105C2×#97C2314yes—
106C2×C4×C8C2314yes—
107#399—28no—
108#398—24no—
109#398—24no—
110#396—28no—
111#395—24no—
112#395—24no—
113#391—22no—
114#391—22no—
115#389—22no—
116#389—22no—
117#389—22no—
118#389—22no—
119#389—28no—
120#389C2314yes—
121#389C2314yes—
122#389C2314yes—
123#389C2314yes—
124#387—28no—
125#385—24no—
126#385—24no—
127#384—22no—
128#384C2314yes—
129#384C2312yes—
130#384C2312yes—
131#380—22no—
132#380—22no—
133#380—22no—
134#380—22no—
135#380—22no—
136#380C2314yes—
137#380C2314yes—
138#380C2314yes—
139C4×C22⋊C4D414yes—
140C4×C22⋊C4C2314yes—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—22no—
145C2×#95D411yescharacteristic
146C2×#95D411yescharacteristic
147#340C2318yes—
148#340C2318yes—
149#340C2318yes—
150#340C2318yes—
151#336C2314yes—
152#336C2314yes—
153C25C2×D411yescharacteristic
154C22×Q8—22no—
155C22×Q8—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×Q818yes—
171C23×C4C2411yescharacteristic
172C23×C4C2412yes—
173C2×M16—44no—
174C2×M16—44no—
175C2×M16C2×Q8116yes—
176C22×C8C2418yes—
177C4⋊Q8—44no—
178C4⋊Q8—44no—
179#125—28no—
180#125—44no—
181#125—48no—
182C22⋊Q8—416no—
183C22⋊Q8—416no—
184C22⋊Q8—48no—
185C22⋊Q8—48no—
186#117—28no—
187#117—44no—
188#117—48no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4C2412yes—
222C2×C4⋊C4C2412yes—
223C2×C4⋊C4C2414yes—
224C2×C4⋊C4C2414yes—
225C2×C4⋊C4C2412yes—
226C2×C4⋊C4C2412yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—48no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4C2×D414yes—
237C2×C22⋊C4C2×D414yes—
238C2×C42—24no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—44no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2412yes—
245C2×C42C2412yes—
246#107—48no—
247#107—48no—
248#106—216no—
249#106—48no—
250C22⋊C8C2×Q8116yes—
251C22⋊C8C2×Q8116yes—
252#97—44no—
253C4×C8—44no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95C2×D414yes—
262#95C2×D414yes—
263C24—22no—
264C24—22no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24C4⋊Q814yes—
270C24C22×D412yes—
271C24C22×D411yescharacteristic
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—82no—
275C2×Q8—82no—
276C2×Q8—82no—
277C2×Q8—82no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—416no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
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—44no—
316C22×C4—44no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4C4⋊Q818yes—
322C22×C4C4⋊Q814yes—
323C22×C4AGL(1,ℤ/8)14yes—
324C22×C4#13714yes—
325C22×C4C22×D412yes—
326C22×C4C22×D412yes—
327C22×C4C22×D412yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D412yes—
330C22×C4C22×D412yes—
331C22×C4C22×Q814yes—
332C22×C4C22×Q814yes—
333C22×C4C22×Q814yes—
334C22×C4C2511yescharacteristic
335M16—416no—
336C2×C8—28no—
337C2×C8—28no—
338C2×C8C22×Q8116yes—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—28no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—28no—
346C4⋊C4—48no—
347C4⋊C4—48no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—48no—
353C4⋊C4—48no—
354C4⋊C4—44no—
355C4⋊C4—48no—
356C4⋊C4—48no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4H(5,𝔽2)14yes—
366C22⋊C4—28no—
367C22⋊C4—28no—
368C22⋊C4—48no—
369C22⋊C4—48no—
370C22⋊C4—48no—
371C22⋊C4—84no—
372C42—24no—
373C42—24no—
374C42—42no—
375C42—42no—
376C42—48no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—28no—
388C23—44no—
389C23—44no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23C2×C4⋊Q814yes—
395C23C2×C4⋊Q812yes—
396C23C2×AGL(1,ℤ/8)12yes—
397C23C2×#13712yes—
398C23C23×D411yescharacteristic
399Q8—164no—
400Q8—164no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—28no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—44no—
420C2×C4—48no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—48no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—84no—
446C2×C4—164no—
447C2×C4—164no—
448C2×C4C2×C4⋊Q814yes—
449C2×C4C2×C4⋊Q812yes—
450C2×C4C2×AGL(1,ℤ/8)14yes—
451C2×C4C2×#13714yes—
452C2×C4C23×Q812yes—
453C2×C4C2×H(5,𝔽2)12yes—
454C2×C4#58512yes—
455C8—216no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—24no—
463C22—24no—
464C22—24no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—88no—
470C22#316514yes—
471C22C22×C4⋊Q811yescharacteristic
472C22#348411yescharacteristic
473C22#348511yescharacteristic
474C22C22×AGL(1,ℤ/8)11yescharacteristic
475C22C22×#13711yescharacteristic
476C22#358511yescharacteristic
477C22#358611yescharacteristic
478C4—24no—
479C4—24no—
480C4—24no—
481C4—22no—
482C4—42no—
483C4—44no—
484C4—42no—
485C4—84no—
486C4—82no—
487C4—82no—
488C4—82no—
489C4—84no—
490C4#350114yes—
491C2—24no—
492C2—44no—
493C2#3539111yescharacteristic
494C2C2×#316511yescharacteristic
495C2#6066411yescharacteristic
496C2#6158611yescharacteristic
497C2#6158711yescharacteristic
498C2#6170411yescharacteristic
499C2#6170811yescharacteristic
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. #29302C2semidirect
4. #29297C2semidirect
5. #28783C2semidirect
6. #27950C2semidirect
7. #26077C2semidirect
8. #25819C2semidirect
15. C2×#501C22semidirect
38. #2084C22semidirect
39. #2084C22semidirect
40. #2083C22semidirect
41. #2059C22semidirect
42. #2058C22semidirect
43. #2057C22semidirect
44. #1992C22semidirect
45. #1990C22semidirect
46. #1944C22semidirect
47. #1941C22semidirect
55. #1845C22semidirect
102. #407Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#24412
#293022
#292972
#287832
#279502
#260772
#258192
#219422
#208462
#205242