Finite Group Databasebeta

← #6484532

Subgroups — #6484532

Order 512.

1906 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic, complemented
2C2×#2367C211yesmaximal, characteristic
3#32956C212yesmaximal
4#32954C212yesmaximal, complemented
5#29953C212yesmaximal, complemented
6#29950C212yesmaximal
7#26542C218yesmaximal, complemented
8#26476C218yesmaximal, complemented
9#21955C211yesmaximal, characteristic, complemented
10#21808C212yesmaximal
11#21365C212yesmaximal, complemented
12C2×#544—21no—
13C22×#124C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#475C2214yescomplemented
17C2×#476C2214yes—
18C2×#432C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2659—21no—
22#2659—21no—
23#2552C2214yescomplemented
24#2551C2214yescomplemented
25#2550—24no—
26#2367—24no—
27#2367C2218yes—
28#2316—21no—
29C2×#391C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#388—21no—
32C2×#388—21no—
33C2×#385—21no—
34#2285C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4—21no—
38#2095C2214yescomplemented
39#2093C2214yes—
40#2068C2218yescomplemented
41#2067C2218yes—
42#2024C2214yescomplemented
43#2023C2214yes—
44#1983C2218yes—
45#1959C2214yes—
46#1959C2214yes—
47#1948C2214yescomplemented
48#1947C2214yescomplemented
49#1940C2218yescomplemented
50#1865C2218yescomplemented
51#1864C2218yes—
52#1853C2214yes—
53#1852C2214yes—
54#1803C2214yescomplemented
55#1802C2214yes—
56#1797C2212yes—
57#1797C2212yes—
58#1787C2214yes—
59#1786C2214yes—
60#1776C2218yescomplemented
61C23×Q8C2311yescharacteristic
62C23×D4C2311yescharacteristic
63C24×C4—21no—
64#544—48no—
65C2×C4⋊Q8—24no—
66C2×C4⋊Q8C2312yes—
67C2×C4⋊Q8C2312yes—
68C2×#125—24no—
69C2×#124—22no—
70C2×#124—24no—
71C2×#124Q818yescomplemented
72C2×#124C2314yes—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—44no—
77C2×C4×Q8C2314yes—
78C2×C4×D4—22no—
79C2×C4×D4—44no—
80C2×C4×D4C2314yes—
81C2×#117—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C42—21no—
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97#476—48no—
98#475—48no—
99#432—28no—
100C2×#107C2318yes—
101C2×#106C2318yes—
102C2×C4⋊C8C2314yes—
103C2×C4⋊C8C2314yes—
104C2×C4⋊C8C2314yes—
105C2×#103C2318yes—
106C2×#103C2318yes—
107C2×#102C2318yes—
108C2×#102C2318yes—
109C2×C22⋊C8C2314yes—
110C2×#97C2314yes—
111#398—24no—
112#398—24no—
113#395—24no—
114#395—24no—
115#391—22no—
116#391—22no—
117#391—22no—
118#391C2314yes—
119#390—22no—
120#390C2314yes—
121#389—24no—
122#389—24no—
123#389—22no—
124#389—22no—
125#389C2314yes—
126#389C2314yes—
127#388—28no—
128#388—28no—
129#385—28no—
130#384C2312yes—
131#384C2312yes—
132#384C2312yes—
133#384C2312yes—
134#382—22no—
135#380—22no—
136#380—22no—
137#380—22no—
138#380C2314yes—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4D414yes—
142C4×C4⋊C4C2314yes—
143C4×C4⋊C4C2314yes—
144C4×C4⋊C4C2314yes—
145C4×C22⋊C4—28no—
146C2×#95—21no—
147C2×#95—21no—
148C2×#95—21no—
149C2×#95D411yescharacteristic
150C2×#95D411yescharacteristic
151#340C2318yes—
152#340C2318yes—
153#336C2314yes—
154#336C2314yes—
155C25—21no—
156C22×Q8—21no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8C2×Q814yes—
163C22×Q8C2412yes—
164C22×D4—21no—
165C22×D4—22no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4C2×Q814yes—
169C22×D4C2412yes—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—24no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4—82no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C22×C8C2418yes—
185C4⋊Q8—44no—
186C4⋊Q8—44no—
187C4⋊Q8—416no—
188#125—416no—
189#124—216no—
190#124—28no—
191#124C2×Q8116yes—
192C22⋊Q8—48no—
193C22⋊Q8—48no—
194C4×Q8—28no—
195C4×Q8—44no—
196C4×Q8—48no—
197C4×D4—28no—
198C4×D4—44no—
199C4×D4—48no—
200#117—48no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4C2×D414yes—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2412yes—
224C2×C4⋊C4C2414yes—
225C2×C4⋊C4C2414yes—
226C2×C4⋊C4C2412yes—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—82no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—44no—
244C2×C42—82no—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×Q814yes—
249C2×C42#4114yes—
250C2×C42C2412yes—
251C2×C42C2412yes—
252C2×C42C2412yes—
253#107—48no—
254#106—216no—
255C4⋊C8—28no—
256C4⋊C8—44no—
257C4⋊C8—44no—
258#103—48no—
259#103—48no—
260#102—48no—
261#102—48no—
262C22⋊C8—28no—
263#97—28no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95C2×D414yes—
272#95C2×D414yes—
273C24—24no—
274C24—22no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24C22×D411yescharacteristic
282C2×Q8—24no—
283C2×Q8—22no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—48no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—48no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—82no—
293C2×Q8—82no—
294C2×Q8—82no—
295C2×Q8—82no—
296C2×Q8C22×Q814yes—
297C2×D4—24no—
298C2×D4—22no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—48no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—48no—
305C2×D4C22×Q814yes—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—48no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—48no—
338C22×C4—84no—
339C22×C4—816no—
340C22×C4C22⋊Q814yes—
341C22×C4C22⋊Q814yes—
342C22×C4AGL(1,ℤ/8)14yes—
343C22×C4#13714yes—
344C22×C4C22×D412yes—
345C22×C4C22×D412yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D412yes—
348C22×C4C22×D412yes—
349C22×C4C22×D412yes—
350C22×C4C22×D412yes—
351C22×C4C22×Q812yes—
352C22×C4C2×#4112yes—
353C22×C4C2511yescharacteristic
354C2×C8—28no—
355C2×C8—28no—
356C2×C8—28no—
357C4⋊C4—24no—
358C4⋊C4—28no—
359C4⋊C4—28no—
360C4⋊C4—22no—
361C4⋊C4—28no—
362C4⋊C4—28no—
363C4⋊C4—24no—
364C4⋊C4—48no—
365C4⋊C4—42no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—48no—
370C4⋊C4—44no—
371C4⋊C4—48no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C4⋊C4H(5,𝔽2)14yes—
379C22⋊C4—216no—
380C22⋊C4—216no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4—48no—
384C22⋊C4—84no—
385C42—24no—
386C42—24no—
387C42—42no—
388C42—42no—
389C42—48no—
390C42—48no—
391C42—84no—
392C42C22⋊Q818yes—
393C42C22×Q814yes—
394C42C2×#4114yes—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—24no—
400C23—48no—
401C23—48no—
402C23—42no—
403C23—42no—
404C23—48no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—84no—
410C23—84no—
411C23—84no—
412C23—84no—
413C23C2×C22⋊Q812yes—
414C23C2×AGL(1,ℤ/8)12yes—
415C23C2×#13712yes—
416C23C23×D411yescharacteristic
417Q8—28no—
418Q8—48no—
419Q8—44no—
420Q8—164no—
421Q8—164no—
422D4—28no—
423D4—48no—
424D4—44no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—28no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—48no—
450C2×C4—48no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—816no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—84no—
468C2×C4C2×C22⋊Q812yes—
469C2×C4C2×C22⋊Q814yes—
470C2×C4C2×C22⋊Q814yes—
471C2×C4C2×AGL(1,ℤ/8)14yes—
472C2×C4C2×#13714yes—
473C2×C4C23×Q811yescharacteristic
474C2×C4C22×#4111yescharacteristic
475C2×C4C2×H(5,𝔽2)12yes—
476C2×C4#58512yes—
477C8—48no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—44no—
485C22—44no—
486C22—44no—
487C22—816no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22#307914yes—
493C22C22×C22⋊Q811yescharacteristic
494C22#348411yescharacteristic
495C22#348711yescharacteristic
496C22C22×AGL(1,ℤ/8)11yescharacteristic
497C22C22×#13711yescharacteristic
498C22#358511yescharacteristic
499C22#358611yescharacteristic
500C4—24no—
501C4—24no—
502C4—22no—
503C4—24no—
504C4—24no—
505C4—44no—
506C4—42no—
507C4—44no—
508C4—42no—
509C4—84no—
510C4—84no—
511C4—82no—
512C4#346012yes—
513C4#350112yes—
514C2—24no—
515C2—84no—
516C2#3517411yescharacteristic
517C2C2×#307911yescharacteristic
518C2#6049211yescharacteristic
519C2#6157011yescharacteristic
520C2#6157211yescharacteristic
521C2#6188911yescharacteristic
522C2#6189111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35167C2semidirect
4. #32954C2semidirect
5. #29953C2semidirect
7. #26542C2semidirect
8. #26476C2semidirect
9. #21955C2semidirect
11. #21365C2semidirect
16. C2×#475C22semidirect
23. #2552C22semidirect
24. #2551C22semidirect
38. #2095C22semidirect
40. #2068C22semidirect
42. #2024C22semidirect
47. #1948C22semidirect
48. #1947C22semidirect
49. #1940C22semidirect
50. #1865C22semidirect
54. #1803C22semidirect
60. #1776C22semidirect
71. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#23672
#329562
#329542
#299532
#299502
#265422
#264762
#219552
#218082
#213652