Finite Group Databasebeta

← #6506424

Subgroups — #6506424

Order 512.

2400 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic, complemented
2C2×#2432C211yesmaximal, characteristic
3#29106C212yesmaximal, complemented
4#29103C212yesmaximal, complemented
5#27261C218yesmaximal, complemented
6#27255C218yesmaximal, complemented
7#25859C212yesmaximal, complemented
8#25856C212yesmaximal, complemented
9#21762C212yesmaximal, complemented
10#20597C212yesmaximal
11#20523C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#124C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C2×#488C2214yescomplemented
17C2×#487C2214yescomplemented
18C2×#416C2214yes—
19C2×#416C2214yes—
20C22×C22⋊C8C2212yes—
21#2857C2212yes—
22#2851C2212yes—
23#2626—22no—
24#2608—22no—
25#2607—22no—
26#2441C2214yescomplemented
27#2439C2214yescomplemented
28#2432—22no—
29#2432—22no—
30#2432C2218yes—
31C2×#394—22no—
32#2329—22no—
33C2×#391C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#385—22no—
36#2280C2212yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38#2074C2214yescomplemented
39#2071C2214yescomplemented
40#2043C2218yescomplemented
41#2042C2218yescomplemented
42#2026C2214yescomplemented
43#2023C2214yescomplemented
44#1958C2214yescomplemented
45#1956C2214yescomplemented
46#1934C2218yescomplemented
47#1933C2218yescomplemented
48#1892C2218yescomplemented
49#1873C2214yescomplemented
50#1873C2214yescomplemented
51#1823C2214yes—
52#1788C2218yes—
53#1787C2214yes—
54#1787C2214yes—
55#1786C2214yes—
56C8×C22⋊C4C2218yescomplemented
57C4×C22⋊C8C2212yes—
58C4×C22⋊C8C2212yes—
59C22×#41—22no—
60C22×#41—28no—
61C22×#41—44no—
62C22×#41—44no—
63C22×#41C2314yes—
64C23×Q8C2311yescharacteristic
65C23×Q8C2311yescharacteristic
66C23×D4—21no—
67C23×D4C2311yescharacteristic
68C24×C4—21no—
69C24×C4C2311yescharacteristic
70C23×C8C2314yes—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—44no—
74C2×C4⋊Q8C2314yes—
75C2×C4⋊Q8C2312yes—
76C2×#125C2312yes—
77C2×#124—22no—
78C2×#124—44no—
79C2×#124C2314yes—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8C2314yes—
84C2×#117C2312yes—
85C2×#117C2312yes—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4D412yes—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#488—48no—
96#487—48no—
97#416—48no—
98#416—48no—
99C2×#103C2318yes—
100C2×#103C2318yes—
101C2×#103C2318yes—
102C2×#103C2318yes—
103C2×#102C2318yes—
104C2×#102C2318yes—
105C2×C22⋊C8—28no—
106C2×C22⋊C8C2314yes—
107C2×C22⋊C8C2314yes—
108C2×C4×C8C2314yes—
109C2×C4×C8C2314yes—
110#399—24no—
111#396—24no—
112#396—24no—
113#394—28no—
114#394—28no—
115#393—24no—
116#391—22no—
117#391—22no—
118#391—24no—
119#391C2314yes—
120#390—21no—
121#390—21no—
122#390C2314yes—
123#389—22no—
124#389—24no—
125#389C2314yes—
126#389C2314yes—
127#387—24no—
128#387—24no—
129#387—24no—
130#385—24no—
131#385—28no—
132#385—28no—
133#384C2312yes—
134#382C2312yes—
135#381—22no—
136#380—22no—
137#380C2314yes—
138#380C2314yes—
139#379—22no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4D412yes—
143C4×C22⋊C4D412yes—
144C4×C22⋊C4C2314yes—
145C2×#95—22no—
146C2×#95D412yes—
147#340C2318yes—
148#340C2318yes—
149#336C2314yes—
150#336C2314yes—
151C25—21no—
152C25C2×D411yescharacteristic
153C2×#41—28no—
154C2×#41—28no—
155C2×#41—28no—
156C2×#41—28no—
157C2×#41—44no—
158C2×#41—44no—
159C2×#41—48no—
160C2×#41—48no—
161C2×#41—464no—
162C2×#41—48no—
163C22×Q8—24no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—22no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—42no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8C2412yes—
173C22×Q8C2412yes—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—21no—
177C22×D4—24no—
178C22×D4—42no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4C2412yes—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—24no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4C2×D412yes—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4#4118yes—
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C22×C8—28no—
200C22×C8—44no—
201C22×C8—44no—
202C22×C8C2418yes—
203C4⋊Q8—28no—
204C4⋊Q8—44no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207C4⋊Q8—44no—
208#125—44no—
209#124—28no—
210#124—44no—
211#124—48no—
212C22⋊Q8—28no—
213C22⋊Q8—28no—
214C22⋊Q8—48no—
215C22⋊Q8—48no—
216#117—44no—
217#117—44no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4C2412yes—
232C2×C4⋊C4C2414yes—
233C2×C4⋊C4C2414yes—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—48no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—48no—
246C2×C22⋊C4C2×D414yes—
247C2×C22⋊C4C2×D414yes—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2412yes—
256C2×C42C2412yes—
257#103—216no—
258#103—48no—
259#103—48no—
260#103—48no—
261#102—216no—
262#102—48no—
263C22⋊C8—28no—
264C22⋊C8—28no—
265C4×C8—44no—
266C4×C8—44no—
267#95—28no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95C2×D414yes—
274#95C2×D414yes—
275C24—24no—
276C24—24no—
277C24—22no—
278C24—41no—
279C24—41no—
280C24—42no—
281C24—42no—
282C24—44no—
283C24—42no—
284C24—44no—
285C24#12414yes—
286C24C22×D412yes—
287C24C22×D411yescharacteristic
288#41—232no—
289#41—416no—
290#41—416no—
291#41—416no—
292#41—464no—
293C2×Q8—24no—
294C2×Q8—22no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—416no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—44no—
307C2×Q8—82no—
308C2×Q8—82no—
309C2×Q8—88no—
310C2×Q8H(5,𝔽2)14yes—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—22no—
315C2×D4—416no—
316C2×D4—48no—
317C2×D4—42no—
318C2×D4—48no—
319C2×D4—432no—
320C2×D4—48no—
321C2×D4—42no—
322C2×D4H(5,𝔽2)14yes—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—416no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—48no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—48no—
345C22×C4—42no—
346C22×C4—432no—
347C22×C4—42no—
348C22×C4—48no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4#12414yes—
355C22×C4#12418yes—
356C22×C4#13518yes—
357C22×C4C22×D414yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D412yes—
360C22×C4C22×D412yes—
361C22×C4C22×D412yes—
362C22×C4C2×#4114yes—
363C22×C4C2×#4114yes—
364C22×C4C2×#4114yes—
365C22×C4C2511yescharacteristic
366C2×C8—28no—
367C2×C8—28no—
368C2×C8—28no—
369C2×C8—88no—
370C4⋊C4—24no—
371C4⋊C4—28no—
372C4⋊C4—28no—
373C4⋊C4—48no—
374C4⋊C4—48no—
375C4⋊C4—42no—
376C4⋊C4—48no—
377C4⋊C4—44no—
378C4⋊C4—48no—
379C4⋊C4—82no—
380C4⋊C4—84no—
381C4⋊C4—84no—
382C4⋊C4—82no—
383C22⋊C4—28no—
384C22⋊C4—28no—
385C22⋊C4—416no—
386C22⋊C4—48no—
387C22⋊C4—48no—
388C22⋊C4—48no—
389C22⋊C4—48no—
390C22⋊C4—84no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C42—24no—
394C42—24no—
395C42—42no—
396C42—41no—
397C42—41no—
398C42—82no—
399C42—82no—
400C23—22no—
401C23—21no—
402C23—21no—
403C23—24no—
404C23—22no—
405C23—22no—
406C23—24no—
407C23—28no—
408C23—24no—
409C23—44no—
410C23—44no—
411C23—48no—
412C23—416no—
413C23—44no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—84no—
418C23—84no—
419C23—88no—
420C23—84no—
421C23—84no—
422C23C2×#12414yes—
423C23C2×#12412yes—
424C23C2×#13514yes—
425C23C23×D411yescharacteristic
426Q8—28no—
427Q8—48no—
428Q8—44no—
429Q8—48no—
430Q8—84no—
431Q8—164no—
432D4—28no—
433D4—48no—
434D4—432no—
435D4—44no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—44no—
452C2×C4—48no—
453C2×C4—48no—
454C2×C4—432no—
455C2×C4—48no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—84no—
472C2×C4C2×#12414yes—
473C2×C4C2×#12412yes—
474C2×C4C2×#13514yes—
475C2×C4C2×#13514yes—
476C2×C4C22×#4112yes—
477C2×C4C2×H(5,𝔽2)12yes—
478C2×C4C2×H(5,𝔽2)12yes—
479C8—48no—
480C22—22no—
481C22—24no—
482C22—24no—
483C22—22no—
484C22—24no—
485C22—24no—
486C22—24no—
487C22—44no—
488C22—44no—
489C22—44no—
490C22—48no—
491C22—88no—
492C22—816no—
493C22—84no—
494C22—82no—
495C22—82no—
496C22#313114yes—
497C22C22×#12411yescharacteristic
498C22#348411yescharacteristic
499C22#348411yescharacteristic
500C22C22×#13512yes—
501C22#358612yes—
502C4—24no—
503C4—22no—
504C4—24no—
505C4—24no—
506C4—44no—
507C4—44no—
508C4—82no—
509C4—84no—
510C4—84no—
511C4—82no—
512C4#349014yes—
513C2—24no—
514C2—44no—
515C2—84no—
516C2#3536711yescharacteristic
517C2C2×#313111yescharacteristic
518C2#6062011yescharacteristic
519C2#6156412yes—
520C2#6156512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
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
1. #35033C2semidirect
3. #29106C2semidirect
4. #29103C2semidirect
5. #27261C2semidirect
6. #27255C2semidirect
7. #25859C2semidirect
8. #25856C2semidirect
9. #21762C2semidirect
11. #20523C2semidirect
16. C2×#488C22semidirect
17. C2×#487C22semidirect
26. #2441C22semidirect
27. #2439C22semidirect
38. #2074C22semidirect
39. #2071C22semidirect
40. #2043C22semidirect
41. #2042C22semidirect
42. #2026C22semidirect
43. #2023C22semidirect
44. #1958C22semidirect
45. #1956C22semidirect
46. #1934C22semidirect
47. #1933C22semidirect
48. #1892C22semidirect
49. #1873C22semidirect
50. #1873C22semidirect
56. C8×C22⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#24322
#291062
#291032
#272612
#272552
#258592
#258562
#217622
#205972
#205232