Finite Group Databasebeta

← #6595142

Subgroups — #6595142

Order 512.

1706 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic, complemented
2#36187C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#33048C212yesmaximal, complemented
5#29206C211yesmaximal, characteristic, complemented
6#28508C211yesmaximal, characteristic, complemented
7#28092C218yesmaximal, complemented
8#27890C212yesmaximal, complemented
9#27878C212yesmaximal, complemented
10#25347C214yesmaximal, complemented
11#23066C214yesmaximal, complemented
12C4×#486C211yesmaximal, characteristic, complemented
13#20824C212yesmaximal, complemented
14C4×#436C211yesmaximal, characteristic, complemented
15C2×#556—21no—
16C2×#555—21no—
17C2×#486C2211yescharacteristic
18C2×#474C2214yes—
19C2×#436C2212yes—
20C2×C4×M16C2212yes—
21#2868C2211yescharacteristic
22#2868C2211yescharacteristic
23#2857C2212yes—
24#2856C2212yes—
25#2568—21no—
26#2563—21no—
27#2547—21no—
28#2547C2212yes—
29#2537—21no—
30#2537C2212yes—
31#2500—22no—
32#2499—22no—
33#2496—22no—
34#2496C2214yes—
35#2491—22no—
36#2491C2214yes—
37#2365—21no—
38#2365C2212yes—
39#2354—21no—
40#2354C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×#127C2211yescharacteristic
43#2285C2212yes—
44#2285C2212yes—
45C42×D4C2211yescharacteristic
46#2093C2214yes—
47#2076C2218yes—
48#2058C2214yes—
49#2034C2218yes—
50#1990C2214yes—
51#1990C2214yes—
52#1990C2212yes—
53#1990C2211yescharacteristic
54#1957C2212yes—
55#1957C2211yescharacteristic
56#1948C2214yes—
57#1936C2218yes—
58#1934C2214yes—
59#1926C2218yes—
60#1921C2218yes—
61#1895C2218yes—
62#1845C2212yes—
63#1844C2212yes—
64#1834C2212yes—
65#1834C2212yes—
66#1834C2211yescharacteristic
67#1809C2218yes—
68#1776C2218yes—
69#1770C2212yes—
70C4×C4⋊C8C2211yescharacteristic
71C4×#102C2214yes—
72#1753C2214yes—
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75C22×M16C2314yes—
76#556—48no—
77#555—48no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×#127C2311yescharacteristic
86C2×#126—24no—
87C2×#124—22no—
88C2×#124—24no—
89C2×C22⋊Q8—22no—
90C2×#121—22no—
91C2×#121—22no—
92C2×#121—22no—
93C2×#121C2314yes—
94C2×C4×Q8—22no—
95C2×C4×Q8—21no—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×C4×D4D411yescharacteristic
99C2×C4×D4D411yescharacteristic
100C2×C4×D4C2312yes—
101C2×C4×D4C2311yescharacteristic
102C2×#117—22no—
103C2×#117—22no—
104C2×#117—22no—
105C22×C42C2312yes—
106#486—24no—
107#474—216no—
108#436—28no—
109C2×#107C2318yes—
110C2×#106C2318yes—
111C2×C4⋊C8C2312yes—
112C2×#102C2318yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×C22⋊C8C2314yes—
116C4×M16—28no—
117C2×#97C2314yes—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2312yes—
120C2×C4×C8C2312yes—
121C2×C4×C8C2312yes—
122C2×C4×C8C2311yescharacteristic
123#400—24no—
124#397—24no—
125#397—24no—
126#394—24no—
127#394—24no—
128#393—24no—
129#391—22no—
130#391—22no—
131#391—21no—
132#391—22no—
133#390—21no—
134#390—21no—
135#390—22no—
136#390—22no—
137#390C2312yes—
138#390C2314yes—
139#389—21no—
140#389—22no—
141#389C2312yes—
142#389C2314yes—
143#388—24no—
144#388—24no—
145#387—24no—
146#387—24no—
147#384—21no—
148#384—21no—
149#384—21no—
150#384—21no—
151#384C2311yescharacteristic
152#384C2311yescharacteristic
153#384C2312yes—
154#384C2312yes—
155#384C2312yes—
156#384C2312yes—
157#382—22no—
158#382—22no—
159#380—22no—
160#380—22no—
161#380—22no—
162#380—22no—
163#380—22no—
164#380C2314yes—
165#380C2314yes—
166#380C2314yes—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4D411yescharacteristic
171C4×C4⋊C4D411yescharacteristic
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2314yes—
175C4×C4⋊C4C2311yescharacteristic
176C4×C22⋊C4D412yes—
177C4×C22⋊C4D412yes—
178C4×C22⋊C4C2312yes—
179C43C2311yescharacteristic
180#340C2318yes—
181#340C2314yes—
182#340C2318yes—
183#340C2314yes—
184#340C2318yes—
185#336C2312yes—
186#336C2314yes—
187C22×Q8—21no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×D4—24no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4C2×D411yescharacteristic
194C22×D4C2412yes—
195C23×C4—22no—
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2412yes—
199C2×M16—28no—
200C2×M16—28no—
201C2×M16—44no—
202C22×C8C2412yes—
203C22×C8C2412yes—
204C22×C8C2414yes—
205C4⋊Q8—28no—
206C4⋊Q8—22no—
207C4⋊Q8—22no—
208C4⋊Q8—44no—
209C4⋊Q8—44no—
210C4⋊Q8—44no—
211C4⋊Q8—44no—
212C4⋊Q8—82no—
213#127—22no—
214#127—22no—
215#126—416no—
216#124—416no—
217#124—48no—
218C22⋊Q8—48no—
219#121—48no—
220#121—48no—
221#121—48no—
222#121—84no—
223C4×Q8—48no—
224C4×Q8—44no—
225C4×D4—24no—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—28no—
229C4×D4—44no—
230C4×D4—44no—
231#117—48no—
232#117—48no—
233#117—48no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2414yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2412yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4C2×D412yes—
269C2×C22⋊C4C2×D412yes—
270C2×C22⋊C4C2×D412yes—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2411yescharacteristic
283C2×C42C2412yes—
284C2×C42C2411yescharacteristic
285C2×C42C2411yescharacteristic
286#107—216no—
287#106—48no—
288C4⋊C8#4118yes—
289#102—216no—
290#102—28no—
291#102—44no—
292C22⋊C8—28no—
293#97—28no—
294C4×C8—24no—
295C4×C8—24no—
296C4×C8—22no—
297C4×C8—42no—
298C4×C8#4114yes—
299#95—24no—
300#95—24no—
301#95—22no—
302#95—24no—
303#95—24no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95C2×D412yes—
308#95C2×D412yes—
309#95C2×D412yes—
310C24—22no—
311C24C22×D412yes—
312C2×Q8—44no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—42no—
319C2×Q8—82no—
320C2×Q8—82no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—24no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—24no—
328C2×D4—48no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—84no—
336C2×D4—84no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4C22×D412yes—
356C22×C4C22×D412yes—
357C22×C4C22×D412yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C2511yescharacteristic
365M16—416no—
366C2×C8—24no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—24no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8C2×#4114yes—
374C2×C8C2×#4118yes—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—28no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—28no—
384C4⋊C4—48no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—48no—
405C22⋊C4—48no—
406C22⋊C4—48no—
407C22⋊C4—48no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—84no—
412C22⋊C4—84no—
413C42—24no—
414C42—24no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—48no—
421C42—48no—
422C42—44no—
423C42AGL(1,ℤ/8)14yes—
424C42AGL(1,ℤ/8)14yes—
425C42C2×#4114yes—
426C42C2×#4112yes—
427C422−1+412yes—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23C23×D411yescharacteristic
443Q8—84no—
444Q8—88no—
445Q8—164no—
446D4—44no—
447D4—44no—
448D4—44no—
449D4—84no—
450D4—84no—
451D4—168no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—48no—
468C2×C4—48no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—88no—
487C2×C4#53212yes—
488C2×C4#53212yes—
489C2×C4#54712yes—
490C2×C4#54712yes—
491C2×C4C2×AGL(1,ℤ/8)12yes—
492C2×C4C2×AGL(1,ℤ/8)12yes—
493C2×C4C2×AGL(1,ℤ/8)12yes—
494C2×C4C2×AGL(1,ℤ/8)12yes—
495C2×C4C2×AGL(1,ℤ/8)12yes—
496C2×C4C2×AGL(1,ℤ/8)12yes—
497C2×C4#57712yes—
498C2×C4#57812yes—
499C2×C4C22×#4111yescharacteristic
500C2×C4C22×#4112yes—
501C2×C4C2×2−1+411yescharacteristic
502C2×C4#58512yes—
503C8—28no—
504C8—24no—
505C8—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22C2×#53211yescharacteristic
521C22C2×#54711yescharacteristic
522C22#350011yescharacteristic
523C22C22×AGL(1,ℤ/8)11yescharacteristic
524C22C22×AGL(1,ℤ/8)11yescharacteristic
525C22C2×#57711yescharacteristic
526C22C2×#57811yescharacteristic
527C4—22no—
528C4—22no—
529C4—22no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—82no—
534C4—84no—
535C4—82no—
536C4—82no—
537C4—84no—
538C4#318414yes—
539C4#335314yes—
540C4#335414yes—
541C4#347512yes—
542C4#351812yes—
543C2—82no—
544C2—162no—
545C2#3618511yescharacteristic
546C2C2×#318411yescharacteristic
547C2#6072311yescharacteristic
548C2C2×#335311yescharacteristic
549C2C2×#335411yescharacteristic
550C2#6253811yescharacteristic
551C2#6253911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36188C2semidirect
3. #33289C2semidirect
4. #33048C2semidirect
5. #29206C2semidirect
6. #28508C2semidirect
7. #28092C2semidirect
8. #27890C2semidirect
9. #27878C2semidirect
10. #25347C2semidirect
11. #23066C2semidirect
12. C4×#486C2semidirect
13. #20824C2semidirect
14. C4×#436C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#361872
#332892
#330482
#292062
#285082
#280922
#278902
#278782
#253472
#230662
C4×#4862
#208242
C4×#4362