Finite Group Databasebeta

← #6468462

Subgroups — #6468462

Order 512.

1654 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34280C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#29294C212yesmaximal, complemented
4#29280C212yesmaximal, complemented
5#29254C212yesmaximal, complemented
6#29247C214yesmaximal, complemented
7#28292C212yesmaximal, complemented
8#27677C212yesmaximal, complemented
9#24304C212yesmaximal, complemented
10#23781C212yesmaximal, complemented
11#23774C212yesmaximal, complemented
12#21620C214yesmaximal, complemented
13#21399C212yesmaximal
14#21368C212yesmaximal, complemented
15#20813C211yesmaximal, characteristic, complemented
16C2×#540C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×#478C2214yes—
19C2×#433C2212yes—
20#2374—22no—
21#2364—22no—
22#2363—22no—
23#2360—22no—
24#2360—22no—
25#2359—22no—
26#2355—22no—
27Q8×C4⋊C4—22no—
28#2330C2211yescharacteristic
29#2330C2212yes—
30#2330C2212yes—
31#2328C2211yescharacteristic
32#2324C2212yes—
33#2317C2212yes—
34#2314C2211yescharacteristic
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×#127C2211yescharacteristic
38C4×#125C2211yescharacteristic
39C4×#121C2212yes—
40#2293C2212yes—
41C2×#384C2211yescharacteristic
42#2275C2212yes—
43#2274C2212yes—
44C4×#117C2211yescharacteristic
45#2099C2212yes—
46#2099C2212yes—
47#2093C2212yes—
48#2093C2212yes—
49#2087C2214yes—
50#2086C2214yes—
51#2067C2214yes—
52#2065C2214yes—
53#2058C2212yes—
54#2057C2212yes—
55#2023C2212yes—
56#2023C2212yes—
57#1989C2214yes—
58#1989C2214yes—
59#1982C2214yes—
60#1949C2214yes—
61#1947C2214yes—
62#1947C2214yes—
63#1921C2214yes—
64#1920C2214yes—
65#1875C2214yes—
66#1874C2214yes—
67#1862C2214yes—
68#1862C2214yes—
69#1854C2214yes—
70#1852C2214yes—
71#1843C2212yes—
72#1831C2212yes—
73#1802C2214yes—
74#1798C2212yes—
75#1797C2212yes—
76#1776C2214yes—
77C4×#107C2214yes—
78#1768C2212yes—
79#1767C2211yescharacteristic
80C4×C4⋊C8C2211yescharacteristic
81#1762C2214yes—
82C4×#102C2214yes—
83#540—24no—
84#540Q818yescomplemented
85#533—28no—
86C2×C4⋊Q8C2312yes—
87C2×#127C2311yescharacteristic
88C2×#125C2311yescharacteristic
89C2×#125C2312yes—
90C2×#123—22no—
91C2×C22⋊Q8—22no—
92C2×#121—22no—
93C2×#121C2312yes—
94C2×C4×Q8—22no—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4C2312yes—
100C2×C4×D4C2312yes—
101C2×C4×D4C2312yes—
102C2×#117—22no—
103C2×#117C2312yes—
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C22×C4⋊C4—22no—
107C22×C4⋊C4C2312yes—
108C22×C42C2311yescharacteristic
109#478—216no—
110#433—28no—
111C2×#107C2314yes—
112C2×#107C2314yes—
113C2×#107C2314yes—
114C2×#106C2314yes—
115C2×C4⋊C8C2314yes—
116C2×C4⋊C8C2314yes—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2312yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×C22⋊C8C2314yes—
124C2×#97C2312yes—
125C2×C4×C8C2312yes—
126#391—22no—
127#391—22no—
128#390—22no—
129#390—22no—
130#390C2312yes—
131#390C2312yes—
132#389—22no—
133#389—22no—
134#389—24no—
135#389—24no—
136#389—22no—
137#389C2312yes—
138#389C2314yes—
139#389C2314yes—
140#389C2312yes—
141#388—22no—
142#388—22no—
143#388—22no—
144#388—22no—
145#388—22no—
146#387—22no—
147#387—22no—
148#387—24no—
149#387—24no—
150#387—22no—
151#386—22no—
152#386—22no—
153#386—22no—
154#386—22no—
155#385—22no—
156#385—22no—
157#385—22no—
158#384—22no—
159#384—22no—
160#384C2311yescharacteristic
161#384C2312yes—
162#384C2312yes—
163#382—22no—
164#382C2311yescharacteristic
165#380C2312yes—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4D412yes—
171C4×C4⋊C4D412yes—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2314yes—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2312yes—
177C4×C22⋊C4—24no—
178C4×C22⋊C4—22no—
179C4×C22⋊C4—22no—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—22no—
183C4×C22⋊C4D412yes—
184C4×C22⋊C4D412yes—
185#376C2311yescharacteristic
186#376C2311yescharacteristic
187C2×#95—22no—
188C43C2311yescharacteristic
189C43C2311yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2312yes—
199#336C2312yes—
200#336C2314yes—
201C22×Q8—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4C2412yes—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4C2×D412yes—
210C23×C4C2411yescharacteristic
211C22×C8C2414yes—
212C22×C8C2414yes—
213C4⋊Q8—28no—
214#127—22no—
215#127C2×Q814yes—
216#125—24no—
217#125—22no—
218#125C2×Q814yes—
219#125#4118yes—
220#123—216no—
221C22⋊Q8—216no—
222#121—24no—
223#121—216no—
224#121C2×Q818yes—
225C4×Q8—216no—
226C4×Q8—84no—
227C4×Q8—84no—
228C4×D4—24no—
229C4×D4—28no—
230C4×D4—44no—
231C4×D4—84no—
232C4×D4—84no—
233C4×D4C2×Q818yes—
234#117—28no—
235#117—42no—
236#117—84no—
237#117C2×Q814yes—
238#117#4114yes—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—84no—
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2414yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4C2×D411yescharacteristic
272C2×C22⋊C4C2×D411yescharacteristic
273C2×C22⋊C4C2×D412yes—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292#107—28no—
293#107—44no—
294#107—44no—
295#106—28no—
296C4⋊C8—28no—
297C4⋊C8—28no—
298C4⋊C8—42no—
299C4⋊C8—42no—
300#102—28no—
301#102—28no—
302#102—44no—
303#102—44no—
304C22⋊C8—28no—
305#97—24no—
306C4×C8—24no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—22no—
314#95—22no—
315#95—24no—
316#95—24no—
317#95—22no—
318#95C2×D412yes—
319#95C2×D412yes—
320#95C2×D412yes—
321#95C2×D412yes—
322C24—22no—
323C24C22×D411yescharacteristic
324C2×Q8—28no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×D4—28no—
328C2×D4—28no—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—24no—
332C2×D4—82no—
333C2×D4—82no—
334C2×D4—82no—
335C2×D4—82no—
336C2×D4C22×Q814yes—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—28no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—84no—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×Q812yes—
368C22×C4C2×#4112yes—
369C22×C4C2511yescharacteristic
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C4⋊C4—28no—
377C4⋊C4—28no—
378C4⋊C4—216no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—84no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C4⋊C4C22⋊Q818yes—
397C4⋊C4C22×Q814yes—
398C4⋊C4C2×#4118yes—
399C22⋊C4—28no—
400C22⋊C4—28no—
401C22⋊C4—28no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C22⋊C4C22⋊Q818yes—
412C42—28no—
413C42—24no—
414C42—24no—
415C42—24no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—42no—
420C42—42no—
421C42—82no—
422C42—82no—
423C42—82no—
424C42—84no—
425C42—84no—
426C42C22×Q812yes—
427C42C22×Q812yes—
428C42C2×#4112yes—
429C42C2×#4112yes—
430C42C2×#4114yes—
431C23—22no—
432C23—21no—
433C23—21no—
434C23—22no—
435C23—24no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23C2×C22⋊Q812yes—
443C23C23×D411yescharacteristic
444Q8—416no—
445D4—28no—
446D4—44no—
447D4—88no—
448D4—88no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—48no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—88no—
486C2×C4—164no—
487C2×C4—164no—
488C2×C4—164no—
489C2×C4C2×C22⋊Q814yes—
490C2×C4C2×C22⋊Q814yes—
491C2×C4C2×C22⋊Q812yes—
492C2×C4#52514yes—
493C2×C4#57614yes—
494C2×C4C23×Q811yescharacteristic
495C2×C4C22×#4112yes—
496C2×C4C22×#4111yescharacteristic
497C2×C4#58512yes—
498C8—44no—
499C8—44no—
500C22—24no—
501C22—22no—
502C22—41no—
503C22—41no—
504C22—42no—
505C22—44no—
506C22—44no—
507C22—44no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22C22×C22⋊Q811yescharacteristic
513C22C2×#52511yescharacteristic
514C22#344911yescharacteristic
515C22C2×#57612yes—
516C22#358612yes—
517C4—24no—
518C4—24no—
519C4—22no—
520C4—24no—
521C4—24no—
522C4—44no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—48no—
528C4—82no—
529C4—84no—
530C4—82no—
531C4Q8×#4112yes—
532C4#348012yes—
533C2—42no—
534C2—84no—
535C2#3495211yescharacteristic
536C2#6049412yes—
537C2#6058412yes—
538C2#6101111yescharacteristic
539C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#540C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34214C2semidirect
3. #29294C2semidirect
4. #29280C2semidirect
5. #29254C2semidirect
6. #29247C2semidirect
7. #28292C2semidirect
8. #27677C2semidirect
9. #24304C2semidirect
10. #23781C2semidirect
11. #23774C2semidirect
12. #21620C2semidirect
14. #21368C2semidirect
15. #20813C2semidirect
84. #540Q8semidirect

Maximal subgroups

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

subgroupindex
#342802
#342142
#292942
#292802
#292542
#292472
#282922
#276772
#243042
#237812
#237742
#216202
#213992
#213682
#208132