Finite Group Databasebeta

← #6378566

Subgroups — #6378566

Order 512.

1669 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
1C4×#540C211yesmaximal, characteristic
2#22359C214yesmaximal, complemented
3#22331C214yesmaximal, complemented
4#22299C211yesmaximal, characteristic, complemented
5#22264C211yesmaximal, characteristic, complemented
6#22137C212yesmaximal, complemented
7#22012C212yesmaximal, complemented
8#21926C211yesmaximal, characteristic, complemented
9#21681C211yesmaximal, characteristic, complemented
10C8×#125C211yesmaximal, characteristic, complemented
11#21565C214yesmaximal, complemented
12#21488C214yesmaximal, complemented
13#21488C212yesmaximal, complemented
14#20494C212yesmaximal, complemented
15#20472C211yesmaximal, characteristic, complemented
16C2×#540C411yescharacteristic
17C2×#436C2212yes—
18C2×#436C2212yes—
19C2×#436C2214yes—
20C2×#436C2214yes—
21C2×#431C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23C2×C4×M16C2214yes—
24#2335C414yes—
25#2333C414yes—
26#2330C411yescharacteristic
27#2328C411yescharacteristic
28#2324C412yes—
29#2319C412yes—
30#2314C411yescharacteristic
31C4×#127C2211yescharacteristic
32C4×#125C2211yescharacteristic
33C4×#123C2214yes—
34C4×#121C2214yes—
35C4×#121C2212yes—
36C42×D4C2212yes—
37C4×#117C2211yescharacteristic
38#1954C2214yes—
39#1954C2214yes—
40#1954C2212yes—
41#1954C2212yes—
42#1939C2214yes—
43#1939C2212yes—
44#1939C2212yes—
45#1924C2214yes—
46#1924C2214yes—
47#1924C2214yes—
48#1924C2214yes—
49#1923C2214yes—
50#1917C2214yes—
51#1916C2214yes—
52#1916C2214yes—
53#1870C2214yes—
54#1870C2214yes—
55#1870C2214yes—
56#1870C2214yes—
57#1851C2214yes—
58#1842C2211yescharacteristic
59#1831C2211yescharacteristic
60#1817C2214yes—
61#1797C2214yes—
62#1797C2212yes—
63#1771C2212yes—
64#1771C2212yes—
65C8×C4⋊C4C2212yes—
66C8×C4⋊C4C2212yes—
67C8×C4⋊C4C2214yes—
68#1767C2212yes—
69#1753C2214yes—
70#1753C2214yes—
71#1753C2212yes—
72#1753C2214yes—
73#1753C2212yes—
74#1749C2214yes—
75#1744C2211yescharacteristic
76#1744C2211yescharacteristic
77C4×#97C2211yescharacteristic
78C42×C8C2211yescharacteristic
79C22×M16C2311yescharacteristic
80C22×M16C2314yes—
81C22×M16C2314yes—
82C22×M16C2311yescharacteristic
83#540—28no—
84C2×#127C2×C411yescharacteristic
85C2×#125C2×C411yescharacteristic
86C2×#123C2×C414yes—
87C2×#121C2×C412yes—
88C2×#121C2×C414yes—
89C2×C4×D4C2×C412yes—
90C2×C4×D4C2×C412yes—
91C2×C4×D4C2312yes—
92C2×C4×D4C2314yes—
93C2×C4×D4C2314yes—
94C2×#117C2×C411yescharacteristic
95C2×#117C2×C411yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2314yes—
98#436—28no—
99#436—28no—
100#436—48no—
101#436—48no—
102#431—24no—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2314yes—
107C2×C4⋊C8C2314yes—
108C2×C4⋊C8C2314yes—
109C2×C22⋊C8C2312yes—
110C2×C22⋊C8C2314yes—
111C2×C22⋊C8C2314yes—
112C2×C22⋊C8C2314yes—
113C2×C22⋊C8C2314yes—
114C2×C22⋊C8C2314yes—
115C2×C22⋊C8C2314yes—
116C2×C22⋊C8C2314yes—
117C4×M16—24no—
118C4×M16—48no—
119C2×#97C2312yes—
120C2×#97C2311yescharacteristic
121C2×#97C2311yescharacteristic
122C2×#97C2314yes—
123C2×#97C2312yes—
124C2×C4×C8C2314yes—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2312yes—
128C2×C4×C8C2314yes—
129C2×C4×C8C2312yes—
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132#390C2×C412yes—
133#390C2×C414yes—
134#390C2×C412yes—
135#390C2×C414yes—
136#389C2×C412yes—
137#389C2×C412yes—
138#389C2×C414yes—
139#388C2×C414yes—
140#388C2×C414yes—
141#388C2×C414yes—
142#388C2×C414yes—
143#388C2×C414yes—
144#387C2×C414yes—
145#387C2×C414yes—
146#387C2×C414yes—
147#386C2×C414yes—
148#386C2×C414yes—
149#386C2×C414yes—
150#386C2×C414yes—
151#385C2×C414yes—
152#384C2×C411yescharacteristic
153#382C2×C411yescharacteristic
154#381C2×C414yes—
155#380C2×C412yes—
156#380C2×C414yes—
157C4×C4⋊C4C2×C412yes—
158C4×C4⋊C4C2×C412yes—
159C4×C4⋊C4C2×C412yes—
160C4×C4⋊C4C2314yes—
161C4×C4⋊C4C2312yes—
162C4×C4⋊C4C2312yes—
163C4×C22⋊C4C2×C412yes—
164C4×C22⋊C4C2×C414yes—
165C4×C22⋊C4C2×C414yes—
166C4×C22⋊C4C2314yes—
167C4×C22⋊C4C2312yes—
168C4×C22⋊C4C2314yes—
169#376C2×C411yescharacteristic
170#376C2×C411yescharacteristic
171C43C2311yescharacteristic
172C43C2311yescharacteristic
173#336C2312yes—
174#336C2314yes—
175#336C2314yes—
176#336C2312yes—
177#336C2314yes—
178#336C2314yes—
179#336C2314yes—
180#336C2314yes—
181#336C2312yes—
182#336C2312yes—
183#336C2314yes—
184C22×D4C22×C412yes—
185C22×D4C22×C414yes—
186C22×D4C22×C414yes—
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C414yes—
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C414yes—
191C23×C4C2411yescharacteristic
192C23×C4C2414yes—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—28no—
196C2×M16—28no—
197C2×M16—28no—
198C2×M16—24no—
199C2×M16—28no—
200C2×M16—42no—
201C2×M16—44no—
202C2×M16—44no—
203C22×C8C2411yescharacteristic
204C22×C8C2414yes—
205C22×C8C2411yescharacteristic
206C22×C8C2414yes—
207C22×C8C2412yes—
208C22×C8C2412yes—
209C22×C8C2411yescharacteristic
210C22×C8C2411yescharacteristic
211#127—24no—
212#125M1618yescomplemented
213#123—216no—
214#121—216no—
215#121—28no—
216C4×D4—28no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—44no—
220C4×D4—44no—
221C4×D4—44no—
222C4×D4—44no—
223C4×D4—44no—
224#117—24no—
225#117—42no—
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C414yes—
231C2×C4⋊C4C22×C414yes—
232C2×C4⋊C4C22×C414yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C414yes—
235C2×C22⋊C4C22×C412yes—
236C2×C22⋊C4C22×C414yes—
237C2×C22⋊C4C22×C414yes—
238C2×C22⋊C4C22×C414yes—
239C2×C22⋊C4C22×C412yes—
240C2×C22⋊C4C22×C414yes—
241C2×C22⋊C4C22×C414yes—
242C2×C22⋊C4C22×C414yes—
243C2×C22⋊C4C22×C414yes—
244C2×C22⋊C4C22×C414yes—
245C2×C22⋊C4C22×C414yes—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C412yes—
256C2×C42C22×C412yes—
257C2×C42C22×C414yes—
258C2×C42C22×C414yes—
259C2×C42C22×C411yescharacteristic
260C2×C42C2412yes—
261C2×C42C2414yes—
262C2×C42C2412yes—
263C2×C42C2411yescharacteristic
264C2×C42C2411yescharacteristic
265C4⋊C8—24no—
266C4⋊C8—24no—
267C4⋊C8—24no—
268C4⋊C8—44no—
269C4⋊C8—44no—
270C4⋊C8—44no—
271C22⋊C8—24no—
272C22⋊C8—28no—
273C22⋊C8—28no—
274C22⋊C8—44no—
275C22⋊C8—44no—
276C22⋊C8—44no—
277C22⋊C8—44no—
278C22⋊C8—44no—
279#97—22no—
280#97—22no—
281#97—24no—
282#97—42no—
283#97—44no—
284C4×C8—28no—
285C4×C8—24no—
286C4×C8—22no—
287C4×C8—22no—
288C4×C8—42no—
289C4×C8—44no—
290C4×C8#4114yes—
291C4×C8#4114yes—
292#95C22×C412yes—
293#95C22×C412yes—
294#95C22×C412yes—
295#95C22×C412yes—
296#95C22×C414yes—
297#95C22×C414yes—
298#95C22×C414yes—
299#95C22×C414yes—
300#95C22×C414yes—
301#95C22×C414yes—
302#95C22×C414yes—
303C24C23×C414yes—
304C24C23×C411yescharacteristic
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—28no—
308C2×D4—28no—
309C2×D4—24no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—28no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4C23×C411yescharacteristic
336C22×C4C23×C414yes—
337C22×C4C23×C412yes—
338C22×C4C23×C412yes—
339C22×C4C23×C412yes—
340C22×C4C23×C411yescharacteristic
341C22×C4C23×C414yes—
342C22×C4C23×C411yescharacteristic
343C22×C4C23×C411yescharacteristic
344C22×C4C23×C412yes—
345C22×C4C23×C414yes—
346C22×C4C23×C411yescharacteristic
347C22×C4C2511yescharacteristic
348M16—44no—
349M16—416no—
350M16—44no—
351M16—416no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—22no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—22no—
362C2×C8—24no—
363C2×C8—22no—
364C2×C8—24no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—24no—
368C2×C8C2×#4112yes—
369C2×C8C2×#4114yes—
370C2×C8C2×#4112yes—
371C2×C8H(5,𝔽2)14yes—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4C2×M1618yes—
380C4⋊C4C2×M16116yes—
381C22⋊C4—28no—
382C22⋊C4—24no—
383C22⋊C4—28no—
384C22⋊C4—28no—
385C22⋊C4—44no—
386C22⋊C4—42no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—21no—
398C42—21no—
399C42—24no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—24no—
405C42—44no—
406C42—44no—
407C42—42no—
408C42—42no—
409C42C2×M1614yes—
410C42C2×#4112yes—
411C42C2×#4112yes—
412C23—24no—
413C23—21no—
414C23—22no—
415C23—24no—
416C23—24no—
417C23—21no—
418C23—24no—
419C23—21no—
420C23—24no—
421C23—24no—
422C23—22no—
423C23—24no—
424C23C24×C411yescharacteristic
425D4—48no—
426D4—88no—
427D4—88no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—88no—
465C2×C4—88no—
466C2×C4C4×#4112yes—
467C2×C4C4×#4112yes—
468C2×C4C22×M1614yes—
469C2×C4C22×M1618yes—
470C2×C4C22×M1612yes—
471C2×C4#56812yes—
472C2×C4C22×#4111yescharacteristic
473C2×C4C22×#4111yescharacteristic
474C2×C4C2×H(5,𝔽2)12yes—
475C2×C4#58512yes—
476C8—22no—
477C8—24no—
478C8—41no—
479C8—44no—
480C8—41no—
481C8—42no—
482C8—44no—
483C8#54014yescomplemented
484C22—41no—
485C22—44no—
486C22—41no—
487C22—44no—
488C22—41no—
489C22—44no—
490C22—44no—
491C22—42no—
492C22—44no—
493C22—44no—
494C22—42no—
495C22—44no—
496C22C2×C4×#4111yescharacteristic
497C22C4×H(5,𝔽2)12yes—
498C22C23×M1611yescharacteristic
499C22C2×#56811yescharacteristic
500C22#357312yes—
501C4—22no—
502C4—28no—
503C4—24no—
504C4—41no—
505C4—41no—
506C4—44no—
507C4—41no—
508C4—42no—
509C4—42no—
510C4—44no—
511C4—42no—
512C4#297114yes—
513C4C2×#54011yescharacteristic
514C4#345511yescharacteristic
515C4#346012yes—
516C2—81no—
517C2—84no—
518C2C4×#54011yescharacteristic
519C2C2×#297111yescharacteristic
520C2#6020911yescharacteristic
521C2#6032512yes—
522C2#6032612yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC2×#540C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22359C2semidirect
3. #22331C2semidirect
4. #22299C2semidirect
5. #22264C2semidirect
6. #22137C2semidirect
7. #22012C2semidirect
8. #21926C2semidirect
9. #21681C2semidirect
10. C8×#125C2semidirect
11. #21565C2semidirect
12. #21488C2semidirect
13. #21488C2semidirect
14. #20494C2semidirect
15. #20472C2semidirect
212. #125M16semidirect
483. C8#540semidirect

Maximal subgroups

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

subgroupindex
C4×#5402
#223592
#223312
#222992
#222642
#221372
#220122
#219262
#216812
C8×#1252
#215652
#214882
#204942
#204722