Finite Group Databasebeta

← #6392694

Subgroups — #6392694

Order 512.

2088 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2978C211yesmaximal, characteristic
2#34360C211yesmaximal, characteristic
3#34201C211yesmaximal, characteristic
4#22716C214yesmaximal, complemented
5#22710C214yesmaximal, complemented
6#22663C214yesmaximal, complemented
7#22661C214yesmaximal, complemented
8#21990C212yesmaximal, complemented
9#21946C212yesmaximal, complemented
10#21944C212yesmaximal, complemented
11#21368C212yesmaximal, complemented
12#20847C212yesmaximal, complemented
13#20846C212yesmaximal, complemented
14C2×#525C2211yescharacteristic
15C2×#520C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#2978C4116yescomplemented
18C2×#436C2214yes—
19C2×#435C2214yes—
20C2×#433C2212yes—
21C2×#428C2214yes—
22C2×#428C2212yescomplemented
23C22×#107C2212yescomplemented
24C2×#408C2212yes—
25C2×#407C2212yes—
26#2376—22no—
27#2363—22no—
28#2362—22no—
29#2358—22no—
30#2354C2214yes—
31#2352C2214yes—
32#2352C2214yes—
33D4×C4⋊C4C2214yes—
34#2329—22no—
35#2325—22no—
36#2321—22no—
37#2317C2211yescharacteristic
38#2314C2212yes—
39#2311C2212yes—
40#2311C2211yescharacteristic
41C4×#123—22no—
42#2280C2211yescharacteristic
43C2×#380C2212yes—
44#2274C2211yescharacteristic
45#1949C2214yescomplemented
46#1949C2214yescomplemented
47#1948C2214yescomplemented
48#1947C2214yescomplemented
49#1854C2214yes—
50#1854C2214yes—
51#1854C2214yes—
52#1854C2214yes—
53#1854C2214yescomplemented
54#1854C2214yescomplemented
55#1853C2214yes—
56#1853C2212yes—
57#1853C2212yes—
58#1853C2214yescomplemented
59#1852C2214yes—
60#1852C2212yes—
61#1852C2212yes—
62#1852C2214yescomplemented
63#1847C2212yescomplemented
64#1844C2212yescomplemented
65#1836C2212yescomplemented
66#1835C2212yescomplemented
67#1809C2214yes—
68#1809C2214yes—
69#1809C2214yescomplemented
70#1807C2214yescomplemented
71#1776C2214yescomplemented
72C4×#107C2214yescomplemented
73#1736C2214yes—
74#1736C2212yescomplemented
75#1735C2212yescomplemented
76C22×#41C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C22×M16C2314yes—
79C22×M16C2312yes—
80C23×C8C2312yes—
81#525C2×C418yes—
82#525Q818yes—
83C2×#123D412yes—
84C2×C22⋊Q8D412yes—
85C2×#121D412yes—
86C2×#120D412yes—
87#520—44no—
88C2×C4×Q8—21no—
89C2×C4×D4—22no—
90C2×C4×D4—21no—
91C2×C4×D4—24no—
92C2×C4×D4C2314yes—
93C2×#117—22no—
94C2×#117—44no—
95C2×#117C2314yes—
96C2×#117C2312yes—
97C2×#117C2312yes—
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—22no—
109C22×C42—21no—
110#436C2×C4132yescomplemented
111#435C2×C4132yescomplemented
112#433C2×C4116yescomplemented
113#428—48no—
114#428—44no—
115C2×#107—44no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#107C2314yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2314yes—
125#408C2×C4116yescomplemented
126#407C2×C4116yescomplemented
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2314yes—
131C2×#97C2312yes—
132C2×C4×C8C2312yes—
133#391—22no—
134#391—21no—
135#390—22no—
136#390—21no—
137#390—24no—
138#389—22no—
139#389—24no—
140#388—24no—
141#388—24no—
142#388—24no—
143#388—24no—
144#388—24no—
145#387—24no—
146#387—24no—
147#387—24no—
148#386—22no—
149#386—22no—
150#386—22no—
151#386—24no—
152#386—24no—
153#385—22no—
154#385—24no—
155#385—24no—
156#384C2312yes—
157#384C2312yes—
158#382C2312yes—
159#382C2312yes—
160#381—21no—
161#380—22no—
162#380—22no—
163#380C2314yes—
164#380C2314yes—
165#380C2314yes—
166#380C2314yes—
167#380C2312yes—
168#380C2312yes—
169#379—21no—
170C4×C4⋊C4—24no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4C2314yes—
173C4×C4⋊C4C2314yes—
174C4×C22⋊C4—22no—
175C4×C22⋊C4—22no—
176C4×C22⋊C4—24no—
177C4×C22⋊C4—24no—
178C2×#95—22no—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187C25C2×D411yescharacteristic
188C2×#41—42no—
189C2×#41—42no—
190C2×#41C22×C414yes—
191C2×#41C2×Q814yes—
192C22×Q8C2×D411yescharacteristic
193C22×D4C2×D414yes—
194C22×D4C2×D412yes—
195C22×D4C2×D411yescharacteristic
196C23×C4—21no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—21no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4C22×C414yes—
205C23×C4C2×D412yes—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×Q814yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C2×M16—44no—
212C2×M16—42no—
213C2×M16—42no—
214C2×M16—44no—
215C2×M16C22×C4116yes—
216C2×M16C22×C418yes—
217C22×C8—42no—
218C22×C8—42no—
219C22×C8C22×C418yes—
220C22×C8C2414yes—
221C22×C8C2414yes—
222#123C4⋊C4116yes—
223C22⋊Q8C4⋊C4116yes—
224#121C4⋊C4116yes—
225#120C4⋊C4116yes—
226C4×Q8—44no—
227C4×D4—44no—
228C4×D4—48no—
229C4×D4—88no—
230C4×D4C22×C4116yes—
231C4×D4C2×Q8116yes—
232#117—28no—
233#117—24no—
234#117—44no—
235#117—42no—
236#117—42no—
237#117—42no—
238#117—48no—
239#117—42no—
240#117—42no—
241#117C22×C414yes—
242#117C2×Q814yes—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2411yescharacteristic
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2414yes—
265C2×C4⋊C4C2411yescharacteristic
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—84no—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D414yes—
283C2×C22⋊C4C2×D414yes—
284C2×C42—24no—
285C2×C42—21no—
286C2×C42—22no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—44no—
292C2×C42C2412yes—
293#107—28no—
294#107—28no—
295#107—44no—
296#107—44no—
297#106—44no—
298#106—44no—
299#106—44no—
300#106—44no—
301C4⋊C8C22×C4116yes—
302C22⋊C8C22×C4116yes—
303C22⋊C8C22×C418yes—
304C22⋊C8C22×C418yes—
305C22⋊C8C22×C4116yes—
306#97C22×C418yes—
307C4×C8C22×C418yes—
308#95—24no—
309#95—24no—
310#95—22no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318C24—41no—
319C24—42no—
320C24—41no—
321C24—42no—
322C24C2×C4⋊C414yes—
323C24C22×D412yes—
324C24C22×D411yescharacteristic
325C24C22×D412yes—
326#41—48no—
327#41—48no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8C2×C4⋊C414yes—
331C2×D4—41no—
332C2×D4—42no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—41no—
336C2×D4—42no—
337C2×D4C2×C4⋊C4116yes—
338C2×D4C2×C4⋊C418yes—
339C2×D4C2×C4⋊C414yes—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—21no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4C2×C4⋊C418yes—
376C22×C4C2×C4⋊C414yes—
377C22×C4C23×C412yes—
378C22×C4C23×C414yes—
379C22×C4C23×C414yes—
380C22×C4C22×D414yes—
381C22×C4C22×D412yes—
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×Q812yes—
385C22×C4C22×Q814yes—
386C22×C4C22×Q814yes—
387C22×C4C2511yescharacteristic
388M16—416no—
389M16—48no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—48no—
395C2×C8C23×C418yes—
396C2×C8C23×C418yes—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C4⋊C4—84no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—82no—
413C4⋊C4—84no—
414C4⋊C4C2×C4⋊C418yes—
415C4⋊C4C2×C4⋊C4116yes—
416C4⋊C4H(5,𝔽2)14yes—
417C22⋊C4—48no—
418C22⋊C4—44no—
419C22⋊C4—48no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—82no—
423C22⋊C4—82no—
424C22⋊C4—84no—
425C22⋊C4—84no—
426C22⋊C4—82no—
427C22⋊C4—84no—
428C22⋊C4C2×C4⋊C418yes—
429C22⋊C4C2×C4⋊C4116yes—
430C22⋊C4C2×C4⋊C4116yes—
431C42—48no—
432C42—44no—
433C42—42no—
434C42—42no—
435C42—84no—
436C42—82no—
437C42—84no—
438C42C23×C414yes—
439C42C22×Q814yes—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—22no—
444C23—24no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—21no—
450C23—22no—
451C23—21no—
452C23—22no—
453C23—22no—
454C23—44no—
455C23—48no—
456C23—84no—
457C23—82no—
458C23—82no—
459C23C22×C4⋊C412yes—
460C23C22×C4⋊C414yes—
461C23C22×C4⋊C414yes—
462C23C23×D411yescharacteristic
463Q8—44no—
464D4—416no—
465D4—44no—
466D4—48no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—28no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—44no—
485C2×C4—48no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—84no—
503C2×C4—84no—
504C2×C4—164no—
505C2×C4—164no—
506C2×C4—164no—
507C2×C4—164no—
508C2×C4C22×C4⋊C418yes—
509C2×C4C22×C4⋊C414yes—
510C2×C4C22×C4⋊C414yes—
511C2×C4C22×C4⋊C412yes—
512C2×C4C24×C411yescharacteristic
513C2×C4C23×Q811yescharacteristic
514C2×C4C2×H(5,𝔽2)12yes—
515C2×C4#58512yes—
516C8—28no—
517C8—28no—
518C22—22no—
519C22—24no—
520C22—24no—
521C22—24no—
522C22—28no—
523C22—24no—
524C22—24no—
525C22—42no—
526C22—44no—
527C22—42no—
528C22—42no—
529C22—41no—
530C22—42no—
531C22—42no—
532C22—41no—
533C22—42no—
534C22—42no—
535C22—88no—
536C22C23×C4⋊C411yescharacteristic
537C22#348411yescharacteristic
538C22#348511yescharacteristic
539C22#358512yes—
540C22#358612yes—
541C4—24no—
542C4—22no—
543C4—24no—
544C4—28no—
545C4—44no—
546C4—82no—
547C4—84no—
548C4—84no—
549C4—81no—
550C4—81no—
551C4—82no—
552C4C4×H(5,𝔽2)12yes—
553C4#350112yes—
554C2—44no—
555C2—42no—
556C2—44no—
557C2#3424811yescharacteristic
558C2#6010912yes—
559C2#6164811yescharacteristic
560C2#6165211yescharacteristic
561C2#6177612yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #22716C2semidirect
5. #22710C2semidirect
6. #22663C2semidirect
7. #22661C2semidirect
8. #21990C2semidirect
9. #21946C2semidirect
10. #21944C2semidirect
11. #21368C2semidirect
12. #20847C2semidirect
13. #20846C2semidirect
17. #2978C4semidirect
22. C2×#428C22semidirect
23. C22×#107C22semidirect
45. #1949C22semidirect
46. #1949C22semidirect
47. #1948C22semidirect
48. #1947C22semidirect
53. #1854C22semidirect
54. #1854C22semidirect
58. #1853C22semidirect
62. #1852C22semidirect
63. #1847C22semidirect
64. #1844C22semidirect
65. #1836C22semidirect
66. #1835C22semidirect
69. #1809C22semidirect
70. #1807C22semidirect
71. #1776C22semidirect
72. C4×#107C22semidirect
74. #1736C22semidirect
75. #1735C22semidirect
110. #436C2×C4semidirect
111. #435C2×C4semidirect
112. #433C2×C4semidirect
125. #408C2×C4semidirect
126. #407C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29782
#343602
#342012
#227162
#227102
#226632
#226612
#219902
#219462
#219442
#213682
#208472
#208462