Finite Group Databasebeta

← #6433110

Subgroups — #6433110

Order 512.

2438 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic, complemented
2#34044C211yesmaximal, characteristic
3#29961C218yesmaximal, complemented
4#29960C214yesmaximal, complemented
5#29959C214yesmaximal, complemented
6#23733C212yesmaximal, complemented
7#23617C214yesmaximal, complemented
8#23452C212yesmaximal, complemented
9#20846C212yesmaximal, complemented
10#20627C211yesmaximal, characteristic, complemented
11#20601C212yesmaximal
12C23×#41C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic, complemented
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#428C2214yescomplemented
17C2×#423C2212yescomplemented
18C2×#417C2214yes—
19C2×#417C2214yes—
20C2×#407C2212yes—
21#2628—22no—
22#2628—22no—
23#2628C2214yes—
24#2628C2214yescomplemented
25C2×#400—21no—
26C2×#399—21no—
27C2×#395—21no—
28C2×#393—21no—
29#2311—21no—
30#2311—21no—
31#2311C2214yes—
32#2310C2212yes—
33#2309C2212yes—
34#2308—21no—
35#2308—21no—
36#2297—22no—
37#2297C2212yes—
38#2297C2212yes—
39C2×#389C2212yescomplemented
40C2×#382C2211yescharacteristic, complemented
41#2280—22no—
42C2×#380C2212yes—
43#2097C2218yescomplemented
44#2092C2218yescomplemented
45#2068C2218yescomplemented
46#2067C2218yescomplemented
47#2066C2218yescomplemented
48#2065C2218yescomplemented
49#2024C2218yescomplemented
50#2023C2218yescomplemented
51#1854C2218yes—
52#1853C2214yescomplemented
53#1852C2214yescomplemented
54#1798C2214yes—
55#1798C2214yescomplemented
56#1788C2218yes—
57#1787C2214yes—
58#1786C2214yes—
59#1736C2214yescomplemented
60#1728C2212yescomplemented
61C22×#41—21no—
62C22×#41—21no—
63C22×#41—24no—
64C22×#41—44no—
65C22×#41—44no—
66C22×#41Q818yescomplemented
67C22×#41C2312yes—
68C22×#41C2312yes—
69C23×Q8—21no—
70C23×Q8C2311yescharacteristic
71C23×D4—21no—
72C23×D4C2311yescharacteristic
73C24×C4—21no—
74C24×C4C2311yescharacteristic
75C22×M16C2314yes—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8—44no—
78C2×C4⋊Q8C2314yes—
79C2×#117—21no—
80C2×#117—21no—
81C2×#117—44no—
82C2×#117—44no—
83C2×#117D412yes—
84C2×#117D412yes—
85C2×#117C2312yes—
86C2×#117C2312yes—
87C2×#117C2312yes—
88C2×#117C2312yes—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4D411yescharacteristic
92C22×C4⋊C4D411yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—22no—
98C22×C22⋊C4D411yescharacteristic
99C22×C22⋊C4D411yescharacteristic
100C22×C42—21no—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103#428—48no—
104C2×#107C2318yes—
105C2×#106C2318yes—
106#423—44no—
107C2×C4⋊C8C2314yes—
108C2×C4⋊C8C2314yes—
109#417—48no—
110#417—48no—
111C2×#103C2318yes—
112C2×#103C2318yes—
113C2×#102C2318yes—
114C2×#102C2318yes—
115#407—28no—
116C2×C22⋊C8C2314yes—
117C2×C22⋊C8C2314yes—
118#400—24no—
119#400—24no—
120#399—24no—
121#399—24no—
122#395—24no—
123#395—24no—
124#393—24no—
125#393—24no—
126#391—22no—
127#391—22no—
128#391C2314yes—
129#391C2314yes—
130#390—22no—
131#390—22no—
132#390C2314yes—
133#390C2314yes—
134#389—24no—
135#389C2318yes—
136#386—22no—
137#386—22no—
138#386—22no—
139#386—22no—
140#386—24no—
141#386—24no—
142#382—22no—
143#382C2314yes—
144#381—22no—
145#381—22no—
146#380—22no—
147#380—22no—
148#380—24no—
149#380—24no—
150#380C2314yes—
151#380C2314yes—
152#379—22no—
153#379—22no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—24no—
156C2×#95—21no—
157C2×#95—22no—
158C2×#95—22no—
159C2×#95—21no—
160#340C2318yes—
161#340C2318yes—
162#336C2318yes—
163C25—21no—
164C25C2×D411yescharacteristic
165C2×#41—28no—
166C2×#41—28no—
167C2×#41—24no—
168C2×#41—24no—
169C2×#41—28no—
170C2×#41—28no—
171C2×#41—44no—
172C2×#41—44no—
173C2×#41—432no—
174C2×#41—44no—
175C2×#41—44no—
176C2×#41—44no—
177C2×#41—44no—
178C2×#41—432no—
179C2×#41C2×Q818yes—
180C2×#41C2×Q818yescomplemented
181C22×Q8—21no—
182C22×Q8—24no—
183C22×Q8—22no—
184C22×Q8—42no—
185C22×Q8—42no—
186C22×Q8—44no—
187C22×Q8—84no—
188C22×Q8C2×Q814yescomplemented
189C22×Q8C2412yes—
190C22×D4—28no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—21no—
194C22×D4—22no—
195C22×D4—42no—
196C22×D4—42no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4C2×Q814yescomplemented
200C22×D4C2412yes—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—21no—
204C23×C4—28no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×Q814yes—
220C23×C4#4114yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C2×M16—28no—
225C2×M16—28no—
226C2×M16—28no—
227C22×C8C2418yes—
228C4⋊Q8—44no—
229C4⋊Q8—44no—
230C4⋊Q8—48no—
231#117—28no—
232#117—44no—
233#117—44no—
234#117—42no—
235#117—42no—
236#117—42no—
237#117—42no—
238#117—42no—
239#117—42no—
240#117—44no—
241#117—44no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—84no—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2414yes—
259C2×C4⋊C4C2414yes—
260C2×C4⋊C4C2414yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—84no—
273C2×C22⋊C4C2×D412yes—
274C2×C22⋊C4C2×D412yes—
275C2×C22⋊C4C2×D412yes—
276C2×C22⋊C4C2×D412yes—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—44no—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2412yes—
285C2×C42C2412yes—
286#107—48no—
287#106—48no—
288C4⋊C8—44no—
289C4⋊C8—44no—
290#103—48no—
291#103—48no—
292#102—48no—
293#102—48no—
294C22⋊C8—28no—
295C22⋊C8—28no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—82no—
314C24—82no—
315C24C22⋊Q814yes—
316C24C22×D411yescharacteristic
317C24C22×D411yescharacteristic
318C24C22×D411yescharacteristic
319#41—232no—
320#41—416no—
321#41—432no—
322#41—432no—
323#41—416no—
324#41—416no—
325C2×Q8—24no—
326C2×Q8—22no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—48no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—48no—
333C2×Q8—48no—
334C2×Q8—88no—
335C2×Q8—816no—
336C2×Q8C22×Q814yes—
337C2×D4—28no—
338C2×D4—28no—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—48no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—48no—
347C2×D4—432no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4C22×Q814yes—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—216no—
360C22×C4—216no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—432no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4C22⋊Q814yes—
397C22×C4C22⋊Q814yes—
398C22×C4C22⋊Q814yes—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×Q812yes—
407C22×C4C22×Q812yes—
408C22×C4C22×Q812yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4H(5,𝔽2)14yes—
413C22×C4H(5,𝔽2)14yes—
414C22×C4C2511yescharacteristic
415M16—416no—
416C2×C8—28no—
417C2×C8—28no—
418C2×C8—28no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—48no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—88no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C42—24no—
445C42—24no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—44no—
451C42—44no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—28no—
463C23—24no—
464C23—48no—
465C23—48no—
466C23—44no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23—82no—
471C23—82no—
472C23—84no—
473C23—88no—
474C23—84no—
475C23—84no—
476C23—88no—
477C23C2×C22⋊Q812yes—
478C23C2×C22⋊Q812yes—
479C23C2×C22⋊Q812yes—
480C23#53412yes—
481C23#53512yes—
482C23C23×D411yescharacteristic
483Q8—28no—
484Q8—48no—
485Q8—44no—
486Q8—816no—
487D4—28no—
488D4—48no—
489D4—416no—
490D4—416no—
491D4—44no—
492C2×C4—28no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—416no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—416no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—88no—
531C2×C4—88no—
532C2×C4—88no—
533C2×C4C2×C22⋊Q812yes—
534C2×C4C2×C22⋊Q812yes—
535C2×C4C2×C22⋊Q812yes—
536C2×C4#53412yes—
537C2×C4#53512yes—
538C2×C4#53818yes—
539C2×C4#54318yes—
540C2×C4C23×Q811yescharacteristic
541C2×C4C22×#4111yescharacteristic
542C2×C4C2×H(5,𝔽2)12yes—
543C2×C4C2×H(5,𝔽2)12yes—
544C8—48no—
545C22—24no—
546C22—22no—
547C22—22no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—42no—
558C22—42no—
559C22—48no—
560C22—44no—
561C22—88no—
562C22—816no—
563C22—82no—
564C22—82no—
565C22—84no—
566C22#262614yes—
567C22C22×C22⋊Q811yescharacteristic
568C22C2×#53411yescharacteristic
569C22C2×#53511yescharacteristic
570C22#358512yes—
571C22#358612yes—
572C4—22no—
573C4—22no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—44no—
579C4—84no—
580C4—84no—
581C4—84no—
582C4#262614yes—
583C4C2×#53812yes—
584C4C2×#54312yes—
585C2—24no—
586C2—42no—
587C2—42no—
588C2—84no—
589C2C2×#262611yescharacteristic
590C2#6049312yes—
591C2#6075311yescharacteristic
592C2#6075411yescharacteristic
593C2#6081012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2628C2semidirect
3. #29961C2semidirect
4. #29960C2semidirect
5. #29959C2semidirect
6. #23733C2semidirect
7. #23617C2semidirect
8. #23452C2semidirect
9. #20846C2semidirect
10. #20627C2semidirect
13. C22×C4⋊Q8C22semidirect
16. C2×#428C22semidirect
17. C2×#423C22semidirect
24. #2628C22semidirect
39. C2×#389C22semidirect
40. C2×#382C22semidirect
43. #2097C22semidirect
44. #2092C22semidirect
45. #2068C22semidirect
46. #2067C22semidirect
47. #2066C22semidirect
48. #2065C22semidirect
49. #2024C22semidirect
50. #2023C22semidirect
52. #1853C22semidirect
53. #1852C22semidirect
55. #1798C22semidirect
59. #1736C22semidirect
60. #1728C22semidirect
66. C22×#41Q8semidirect
180. C2×#41C2×Q8semidirect
188. C22×Q8C2×Q8semidirect
199. C22×D4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
#340442
#299612
#299602
#299592
#237332
#236172
#234522
#208462
#206272
#206012