Finite Group Databasebeta

← #6453093

Subgroups — #6453093

Order 512.

2114 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2C22×#428C211yesmaximal, characteristic
3#35311C211yesmaximal, characteristic
4#34178C211yesmaximal, characteristic
5#24590C218yesmaximal, complemented
6#24580C214yesmaximal, complemented
7#24577C214yesmaximal, complemented
8#23661C212yesmaximal, complemented
9#23615C212yesmaximal, complemented
10#21174C212yesmaximal, complemented
11#21064C212yesmaximal, complemented
12#20972C211yesmaximal, characteristic
13#20836C211yesmaximal, characteristic
14#20699C211yesmaximal, characteristic
15C23×M16C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#483—28no—
20C2×#483C2214yes—
21C2×#483C2214yes—
22C2×#428—28no—
23C2×#428C2214yes—
24C2×#428C2212yes—
25C2×#428C2212yes—
26C22×#107C2212yes—
27C22×#106C2212yes—
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C2×#424C2212yes—
31C2×#419C2212yes—
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35#2855—22no—
36#2625—22no—
37#2503C2212yes—
38#2502C2212yes—
39#2499C2214yes—
40#2484—22no—
41#2483—22no—
42#2345—22no—
43#2345C2214yes—
44#2344—22no—
45#2344C2214yes—
46#2317C2212yes—
47#2316C2212yes—
48#2313—22no—
49#2310C2212yes—
50#2280—22no—
51#2280C2211yescharacteristic
52#2280C2211yescharacteristic
53#2274C2212yes—
54#2085C2218yescomplemented
55#2084C2218yescomplemented
56#2068C2218yescomplemented
57#2066C2218yescomplemented
58#1949C2218yescomplemented
59#1948C2218yescomplemented
60#1928C2218yescomplemented
61#1927C2218yescomplemented
62#1894C2214yes—
63#1892C2214yes—
64#1851C2214yes—
65#1850C2214yes—
66#1825C2214yes—
67#1824C2212yes—
68#1823C2212yes—
69#1736C2212yes—
70#1735C2212yes—
71#1735C2212yes—
72#1729C2212yes—
73C23×Q8C2311yescharacteristic
74C24×C4—21no—
75C24×C4C2311yescharacteristic
76C22×M16—24no—
77C22×M16—28no—
78C22×M16C2312yes—
79C22×M16C2312yes—
80C23×C8C2312yes—
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8C2312yes—
88C2×C22⋊Q8C2312yes—
89C2×#117—21no—
90C2×#117—21no—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117C2312yes—
95C2×#117C2312yes—
96C2×#117C2312yes—
97C2×#117C2312yes—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D418yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C42—21no—
114#483—216no—
115#483—216no—
116#428—28no—
117#428—28no—
118#428—216no—
119C2×#107—28no—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#106—24no—
123C2×#106—24no—
124C2×#106—28no—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129#424—28no—
130C2×C4⋊C8C2318yes—
131#419—28no—
132C2×#103—24no—
133C2×#103—24no—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2318yes—
137C2×C22⋊C8—24no—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140#400—24no—
141#398—24no—
142#397—28no—
143#391—24no—
144#391C2314yes—
145#391C2314yes—
146#389—24no—
147#389—24no—
148#389C2318yes—
149#387—28no—
150#387—28no—
151#387—28no—
152#385—24no—
153#385—24no—
154#384C2314yes—
155#382C2314yes—
156#381—21no—
157#381—24no—
158#381—24no—
159#381—24no—
160#381D412yes—
161#380—24no—
162#380—24no—
163#380D412yes—
164#380D412yes—
165#380C2312yes—
166#380C2312yes—
167#379—21no—
168#379D412yes—
169C4×C4⋊C4—24no—
170C4×C4⋊C4C2318yes—
171C4×C22⋊C4—21no—
172C4×C22⋊C4—21no—
173C4×C22⋊C4—24no—
174C4×C22⋊C4—24no—
175C4×C22⋊C4C2312yes—
176C4×C22⋊C4C2312yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2318yes—
182#336C2318yes—
183C25C2×D411yescharacteristic
184C22×Q8—21no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8C2412yes—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—28no—
194C23×C4—21no—
195C23×C4—22no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—84no—
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D414yes—
206C23×C4#4114yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C2×M16—232no—
211C2×M16—24no—
212C2×M16—24no—
213C2×M16—24no—
214C2×M16—28no—
215C2×M16#4118yes—
216C22×C8—24no—
217C22×C8—28no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—24no—
221C22×C8—24no—
222C22×C8C2412yes—
223C22×C8C2412yes—
224C22×C8C2414yes—
225C22⋊Q8—24no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—44no—
230C22⋊Q8—44no—
231#117—24no—
232#117—24no—
233#117—24no—
234#117—24no—
235#117—24no—
236#117—24no—
237#117—24no—
238#117—24no—
239#117—44no—
240#117—44no—
241#117—88no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2414yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4C2×D414yes—
288C2×C22⋊C4C2×D414yes—
289C2×C22⋊C4C2×D414yes—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—28no—
294C2×C42—44no—
295C2×C42C2414yes—
296#107—28no—
297#107—28no—
298#106—28no—
299#106—28no—
300#106—28no—
301#106—28no—
302C4⋊C8—216no—
303#103—216no—
304#103—28no—
305#103—28no—
306C22⋊C8—24no—
307C22⋊C8—24no—
308#95—28no—
309#95—28no—
310#95—22no—
311#95—22no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95C2×D414yes—
316#95C2×D414yes—
317C24—24no—
318C24—24no—
319C24—42no—
320C24#12114yes—
321C24C22×D411yescharacteristic
322C24C22×D411yescharacteristic
323C24C22×D411yescharacteristic
324C2×Q8—22no—
325C2×Q8—24no—
326C2×Q8—22no—
327C2×Q8—24no—
328C2×Q8—42no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—48no—
333C22×C4—24no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—216no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—28no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—48no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4—816no—
364C22×C4—82no—
365C22×C4—84no—
366C22×C4—82no—
367C22×C4#12114yes—
368C22×C4#12114yes—
369C22×C4#12114yes—
370C22×C4C22×D418yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D412yes—
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4112yes—
380C22×C4C2×#4114yes—
381C22×C42−1+414yes—
382C22×C4C2511yescharacteristic
383M16—216no—
384M16—216no—
385C2×C8—216no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—44no—
393C2×C8—44no—
394C2×C8C2×#4118yes—
395C4⋊C4—28no—
396C4⋊C4—28no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—28no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—88no—
416C4⋊C4—88no—
417C4⋊C4C22×D414yes—
418C4⋊C4C22×D414yes—
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C42—24no—
430C42—24no—
431C42—44no—
432C42—44no—
433C42—44no—
434C42—88no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—28no—
441C23—22no—
442C23—28no—
443C23—22no—
444C23—44no—
445C23—416no—
446C23C2×#12112yes—
447C23C2×#12112yes—
448C23C2×#12112yes—
449C23#54714yes—
450C23#57714yes—
451C23C23×D411yescharacteristic
452Q8—42no—
453Q8—42no—
454Q8—44no—
455Q8—84no—
456C2×C4—28no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—416no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4—88no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—84no—
493C2×C4C2×#12112yes—
494C2×C4C2×#12112yes—
495C2×C4C2×#12112yes—
496C2×C4#52514yes—
497C2×C4#53614yes—
498C2×C4#54714yes—
499C2×C4#55218yes—
500C2×C4C23×D411yescharacteristic
501C2×C4C22×#4112yes—
502C2×C4C22×#4111yescharacteristic
503C2×C4C2×2−1+412yes—
504C8—28no—
505C8—42no—
506C8—42no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—28no—
511C22—28no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—416no—
517C22—44no—
518C22#241314yes—
519C22#304214yes—
520C22C22×#12111yescharacteristic
521C22C2×#54712yes—
522C22C2×#57712yes—
523C22#358612yes—
524C4—22no—
525C4—22no—
526C4—44no—
527C4—44no—
528C4—48no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4—84no—
533C4—82no—
534C4#241314yes—
535C4C2×#52511yescharacteristic
536C4C2×#53611yescharacteristic
537C4C2×#55212yes—
538C2—24no—
539C2—24no—
540C2—42no—
541C2C2×#241311yescharacteristic
542C2C2×#304211yescharacteristic
543C2#6042411yescharacteristic
544C2#6127114yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #24590C2semidirect
6. #24580C2semidirect
7. #24577C2semidirect
8. #23661C2semidirect
9. #23615C2semidirect
10. #21174C2semidirect
11. #21064C2semidirect
54. #2085C22semidirect
55. #2084C22semidirect
56. #2068C22semidirect
57. #2066C22semidirect
58. #1949C22semidirect
59. #1948C22semidirect
60. #1928C22semidirect
61. #1927C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
C22×#4282
#353112
#341782
#245902
#245802
#245772
#236612
#236152
#211742
#210642
#209722
#208362
#206992