Finite Group Databasebeta

← #6457215

Subgroups — #6457215

Order 512.

2178 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3095C211yesmaximal, characteristic
2C22×#483C211yesmaximal, characteristic
3#34203C211yesmaximal, characteristic
4C2×#2317C211yesmaximal, characteristic
5#27657C218yesmaximal, complemented
6#27653C214yesmaximal, complemented
7#27647C214yesmaximal, complemented
8#23661C212yesmaximal, complemented
9#23633C212yesmaximal, complemented
10#21174C212yesmaximal, complemented
11#20700C211yesmaximal, characteristic
12#20699C211yesmaximal, characteristic
13#20651C212yesmaximal, complemented
14#20628C211yesmaximal, characteristic
15C22×C22⋊Q8C2211yescharacteristic
16C2×#519C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#3095—28no—
21C2×#484C2212yes—
22C2×#485C2214yes—
23C2×#483—28no—
24C2×#483C2214yes—
25C2×#483C2214yes—
26C2×#483C2212yes—
27C2×#428C2212yes—
28C22×#106C2212yes—
29C2×#424C2212yes—
30C2×#419C2212yes—
31C22×#103C2212yes—
32C2×#407C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34#2362—22no—
35#2360—22no—
36#2344—22no—
37#2342—22no—
38#2323C2214yes—
39#2317—22no—
40#2317—22no—
41#2317C2214yes—
42#2317C2212yes—
43#2317C2212yes—
44#2316C2211yescharacteristic
45#2310C2211yescharacteristic
46C4×C22⋊Q8C2214yes—
47C2×#389—22no—
48C2×#385—22no—
49C2×#384C2212yes—
50C2×#382C2212yes—
51#2280C2211yescharacteristic
52#2274C2211yescharacteristic
53#2085C2218yescomplemented
54#2084C2218yescomplemented
55#2074C2218yescomplemented
56#2072C2218yescomplemented
57#1949C2218yescomplemented
58#1948C2218yescomplemented
59#1825C2214yes—
60#1824C2212yes—
61#1824C2214yes—
62#1823C2214yes—
63#1823C2212yes—
64#1805C2218yescomplemented
65#1803C2218yescomplemented
66#1798C2214yes—
67#1797C2214yes—
68#1736C2212yes—
69#1736C2212yes—
70#1735C2212yes—
71#1735C2212yes—
72#1729C2212yes—
73C23×Q8C2311yescharacteristic
74C24×C4—21no—
75C24×C4C2311yescharacteristic
76C22×M16C2312yes—
77C23×C8C2312yes—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8D418yes—
82C2×C22⋊Q8C2312yes—
83C2×C22⋊Q8C2312yes—
84#519—28no—
85C2×C4×Q8—22no—
86C2×C4×Q8C2314yes—
87C2×#117—28no—
88C2×#117—21no—
89C2×#117—21no—
90C2×#117—22no—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117D412yes—
94C2×#117D412yes—
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C2×#117C2312yes—
98C2×#117C2312yes—
99C2×#117C2312yes—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4—21no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4D412yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4D412yes—
119C22×C42—21no—
120C22×C42C2312yes—
121#485—216no—
122#484—28no—
123#483—216no—
124#483—216no—
125#483—28no—
126#428—28no—
127C2×#107C2314yes—
128C2×#106—28no—
129C2×#106C2318yes—
130C2×#106C2314yes—
131#424—28no—
132C2×C4⋊C8C2318yes—
133#419—28no—
134C2×#103—28no—
135C2×#103C2314yes—
136C2×#103C2318yes—
137C2×#103C2314yes—
138#407—24no—
139C2×C22⋊C8—24no—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144#391—22no—
145#391C2314yes—
146#389—22no—
147#389—28no—
148#389—24no—
149#389—24no—
150#389—24no—
151#389C2314yes—
152#388—24no—
153#388—24no—
154#387—24no—
155#387—24no—
156#385—24no—
157#385—24no—
158#385—28no—
159#385—24no—
160#384—24no—
161#384C2314yes—
162#384C2314yes—
163#382—24no—
164#382C2314yes—
165#382C2314yes—
166#381—21no—
167#381—22no—
168#380—22no—
169#380C2312yes—
170#380C2312yes—
171#379—21no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—24no—
174C4×C4⋊C4C2314yes—
175C4×C22⋊C4—24no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4—22no—
179C2×#95—24no—
180#340C2314yes—
181#340C2318yes—
182#340C2314yes—
183#340C2314yes—
184#340C2318yes—
185#340C2314yes—
186#336C2318yes—
187C25C2×D411yescharacteristic
188C22×Q8—21no—
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8C2×D414yes—
193C22×Q8C2412yes—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—44no—
204C23×C4—84no—
205C23×C4C2×D414yes—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D414yes—
210C23×C4#4114yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C2×M16—24no—
215C2×M16—24no—
216C2×M16—24no—
217C22×C8—24no—
218C22×C8—24no—
219C22×C8—24no—
220C22×C8C2414yes—
221C22×C8C2414yes—
222C22⋊Q8—28no—
223C22⋊Q8—28no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8C2×D418yes—
227C22⋊Q8C2×D418yes—
228C4×Q8—216no—
229C4×Q8—48no—
230#117—24no—
231#117—28no—
232#117—28no—
233#117—28no—
234#117—24no—
235#117—24no—
236#117—24no—
237#117—24no—
238#117—24no—
239#117—24no—
240#117—44no—
241#117—44no—
242#117—48no—
243#117#4118yes—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—88no—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2414yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4C2×D412yes—
295C2×C22⋊C4C2×D414yes—
296C2×C22⋊C4C2×D412yes—
297C2×C42—28no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—24no—
304C2×C42—44no—
305C2×C42C2×D414yes—
306C2×C42C2412yes—
307C2×C42C2412yes—
308#107—28no—
309#106—216no—
310#106—28no—
311C4⋊C8—216no—
312#103—216no—
313#103—28no—
314#103—28no—
315C22⋊C8—24no—
316C22⋊C8—24no—
317C22⋊C8—24no—
318C22⋊C8—24no—
319#95—28no—
320#95—28no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—22no—
325#95—24no—
326#95—24no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—42no—
331C24#12114yes—
332C24C22×D411yescharacteristic
333C24C22×D411yescharacteristic
334C24C22×D411yescharacteristic
335C2×Q8—24no—
336C2×Q8—22no—
337C2×Q8—216no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—48no—
343C2×Q8C22×D414yes—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—216no—
351C22×C4—216no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—28no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—48no—
378C22×C4—82no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4—816no—
382C22×C4#12114yes—
383C22×C4#12114yes—
384C22×C4#12114yes—
385C22×C4C22×D414yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D414yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C2×#4112yes—
394C22×C4C2×#4112yes—
395C22×C4C2×#4112yes—
396C22×C4C2×#4114yes—
397C22×C42−1+414yes—
398C22×C4C2511yescharacteristic
399M16—48no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—48no—
407C4⋊C4—28no—
408C4⋊C4—28no—
409C4⋊C4—24no—
410C4⋊C4—28no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—48no—
427C4⋊C4—84no—
428C4⋊C4—88no—
429C4⋊C4—88no—
430C4⋊C4#12318yes—
431C4⋊C4C22×D414yes—
432C22⋊C4—28no—
433C22⋊C4—28no—
434C22⋊C4—24no—
435C22⋊C4—28no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—44no—
441C22⋊C4—82no—
442C22⋊C4#12318yes—
443C42—28no—
444C42—24no—
445C42—44no—
446C42—44no—
447C42—42no—
448C42—48no—
449C42—42no—
450C42—84no—
451C42C2×#4118yes—
452C23—22no—
453C23—22no—
454C23—21no—
455C23—21no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—28no—
461C23—28no—
462C23—22no—
463C23—44no—
464C23—48no—
465C23—48no—
466C23C2×#12112yes—
467C23C2×#12112yes—
468C23C2×#12112yes—
469C23C2×#12312yes—
470C23#53612yes—
471C23#57714yes—
472C23C23×D411yescharacteristic
473Q8—28no—
474Q8—44no—
475Q8—48no—
476C2×C4—24no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—416no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—48no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4—88no—
514C2×C4—88no—
515C2×C4—84no—
516C2×C4—88no—
517C2×C4C2×#12112yes—
518C2×C4C2×#12112yes—
519C2×C4C2×#12112yes—
520C2×C4C2×#12318yes—
521C2×C4C2×#12312yes—
522C2×C4#53612yes—
523C2×C4#54718yes—
524C2×C4#55218yes—
525C2×C4C23×D411yescharacteristic
526C2×C4C22×#4112yes—
527C2×C4C22×#4111yescharacteristic
528C2×C4C2×2−1+412yes—
529C8—44no—
530C8—44no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—24no—
535C22—24no—
536C22—22no—
537C22—24no—
538C22—24no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—416no—
543C22—42no—
544C22—42no—
545C22#248614yes—
546C22#304314yes—
547C22C22×#12111yescharacteristic
548C22C22×#12311yescharacteristic
549C22C2×#53611yescharacteristic
550C22C2×#57712yes—
551C22#358612yes—
552C4—28no—
553C4—22no—
554C4—22no—
555C4—44no—
556C4—44no—
557C4—44no—
558C4—82no—
559C4—84no—
560C4—82no—
561C4—82no—
562C4—82no—
563C4#248614yes—
564C4C2×#54712yes—
565C4C2×#55212yes—
566C2—24no—
567C2—24no—
568C2—42no—
569C2C2×#248611yescharacteristic
570C2C2×#304311yescharacteristic
571C2#6042711yescharacteristic
572C2#6052212yes—
573C2#6087212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
5. #27657C2semidirect
6. #27653C2semidirect
7. #27647C2semidirect
8. #23661C2semidirect
9. #23633C2semidirect
10. #21174C2semidirect
13. #20651C2semidirect
53. #2085C22semidirect
54. #2084C22semidirect
55. #2074C22semidirect
56. #2072C22semidirect
57. #1949C22semidirect
58. #1948C22semidirect
64. #1805C22semidirect
65. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30952
C22×#4832
#342032
C2×#23172
#276572
#276532
#276472
#236612
#236332
#211742
#207002
#206992
#206512
#206282