Finite Group Databasebeta

← #6479551

Subgroups — #6479551

Order 512.

1500 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35311C211yesmaximal, characteristic
2#34048C211yesmaximal, characteristic
3#33101C214yesmaximal, complemented
4#29915C214yesmaximal, complemented
5#27872C214yesmaximal, complemented
6#27867C214yesmaximal, complemented
7#27622C214yesmaximal, complemented
8#27596C214yesmaximal, complemented
9#21943C211yesmaximal, characteristic
10#21939C211yesmaximal, characteristic
11#21768C211yesmaximal, characteristic
12#21762C211yesmaximal, characteristic
13#20685C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#407C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18#2855C2212yes—
19#2855C2212yes—
20#2625C2214yes—
21#2503—21no—
22#2502—21no—
23#2499—22no—
24#2484—22no—
25#2483—21no—
26#2483—21no—
27#2317C2212yes—
28#2314C2214yes—
29#2314C2214yes—
30#2313—21no—
31#2313—21no—
32#2312—21no—
33#2312—21no—
34#2310C2212yes—
35#2298—22no—
36#2298—22no—
37#2285C2212yes—
38#2280C2211yescharacteristic
39#2280C2211yescharacteristic
40#2280C2211yescharacteristic
41C4×#117C2212yes—
42#2100C2214yescomplemented
43#2099C2214yescomplemented
44#2096C2214yescomplemented
45#2095C2214yescomplemented
46#2085C2214yescomplemented
47#2085C2214yescomplemented
48#2074C2214yescomplemented
49#2074C2214yescomplemented
50#1928C2214yescomplemented
51#1928C2214yescomplemented
52#1927C2214yescomplemented
53#1927C2214yescomplemented
54#1921C2214yescomplemented
55#1920C2214yescomplemented
56#1919C2214yescomplemented
57#1918C2214yescomplemented
58#1894C2214yes—
59#1892C2212yes—
60#1892C2212yes—
61#1873C2214yes—
62#1873C2214yes—
63#1854C2214yes—
64#1854C2214yes—
65#1854C2214yes—
66#1854C2214yes—
67#1824C2212yes—
68#1823C2212yes—
69#1818C2214yes—
70#1817C2212yes—
71#1817C2212yes—
72#1809C2214yes—
73#1808C2212yes—
74#1807C2212yes—
75#1787C2212yes—
76#1787C2212yes—
77#1729C2212yes—
78#1729C2212yes—
79C23×Q8C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C22×M16C2312yes—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8C2312yes—
89C2×C22⋊Q8C2312yes—
90C2×#117—24no—
91C2×#117—24no—
92C2×#117D412yes—
93C2×#117D412yes—
94C2×#117C2312yes—
95C2×#117C2312yes—
96C2×#117C2312yes—
97C2×#117C2312yes—
98C2×#117C2312yes—
99C22×C4⋊C4—21no—
100C22×C4⋊C4D411yescharacteristic
101C22×C4⋊C4D411yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—21no—
104C22×C22⋊C4D411yescharacteristic
105C22×C22⋊C4D411yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C2×#107C2314yes—
109C2×#107C2314yes—
110C2×#106C2314yes—
111C2×#106C2314yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116#407—24no—
117C2×C22⋊C8—24no—
118C2×C22⋊C8C2312yes—
119C2×C22⋊C8C2312yes—
120C2×C22⋊C8C2312yes—
121C2×C22⋊C8C2312yes—
122#400—24no—
123#398—24no—
124#397—24no—
125#397—24no—
126#391—22no—
127#391—22no—
128#391C2314yes—
129#391C2314yes—
130#389C2314yes—
131#389C2314yes—
132#388—24no—
133#388—24no—
134#388—24no—
135#388—24no—
136#387—24no—
137#387—24no—
138#387—24no—
139#387—24no—
140#387—24no—
141#387—24no—
142#385—22no—
143#385—22no—
144#385—22no—
145#385—22no—
146#384—22no—
147#384C2314yes—
148#382—22no—
149#382C2314yes—
150#381—21no—
151#381—21no—
152#381—22no—
153#381—22no—
154#381—21no—
155#381—22no—
156#381—22no—
157#380—22no—
158#380—22no—
159#380C2312yes—
160#380C2312yes—
161#380C2312yes—
162#380C2312yes—
163#380C2312yes—
164#380C2312yes—
165#379—21no—
166#379—21no—
167#379—21no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4C2314yes—
171C4×C4⋊C4C2314yes—
172C4×C22⋊C4—22no—
173C4×C22⋊C4—22no—
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4—21no—
179C4×C22⋊C4—21no—
180#376—22no—
181C43—22no—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25C2×D411yescharacteristic
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—42no—
199C22×Q8—42no—
200C22×Q8C2412yes—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4#4114yes—
213C23×C4#4114yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C2×M16—24no—
218C2×M16—42no—
219C2×M16—42no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—42no—
223C22×C8—42no—
224C22×C8C2412yes—
225C22×C8C2412yes—
226C22×C8C2414yes—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—24no—
231C22⋊Q8—44no—
232C22⋊Q8—44no—
233#117—28no—
234#117—28no—
235#117—24no—
236#117—24no—
237#117—24no—
238#117—24no—
239#117—42no—
240#117—42no—
241#117—42no—
242#117—42no—
243#117—42no—
244#117—42no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2414yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4C2×D412yes—
280C2×C22⋊C4C2×D412yes—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D412yes—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2412yes—
295C2×C42C2412yes—
296#107—44no—
297#107—44no—
298#106—44no—
299#106—44no—
300#103—44no—
301#103—44no—
302#103—44no—
303#103—44no—
304C22⋊C8—24no—
305C22⋊C8—24no—
306C22⋊C8—24no—
307C22⋊C8—24no—
308#95—24no—
309#95—24no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24#12314yes—
328C24C22×D411yescharacteristic
329C24C22×D411yescharacteristic
330C24C22×D411yescharacteristic
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—44no—
336C2×Q8—42no—
337C2×Q8—82no—
338C2×Q8—82no—
339C2×Q8—84no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—48no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—82no—
369C22×C4#12314yes—
370C22×C4#12314yes—
371C22×C4#12314yes—
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D412yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C2×#4112yes—
380C22×C4C2×#4112yes—
381C22×C4C2×#4112yes—
382C22×C4C2×#4112yes—
383C22×C4C2×#4112yes—
384C22×C4C2×#4112yes—
385C22×C4C2511yescharacteristic
386M16—84no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—82no—
395C2×C8—82no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—42no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—84no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—82no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C22⋊C4—82no—
429C22⋊C4—82no—
430C42—24no—
431C42—24no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—44no—
437C42—44no—
438C42—44no—
439C42—84no—
440C42—84no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—88no—
454C23C2×#12312yes—
455C23C2×#12312yes—
456C23C2×#12312yes—
457C23C23×D411yescharacteristic
458Q8—42no—
459Q8—42no—
460Q8—44no—
461Q8—84no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—84no—
503C2×C4—84no—
504C2×C4C2×#12312yes—
505C2×C4C2×#12312yes—
506C2×C4C2×#12312yes—
507C2×C4C22×#4111yescharacteristic
508C2×C4C22×#4111yescharacteristic
509C2×C4#58512yes—
510C2×C4#58512yes—
511C8—42no—
512C8—42no—
513C8—44no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22—88no—
527C22C22×#12311yescharacteristic
528C22#344912yes—
529C22#358412yes—
530C22#358612yes—
531C4—22no—
532C4—22no—
533C4—22no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—82no—
539C4—84no—
540C4—84no—
541C4—84no—
542C4—82no—
543C4#349111yescharacteristic
544C4#349111yescharacteristic
545C4#349112yes—
546C2—42no—
547C2—42no—
548C2—42no—
549C2#3509711yescharacteristic
550C2#6051611yescharacteristic
551C2#6051911yescharacteristic
552C2#6098714yes—
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
3. #33101C2semidirect
4. #29915C2semidirect
5. #27872C2semidirect
6. #27867C2semidirect
7. #27622C2semidirect
8. #27596C2semidirect
42. #2100C22semidirect
43. #2099C22semidirect
44. #2096C22semidirect
45. #2095C22semidirect
46. #2085C22semidirect
47. #2085C22semidirect
48. #2074C22semidirect
49. #2074C22semidirect
50. #1928C22semidirect
51. #1928C22semidirect
52. #1927C22semidirect
53. #1927C22semidirect
54. #1921C22semidirect
55. #1920C22semidirect
56. #1919C22semidirect
57. #1918C22semidirect

Maximal subgroups

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

subgroupindex
#353112
#340482
#331012
#299152
#278722
#278672
#276222
#275962
#219432
#219392
#217682
#217622
#206852