Finite Group Databasebeta

← #6436168

Subgroups — #6436168

Order 512.

1820 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3081C212yesmaximal
2#35330C211yesmaximal, characteristic, complemented
3#34255C211yesmaximal, characteristic
4#29874C214yesmaximal, complemented
5#29872C214yesmaximal, complemented
6#29870C214yesmaximal, complemented
7#29865C214yesmaximal, complemented
8#23662C212yesmaximal, complemented
9#23659C211yesmaximal, characteristic, complemented
10#23656C211yesmaximal, characteristic, complemented
11#23632C212yesmaximal
12#23628C211yesmaximal, characteristic
13C2×#2073C211yesmaximal, characteristic
14#21315C212yesmaximal, complemented
15#20754C211yesmaximal, characteristic, complemented
16C2×#544C2212yes—
17C2×#518C2211yescharacteristic
18#3081—216no—
19C2×#478C2214yes—
20C2×#478C2214yes—
21C2×#476C2214yes—
22C2×#474C2214yes—
23C2×#428C2212yes—
24C2×#427C2212yes—
25C2×#424C2212yes—
26C2×#418C2212yes—
27C22×#102C2212yes—
28#2862C2212yescomplemented
29#2859C2212yescomplemented
30#2660C2214yescomplemented
31#2550—22no—
32#2550—22no—
33#2549—24no—
34#2357—22no—
35#2354—22no—
36#2342—24no—
37#2319C2214yes—
38#2317C2211yescharacteristic
39#2317C2212yes—
40#2309C2211yescharacteristic
41#2296C2211yescharacteristic
42#2293C2214yes—
43C2×#390C2211yescharacteristic
44#2285C2211yescharacteristic
45C2×#382C2211yescharacteristic
46#2274C2211yescharacteristic
47C4×#117C2211yescharacteristic
48#2085C2212yescomplemented
49#2085C2212yescomplemented
50#2085C2214yescomplemented
51#2085C2214yescomplemented
52#2085C2214yescomplemented
53#2085C2214yescomplemented
54#2084C2212yescomplemented
55#2084C2214yescomplemented
56#2083C2212yescomplemented
57#2083C2214yescomplemented
58#2073C2214yes—
59#2073C2212yes—
60#2073C2212yes—
61#2073C2212yes—
62#2073C2212yes—
63#2073C2214yes—
64#2073C2214yes—
65#2071C2214yes—
66#2071C2214yes—
67#2071C2214yes—
68#1989C2214yescomplemented
69#1989C2214yescomplemented
70#1988C2214yescomplemented
71#1987C2214yescomplemented
72#1832C2212yescomplemented
73#1832C2212yescomplemented
74#1831C2214yescomplemented
75#1736C2212yes—
76#1736C2212yes—
77#1735C2212yes—
78C2×#340C2212yes—
79#1729C2212yes—
80#1728C2211yescharacteristic
81C2×#336C2211yescharacteristic
82C23×D4C2311yescharacteristic
83C22×M16C2312yes—
84C23×C8C2312yes—
85#544—28no—
86#544—28no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8C2314yes—
89C2×#125—22no—
90C2×#125C2314yes—
91C2×C22⋊Q8—24no—
92#518—24no—
93#518D418yes—
94C2×C4×D4—22no—
95C2×C4×D4C2314yes—
96C2×#117—24no—
97C2×#117C2311yescharacteristic
98C2×#117C2312yes—
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4—22no—
105C22×C22⋊C4—22no—
106C22×C42C2311yescharacteristic
107#478—216no—
108#478—216no—
109#476—216no—
110#474—216no—
111#428—28no—
112#427—28no—
113C2×#107C2312yes—
114C2×#107C2314yes—
115C2×#107C2312yes—
116C2×#106C2312yes—
117C2×#106C2314yes—
118C2×#106C2312yes—
119#424—28no—
120C2×C4⋊C8C2314yes—
121C2×C4⋊C8C2314yes—
122#418—28no—
123C2×#102—28no—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128#398—24no—
129#398—24no—
130#395—24no—
131#395—24no—
132#391—24no—
133#391—24no—
134#390—21no—
135#390—21no—
136#390—22no—
137#390C2314yes—
138#390C2312yes—
139#390C2312yes—
140#389—22no—
141#389—22no—
142#389—24no—
143#389—24no—
144#389C2314yes—
145#389C2312yes—
146#389C2312yes—
147#388—24no—
148#388—24no—
149#388—24no—
150#388—24no—
151#385—22no—
152#385—22no—
153#385—24no—
154#384—21no—
155#384C2314yes—
156#384C2312yes—
157#382—21no—
158#382—22no—
159#382C2314yes—
160#382C2311yescharacteristic
161#382C2312yes—
162#382C2311yescharacteristic
163#382C2312yes—
164#381—24no—
165#380—22no—
166#380—24no—
167#379—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4D412yes—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4C2314yes—
174C4×C22⋊C4—22no—
175C4×C22⋊C4—24no—
176C4×C22⋊C4D414yes—
177#376C2312yes—
178C2×#95—22no—
179C43C2312yes—
180#340C2312yes—
181#340C2312yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2312yes—
187#340C2314yes—
188#340C2312yes—
189#340C2314yes—
190#336C2312yes—
191#336C2314yes—
192#336C2314yes—
193#336C2311yescharacteristic
194#336C2311yescharacteristic
195#336C2312yes—
196#336C2312yes—
197C25—21no—
198C22×Q8—24no—
199C22×D4—21no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4C2×D414yes—
204C22×D4C2412yes—
205C23×C4—24no—
206C23×C4—21no—
207C23×C4—22no—
208C23×C4C2×D412yes—
209C23×C4C2411yescharacteristic
210C2×M16—24no—
211C2×M16—24no—
212C2×M16—24no—
213C22×C8—24no—
214C22×C8—24no—
215C22×C8—24no—
216C22×C8C2412yes—
217C22×C8C2412yes—
218C22×C8C2414yes—
219C4⋊Q8—28no—
220C4⋊Q8—28no—
221C4⋊Q8—48no—
222#125—28no—
223#125—28no—
224#125—48no—
225C22⋊Q8—416no—
226C4×D4—216no—
227C4×D4—28no—
228C4×D4C2×D4116yes—
229#117—22no—
230#117—24no—
231#117—24no—
232#117—24no—
233#117—24no—
234#117—88no—
235#117C2×D414yes—
236#117C2×D414yes—
237#117#4114yes—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—216no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—84no—
272C2×C22⋊C4C2×D412yes—
273C2×C22⋊C4C2×D412yes—
274C2×C42—21no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—21no—
278C2×C42—24no—
279C2×C42C2×D412yes—
280C2×C42C2×D414yes—
281C2×C42C2×D414yes—
282C2×C42C2412yes—
283C2×C42C2411yescharacteristic
284C2×C42C2411yescharacteristic
285C2×C42C2412yes—
286#107—24no—
287#107—24no—
288#107—28no—
289#106—24no—
290#106—24no—
291#106—28no—
292C4⋊C8—28no—
293C4⋊C8—28no—
294#102—28no—
295#102—28no—
296#102—28no—
297#102—28no—
298#95—24no—
299#95—24no—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—24no—
307#95—24no—
308#95C2×D414yes—
309#95C2×D414yes—
310C24—22no—
311C24—24no—
312C24—42no—
313C24—42no—
314C24—82no—
315C24—82no—
316C24C22×D411yescharacteristic
317C2×Q8—48no—
318C2×Q8—84no—
319C2×Q8—84no—
320C2×D4—24no—
321C2×D4—216no—
322C2×D4—22no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—48no—
328C2×D4C22×D414yes—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—216no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—84no—
342C22×C4—84no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4C22×D412yes—
347C22×C4C22×D412yes—
348C22×C4C22×D412yes—
349C22×C4C22×D412yes—
350C22×C4C22×D414yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D412yes—
353C22×C4C2×#4112yes—
354C22×C42−1+414yes—
355C22×C42−1+414yes—
356C22×C4C2511yescharacteristic
357M16—48no—
358C2×C8—22no—
359C2×C8—24no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—48no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—48no—
384C4⋊C4—42no—
385C4⋊C4—84no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C4⋊C4—84no—
389C4⋊C4—82no—
390C4⋊C4—84no—
391C4⋊C4#12118yes—
392C4⋊C4C22×D414yes—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—216no—
396C22⋊C4—48no—
397C22⋊C4—44no—
398C22⋊C4—42no—
399C22⋊C4—84no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C22⋊C4#12118yes—
403C42—24no—
404C42—24no—
405C42—22no—
406C42—22no—
407C42—24no—
408C42—44no—
409C42—44no—
410C42—44no—
411C42—42no—
412C42—84no—
413C42C22×D414yes—
414C42C2×#4114yes—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—44no—
419C23—44no—
420C23—42no—
421C23—42no—
422C23—48no—
423C23—88no—
424C23—164no—
425C23—162no—
426C23—162no—
427C23C2×#12112yes—
428C23#53612yes—
429C23#53612yes—
430C23C23×D411yescharacteristic
431Q8—168no—
432D4—28no—
433D4—48no—
434D4—44no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—816no—
465C2×C4—84no—
466C2×C4—164no—
467C2×C4—162no—
468C2×C4C2×#12118yes—
469C2×C4C2×#12112yes—
470C2×C4#53612yes—
471C2×C4#53612yes—
472C2×C4#53618yes—
473C2×C4#54118yes—
474C2×C4C23×D411yescharacteristic
475C2×C4C22×#4111yescharacteristic
476C2×C4C2×2−1+412yes—
477C2×C4C2×2−1+412yes—
478C8—42no—
479C8—42no—
480C8—44no—
481C22—24no—
482C22—24no—
483C22—22no—
484C22—22no—
485C22—48no—
486C22—44no—
487C22—84no—
488C22—82no—
489C22—82no—
490C22—168no—
491C22#262114yes—
492C22C22×#12111yescharacteristic
493C22C2×#53611yescharacteristic
494C22C2×#53611yescharacteristic
495C22#358412yes—
496C22#358512yes—
497C4—28no—
498C4—44no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—82no—
504C4—84no—
505C4—82no—
506C4—82no—
507C4—82no—
508C4#262114yes—
509C4C2×#53612yes—
510C4C2×#54112yes—
511C2—24no—
512C2—84no—
513C2C2×#262111yescharacteristic
514C2#6043012yes—
515C2#6087512yes—
516C2#6087911yescharacteristic
517C2#6088311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35330C2semidirect
4. #29874C2semidirect
5. #29872C2semidirect
6. #29870C2semidirect
7. #29865C2semidirect
8. #23662C2semidirect
9. #23659C2semidirect
10. #23656C2semidirect
14. #21315C2semidirect
15. #20754C2semidirect
28. #2862C22semidirect
29. #2859C22semidirect
30. #2660C22semidirect
48. #2085C22semidirect
49. #2085C22semidirect
50. #2085C22semidirect
51. #2085C22semidirect
52. #2085C22semidirect
53. #2085C22semidirect
54. #2084C22semidirect
55. #2084C22semidirect
56. #2083C22semidirect
57. #2083C22semidirect
68. #1989C22semidirect
69. #1989C22semidirect
70. #1988C22semidirect
71. #1987C22semidirect
72. #1832C22semidirect
73. #1832C22semidirect
74. #1831C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30812
#353302
#342552
#298742
#298722
#298702
#298652
#236622
#236592
#236562
#236322
#236282
C2×#20732
#213152
#207542