Finite Group Databasebeta

← #6470701

Subgroups — #6470701

Order 512.

2136 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C212yesmaximal
2#35011C211yesmaximal, characteristic
3C2×#2362C211yesmaximal, characteristic
4#33108C212yesmaximal, complemented
5#32906C212yesmaximal, complemented
6#29756C212yesmaximal, complemented
7#29715C212yesmaximal, complemented
8#25219C218yesmaximal, complemented
9#25095C218yesmaximal, complemented
10#21934C211yesmaximal, characteristic
11#20911C212yesmaximal
12C22×C22⋊Q8C2212yes—
13C22×C22⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16#3195—216no—
17C2×#483C2214yes—
18C2×#483C2214yes—
19C2×#470C2218yes—
20C2×#462C2214yescomplemented
21C2×#451C2214yes—
22C2×#451C2214yes—
23C2×#419C2214yes—
24C2×#407C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2659—21no—
28#2659—21no—
29#2583—24no—
30#2502C2214yes—
31#2434C2214yes—
32#2362—24no—
33#2362C2218yes—
34#2315—21no—
35#2315—21no—
36C2×#391C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#388—21no—
39C2×#388—21no—
40C2×#387—21no—
41C2×C4×C4⋊C4—21no—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44#2084C2214yescomplemented
45#2072C2214yescomplemented
46#2066C2218yescomplemented
47#2056C2218yescomplemented
48#2043C2214yescomplemented
49#2016C2214yescomplemented
50#1981C2214yescomplemented
51#1948C2214yescomplemented
52#1927C2218yescomplemented
53#1878C2218yescomplemented
54#1873C2218yes—
55#1872C2214yes—
56#1872C2214yes—
57#1851C2218yes—
58#1803C2214yescomplemented
59#1797C2212yes—
60#1797C2212yes—
61#1736C2214yes—
62C23×Q8C2312yes—
63C23×Q8C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C22×Q16C2318yes—
66C22×M16C2314yes—
67C2×C4⋊Q8C2312yes—
68C2×C4⋊Q8C2312yes—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—28no—
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8D418yes—
76C2×C22⋊Q8C2314yes—
77C2×C22⋊Q8C2314yes—
78C2×C22⋊Q8C2314yes—
79C2×C4×Q8—22no—
80C2×C4×Q8—44no—
81C2×C4×Q8C2314yes—
82C2×#117—22no—
83C2×#117—44no—
84C2×#117C2314yes—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4—21no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4D411yescharacteristic
97C22×C22⋊C4D411yescharacteristic
98C22×C42—21no—
99C22×C42—21no—
100C22×C42—21no—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103#483—216no—
104#483—216no—
105#470—232no—
106#462—48no—
107#451—216no—
108#451—216no—
109C2×#106C2318yes—
110C2×C4⋊C8C2314yes—
111C2×C4⋊C8C2314yes—
112#419—48no—
113C2×#103C2318yes—
114C2×#103C2318yes—
115C2×#103C2318yes—
116C2×#103C2318yes—
117#407—28no—
118C2×C22⋊C8C2314yes—
119C2×C22⋊C8C2314yes—
120#398—24no—
121#398—24no—
122#395—24no—
123#395—24no—
124#391—22no—
125#391C2314yes—
126#391C2314yes—
127#391C2314yes—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389C2314yes—
134#388—28no—
135#388—28no—
136#387—28no—
137#385—24no—
138#385—24no—
139#385—24no—
140#385—24no—
141#384C2312yes—
142#384C2312yes—
143#380—22no—
144#380—22no—
145#380—22no—
146#380—22no—
147C4×C4⋊C4—22no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4C2314yes—
150C4×C22⋊C4D414yes—
151C4×C22⋊C4C2314yes—
152C2×#95—21no—
153C2×#95—21no—
154C2×#95—21no—
155C2×#95—21no—
156C2×#95D411yescharacteristic
157C2×#95D411yescharacteristic
158#340C2318yes—
159#340C2318yes—
160#336C2314yes—
161#336C2314yes—
162C25C2×D411yescharacteristic
163C22×Q8—21no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—42no—
169C22×Q8—42no—
170C22×Q8—44no—
171C22×Q8—44no—
172C22×Q8C2×D414yes—
173C22×Q8C2414yes—
174C22×Q8C2412yes—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D414yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4#4114yes—
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C2×Q16—216no—
194C2×Q16—48no—
195C2×Q16—48no—
196C2×M16—28no—
197C2×M16—44no—
198C2×M16—44no—
199C22×C8C2418yes—
200C4⋊Q8—44no—
201C4⋊Q8—44no—
202C22⋊Q8—216no—
203C22⋊Q8—28no—
204C22⋊Q8—28no—
205C22⋊Q8—28no—
206C22⋊Q8—28no—
207C22⋊Q8—28no—
208C22⋊Q8—48no—
209C22⋊Q8—48no—
210C22⋊Q8C2×D4116yes—
211C4×Q8—28no—
212C4×Q8—44no—
213C4×Q8—48no—
214#117—28no—
215#117—44no—
216#117—48no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2414yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4C2×D414yes—
258C2×C22⋊C4C2×D414yes—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42—44no—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2412yes—
270C2×C42C2412yes—
271#106—216no—
272C4⋊C8—44no—
273C4⋊C8—44no—
274#103—216no—
275#103—216no—
276#103—216no—
277#103—48no—
278C22⋊C8—28no—
279C22⋊C8—28no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95C2×D414yes—
289#95C2×D414yes—
290C24—22no—
291C24—22no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24#12114yes—
297C24C22×D412yes—
298C24C22×D411yescharacteristic
299C2×Q8—24no—
300C2×Q8—22no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—28no—
305C2×Q8—24no—
306C2×Q8—28no—
307C2×Q8—48no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—48no—
311C2×Q8—416no—
312C2×Q8—44no—
313C2×Q8—82no—
314C2×Q8—82no—
315C2×Q8—88no—
316C2×Q8C22×D414yes—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—48no—
333C22×C4—48no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4—82no—
361C22×C4#12114yes—
362C22×C4#12118yes—
363C22×C4#13714yes—
364C22×C4#13714yes—
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D412yes—
370C22×C4C22×D414yes—
371C22×C4C22×D412yes—
372C22×C4C22×D412yes—
373C22×C4C2×#4114yes—
374C22×C4C2×#4112yes—
375C22×C4C2511yescharacteristic
376Q16—816no—
377M16—88no—
378C2×C8—28no—
379C2×C8—28no—
380C2×C8—28no—
381C4⋊C4—24no—
382C4⋊C4—216no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—48no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4C22×D414yes—
404C4⋊C4H(5,𝔽2)14yes—
405C22⋊C4—216no—
406C22⋊C4—28no—
407C22⋊C4—28no—
408C22⋊C4—48no—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C42—24no—
412C42—24no—
413C42—42no—
414C42—42no—
415C42—48no—
416C42—48no—
417C42—48no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—28no—
429C23—44no—
430C23—44no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23C2×#12114yes—
436C23C2×#12112yes—
437C23C2×#13712yes—
438C23C2×#13712yes—
439C23C23×D411yescharacteristic
440Q8—28no—
441Q8—48no—
442Q8—44no—
443Q8—48no—
444Q8—48no—
445Q8—88no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—48no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—48no—
472C2×C4—48no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—48no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—84no—
492C2×C4—88no—
493C2×C4—88no—
494C2×C4C2×#12114yes—
495C2×C4C2×#12112yes—
496C2×C4C2×#13714yes—
497C2×C4C2×#13714yes—
498C2×C4C23×D411yescharacteristic
499C2×C4C22×#4111yescharacteristic
500C2×C4C2×H(5,𝔽2)12yes—
501C2×C4#58512yes—
502C8—48no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—88no—
517C22#303914yes—
518C22C22×#12111yescharacteristic
519C22#348411yescharacteristic
520C22#348511yescharacteristic
521C22C22×#13711yescharacteristic
522C22C22×#13711yescharacteristic
523C22#358611yescharacteristic
524C22#358611yescharacteristic
525C4—24no—
526C4—24no—
527C4—22no—
528C4—24no—
529C4—42no—
530C4—42no—
531C4—44no—
532C4—44no—
533C4—44no—
534C4—84no—
535C4—82no—
536C4—84no—
537C4#345912yes—
538C4#349612yes—
539C2—24no—
540C2—44no—
541C2#3498411yescharacteristic
542C2C2×#303911yescharacteristic
543C2#6042411yescharacteristic
544C2#6157311yescharacteristic
545C2#6158611yescharacteristic
546C2#6170511yescharacteristic
547C2#6172311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #33108C2semidirect
5. #32906C2semidirect
6. #29756C2semidirect
7. #29715C2semidirect
8. #25219C2semidirect
9. #25095C2semidirect
20. C2×#462C22semidirect
44. #2084C22semidirect
45. #2072C22semidirect
46. #2066C22semidirect
47. #2056C22semidirect
48. #2043C22semidirect
49. #2016C22semidirect
50. #1981C22semidirect
51. #1948C22semidirect
52. #1927C22semidirect
53. #1878C22semidirect
58. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
#350112
C2×#23622
#331082
#329062
#297562
#297152
#252192
#250952
#219342
#209112