Finite Group Databasebeta

← #6632012

Subgroups — #6632012

Order 512.

2452 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2C2×#2409C211yesmaximal, characteristic
3#32882C212yesmaximal, complemented
4#31263C214yesmaximal, complemented
5#30507C214yesmaximal, complemented
6#30407C218yesmaximal, complemented
7#29895C212yesmaximal, complemented
8#27586C212yesmaximal, complemented
9#24380C214yesmaximal, complemented
10#23934C212yesmaximal, complemented
11#21738C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C2×#533C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C22×C4×D4C2211yescharacteristic
16C2×#467C2218yescomplemented
17C2×#451C2214yes—
18C2×#451C2218yescomplemented
19C22×C22⋊C8C2212yes—
20#2848C2212yes—
21#2848C2212yes—
22#2626—21no—
23#2626—21no—
24#2580—22no—
25#2566—22no—
26#2563C2212yes—
27#2542C2212yes—
28#2538C2214yes—
29#2409—22no—
30#2409—22no—
31#2409C2214yes—
32#2409C2214yes—
33C2×#395—21no—
34C2×#393—21no—
35#2317—21no—
36#2309—21no—
37C2×#391C2212yes—
38C2×#386—21no—
39C2×#385—21no—
40#2285C2211yescharacteristic
41C2×#380C2211yescharacteristic
42#2095C2214yescomplemented
43#2076C2218yescomplemented
44#2072C2214yescomplemented
45#2036C2218yescomplemented
46#2036C2218yescomplemented
47#2024C2214yescomplemented
48#2016C2214yescomplemented
49#2001C2218yescomplemented
50#1928C2218yescomplemented
51#1892C2218yescomplemented
52#1878C2218yescomplemented
53#1872C2214yes—
54#1869C2214yes—
55#1869C2214yes—
56#1852C2214yes—
57#1817C2212yes—
58#1787C2214yes—
59#1787C2218yescomplemented
60#1779C2212yes—
61C23×Q8—21no—
62C23×Q8—22no—
63C23×Q8—44no—
64C23×Q8—48no—
65C23×Q8C2312yes—
66C23×Q8C2314yes—
67C23×D4D411yescharacteristic
68C24×C4—21no—
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C22×Q16C2318yes—
72C23×C8C2314yes—
73#533—44no—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×#125—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—48no—
80C2×C22⋊Q8C2312yes—
81C2×C22⋊Q8C2318yes—
82C2×C22⋊Q8C2314yes—
83C2×C22⋊Q8C2314yes—
84C2×C4×Q8—21no—
85C2×C4×Q8C2312yes—
86C2×C4×D4—44no—
87C2×C4×D4D412yes—
88C2×C4×D4D412yes—
89C2×C4×D4C2312yes—
90C2×C4×D4C2312yes—
91C2×#117—21no—
92C2×#117—21no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4D411yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4—22no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4D411yescharacteristic
99C22×C22⋊C4D411yescharacteristic
100C22×C42C2311yescharacteristic
101#467—416no—
102#451—48no—
103#451—416no—
104C2×#107C2318yes—
105C2×#103C2318yes—
106C2×#103C2318yes—
107C2×#103C2318yes—
108C2×#103C2318yes—
109C2×C22⋊C8—44no—
110C2×C22⋊C8C2314yes—
111C2×C22⋊C8C2318yes—
112C2×C22⋊C8C2314yes—
113#399—22no—
114#399—22no—
115#396—24no—
116#396—24no—
117#395—24no—
118#395—24no—
119#393—22no—
120#393—22no—
121#393—24no—
122#393—24no—
123#391—22no—
124#391—22no—
125#391C2314yes—
126#391C2314yes—
127#390—22no—
128#390—22no—
129#390—24no—
130#389—24no—
131#386—24no—
132#386—24no—
133#385—24no—
134#385—24no—
135#385—22no—
136#385—22no—
137#385—24no—
138#385—24no—
139#384—22no—
140#384C2312yes—
141#382—22no—
142#382—21no—
143#380—21no—
144#380—22no—
145#380C2314yes—
146#380C2314yes—
147#380C2314yes—
148#380C2312yes—
149#379—22no—
150#379—22no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4C2312yes—
153C2×#95—21no—
154C2×#95—22no—
155C2×#95—22no—
156#340C2318yes—
157#340C2318yes—
158#336C2314yes—
159#336C2314yes—
160C25C2×D411yescharacteristic
161C25C2×D411yescharacteristic
162C22×Q8—28no—
163C22×Q8—22no—
164C22×Q8—24no—
165C22×Q8—21no—
166C22×Q8—216no—
167C22×Q8—28no—
168C22×Q8—44no—
169C22×Q8—48no—
170C22×Q8—44no—
171C22×Q8—44no—
172C22×Q8—48no—
173C22×Q8—44no—
174C22×Q8—48no—
175C22×Q8—416no—
176C22×Q8—464no—
177C22×Q8C2412yes—
178C22×Q8C2414yes—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4Q1618yescomplemented
182C22×D4C2×D412yes—
183C22×D4C2×D412yes—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—48no—
192C23×C4—44no—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D412yes—
198C23×C4C2×D412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C2×Q16—216no—
203C2×Q16—48no—
204C2×Q16—48no—
205C22×C8—44no—
206C22×C8—48no—
207C22×C8C2418yes—
208C4⋊Q8—42no—
209C4⋊Q8—42no—
210#125—44no—
211C22⋊Q8—216no—
212C22⋊Q8—28no—
213C22⋊Q8—48no—
214C22⋊Q8—44no—
215C22⋊Q8—416no—
216C22⋊Q8—48no—
217C22⋊Q8—48no—
218C4×Q8—44no—
219C4×Q8—44no—
220C4×D4—44no—
221C4×D4—44no—
222C4×D4—42no—
223C4×D4—42no—
224C4×D4—42no—
225C4×D4—42no—
226#117—44no—
227#117—44no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2414yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2414yes—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—48no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4C2×D412yes—
260C2×C22⋊C4C2×D412yes—
261C2×C22⋊C4C2×D412yes—
262C2×C22⋊C4C2×D412yes—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2411yescharacteristic
269C2×C42C2411yescharacteristic
270#107—48no—
271#103—216no—
272#103—48no—
273#103—48no—
274#103—48no—
275C22⋊C8—28no—
276C22⋊C8—44no—
277C22⋊C8—48no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—22no—
284#95—22no—
285#95—24no—
286#95—24no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24C2×Q1614yes—
296C24C2×Q1614yes—
297C24C22×D412yes—
298C24C22×D412yes—
299C24C22×D411yescharacteristic
300C2×Q8—28no—
301C2×Q8—24no—
302C2×Q8—28no—
303C2×Q8—28no—
304C2×Q8—28no—
305C2×Q8—24no—
306C2×Q8—416no—
307C2×Q8—432no—
308C2×Q8—44no—
309C2×Q8—48no—
310C2×Q8—42no—
311C2×Q8—416no—
312C2×Q8—432no—
313C2×Q8—464no—
314C2×Q8—432no—
315C2×Q8—416no—
316C2×Q8—44no—
317C2×Q8—82no—
318C2×Q8H(5,𝔽2)18yes—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—416no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4C2×Q1618yes—
325C2×D4C2×Q1618yes—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—28no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—28no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—416no—
352C22×C4—416no—
353C22×C4—44no—
354C22×C4—48no—
355C22×C4—432no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—88no—
361C22×C4C2×Q1614yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D412yes—
365C22×C4C22×D412yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4H(5,𝔽2)14yes—
370C22×C4C2511yescharacteristic
371Q16—816no—
372C2×C8—28no—
373C2×C8—28no—
374C2×C8—28no—
375C2×C8—88no—
376C4⋊C4—28no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—416no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—82no—
390C4⋊C4—84no—
391C4⋊C4—84no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—416no—
397C22⋊C4—48no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C22⋊C4—84no—
403C22⋊C4—84no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—44no—
409C42—44no—
410C23—22no—
411C23—22no—
412C23—24no—
413C23—24no—
414C23—28no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—28no—
422C23—48no—
423C23—48no—
424C23—44no—
425C23—44no—
426C23—48no—
427C23—82no—
428C23—82no—
429C23—84no—
430C23#53412yes—
431C23C22×Q1614yes—
432C23C22×Q1614yes—
433C23C22×Q1612yes—
434C23#57512yes—
435C23C23×D411yescharacteristic
436Q8—216no—
437Q8—416no—
438Q8—432no—
439Q8—48no—
440Q8—48no—
441Q8—84no—
442D4—48no—
443D4—48no—
444C2×C4—28no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—28no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—28no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—48no—
461C2×C4—44no—
462C2×C4—416no—
463C2×C4—432no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—84no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—88no—
484C2×C4#53412yes—
485C2×C4C22×Q1612yes—
486C2×C4C22×Q1612yes—
487C2×C4C2×H(5,𝔽2)12yes—
488C2×C4C2×H(5,𝔽2)14yes—
489C8—48no—
490C22—22no—
491C22—22no—
492C22—24no—
493C22—24no—
494C22—24no—
495C22—24no—
496C22—28no—
497C22—28no—
498C22—416no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—44no—
504C22—44no—
505C22—88no—
506C22#318914yes—
507C22C2×#53411yescharacteristic
508C22#348412yes—
509C22C23×Q1611yescharacteristic
510C22C2×#57511yescharacteristic
511C22#358612yes—
512C4—22no—
513C4—22no—
514C4—22no—
515C4—41no—
516C4—41no—
517C4—42no—
518C4—42no—
519C4—48no—
520C4—82no—
521C4—82no—
522C4—84no—
523C4—84no—
524C4#353214yes—
525C2—24no—
526C2—44no—
527C2—44no—
528C2#3670211yescharacteristic
529C2C2×#318911yescharacteristic
530C2#6075411yescharacteristic
531C2#6156312yes—
532C2#6157812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32882C2semidirect
4. #31263C2semidirect
5. #30507C2semidirect
6. #30407C2semidirect
7. #29895C2semidirect
8. #27586C2semidirect
9. #24380C2semidirect
10. #23934C2semidirect
16. C2×#467C22semidirect
18. C2×#451C22semidirect
42. #2095C22semidirect
43. #2076C22semidirect
44. #2072C22semidirect
45. #2036C22semidirect
46. #2036C22semidirect
47. #2024C22semidirect
48. #2016C22semidirect
49. #2001C22semidirect
50. #1928C22semidirect
51. #1892C22semidirect
52. #1878C22semidirect
59. #1787C22semidirect
181. C22×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#351722
C2×#24092
#328822
#312632
#305072
#304072
#298952
#275862
#243802
#239342
#217382