Finite Group Databasebeta

← #6506639

Subgroups — #6506639

Order 512.

3140 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C212yesmaximal, complemented
2C2×#3007C212yesmaximal
3#35011C211yesmaximal, characteristic, complemented
4C2×#2432C211yesmaximal, characteristic
5#28362C218yesmaximal, complemented
6#28313C212yesmaximal, complemented
7#28302C212yesmaximal, complemented
8#25795C218yesmaximal, complemented
9#25730C212yesmaximal, complemented
10#25722C212yesmaximal, complemented
11#20524C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C22×#137C2214yes—
14C22×C4⋊Q8C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18#3195—216no—
19C2×#497C2214yescomplemented
20C2×#494C2214yescomplemented
21C2×#483C2214yes—
22C2×#483C2214yescomplemented
23C2×#470C2218yescomplemented
24#3007—216no—
25C2×#452C2218yes—
26C2×#451C2214yes—
27C2×#451C2214yes—
28C2×#451C2214yescomplemented
29C2×#450C2214yes—
30C2×#417C2214yes—
31C2×#407C2212yes—
32#2857C2212yes—
33#2857C2212yes—
34#2659—22no—
35#2583—24no—
36#2502C2214yescomplemented
37#2434C2214yescomplemented
38#2432—24no—
39#2432C2218yes—
40C2×#394—22no—
41#2315—22no—
42C2×#391C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#385—22no—
45C2×C4×C22⋊C4C2211yescharacteristic
46#2096C2214yescomplemented
47#2094C2214yescomplemented
48#2016C2218yescomplemented
49#2016C2214yescomplemented
50#2015C2218yescomplemented
51#2013C2214yescomplemented
52#1971C2218yescomplemented
53#1968C2218yescomplemented
54#1959C2214yescomplemented
55#1956C2214yescomplemented
56#1753C2218yescomplemented
57#1749C2212yes—
58C4×C22⋊C8C2212yes—
59C22×#41—22no—
60C22×#41—28no—
61C22×#41—44no—
62C22×#41—44no—
63C22×#41C2314yes—
64C23×Q8C2311yescharacteristic
65C23×Q8C2311yescharacteristic
66C23×Q8C2312yes—
67C23×D4—21no—
68C23×D4C2311yescharacteristic
69C24×C4—21no—
70C24×C4C2311yescharacteristic
71C2×#137—216no—
72C2×#137—48no—
73C2×#137—48no—
74C22×Q16C2318yes—
75C22×Q16C2318yes—
76C22×SD16C2318yes—
77C22×M16C2314yes—
78C2×C4⋊Q8—22no—
79C2×C4⋊Q8—44no—
80C2×C4⋊Q8C2314yes—
81C2×C4⋊Q8C2312yes—
82C2×C4⋊Q8C2312yes—
83C2×#124—22no—
84C2×#124—44no—
85C2×#124C2314yes—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8C2314yes—
93C2×C22⋊Q8C2314yes—
94C2×C22⋊Q8C2314yes—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4—22no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—22no—
103C22×C22⋊C4—22no—
104C22×C22⋊C4D412yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#497—48no—
108#494—48no—
109#483—216no—
110#483—216no—
111#470—232no—
112#452—232no—
113#451—216no—
114#451—216no—
115#451—216no—
116#450—216no—
117C2×#106C2318yes—
118#417—216no—
119C2×#103C2318yes—
120C2×#103C2318yes—
121C2×#103C2318yes—
122C2×#102C2318yes—
123#407D4116yes—
124C2×C22⋊C8C2314yes—
125C2×C22⋊C8C2314yes—
126C2×#97C2314yes—
127C2×C4×C8C2314yes—
128#398—28no—
129#395—28no—
130#394—216no—
131#391—22no—
132#391C2314yes—
133#391C2314yes—
134#391C2314yes—
135#390—22no—
136#390C2314yes—
137#389—24no—
138#389—24no—
139#385—28no—
140#385—28no—
141#385—216no—
142#384C2312yes—
143#384C2312yes—
144#380—24no—
145#380—24no—
146C4×C22⋊C4D414yes—
147C4×C22⋊C4C2314yes—
148C2×#95—22no—
149C2×#95D412yes—
150#336C2314yes—
151#336C2314yes—
152C25—21no—
153C25C2×D411yescharacteristic
154C2×#41—28no—
155C2×#41—28no—
156C2×#41—28no—
157C2×#41—28no—
158C2×#41—44no—
159C2×#41—44no—
160C2×#41—48no—
161C2×#41—48no—
162C2×#41—464no—
163C2×#41—48no—
164C22×Q8—24no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—22no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8—44no—
176C22×Q8—44no—
177C22×Q8C2412yes—
178C22×Q8C2414yes—
179C22×Q8C2412yes—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—21no—
183C22×D4—24no—
184C22×D4—42no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4C2412yes—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4C2×D412yes—
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D412yes—
203C23×C4C2×D418yes—
204C23×C4C2×D412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207#137—464no—
208C2×Q16—216no—
209C2×Q16—216no—
210C2×Q16—216no—
211C2×Q16—48no—
212C2×Q16—48no—
213C2×Q16—48no—
214C2×SD16—216no—
215C2×SD16—216no—
216C2×SD16—48no—
217C2×M16—44no—
218C2×M16—44no—
219C2×M16C2×D4116yes—
220C22×C8C2418yes—
221C4⋊Q8—28no—
222C4⋊Q8—44no—
223C4⋊Q8—44no—
224C4⋊Q8—44no—
225C4⋊Q8—48no—
226#124—28no—
227#124—44no—
228#124—48no—
229C22⋊Q8—28no—
230C22⋊Q8—28no—
231C22⋊Q8—28no—
232C22⋊Q8—28no—
233C22⋊Q8—28no—
234C22⋊Q8—28no—
235C22⋊Q8—48no—
236C22⋊Q8—48no—
237C22⋊Q8—48no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—48no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4C2414yes—
255C2×C4⋊C4C2412yes—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4C2×D418yes—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2412yes—
272C2×C42C2412yes—
273#106—216no—
274#103—216no—
275#103—216no—
276#103—216no—
277#102—216no—
278C22⋊C8C2×D4116yes—
279C22⋊C8C2×D4116yes—
280#97—44no—
281C4×C8—44no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95C2×D418yes—
286C24—24no—
287C24—24no—
288C24—22no—
289C24—41no—
290C24—41no—
291C24—42no—
292C24—42no—
293C24—44no—
294C24—42no—
295C24—44no—
296C24#12714yes—
297C24C22×D412yes—
298C24C22×D411yescharacteristic
299#41—232no—
300#41—416no—
301#41—416no—
302#41—416no—
303#41—464no—
304C2×Q8—24no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—28no—
314C2×Q8—28no—
315C2×Q8—416no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—416no—
320C2×Q8—42no—
321C2×Q8—44no—
322C2×Q8—42no—
323C2×Q8—82no—
324C2×Q8—82no—
325C2×Q8—88no—
326C2×Q8—88no—
327C2×Q8H(5,𝔽2)14yes—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—416no—
333C2×D4—48no—
334C2×D4—42no—
335C2×D4—48no—
336C2×D4—432no—
337C2×D4—48no—
338C2×D4—42no—
339C2×D4H(5,𝔽2)14yes—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—416no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—48no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—48no—
362C22×C4—42no—
363C22×C4—432no—
364C22×C4—42no—
365C22×C4—48no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—84no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4#12714yes—
377C22×C4#12718yes—
378C22×C4#13718yescomplemented
379C22×C4C22×D414yes—
380C22×C4C22×D412yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D414yes—
383C22×C4C22×D414yes—
384C22×C4C22×D414yes—
385C22×C4C22×D414yes—
386C22×C4C2511yescharacteristic
387Q16—432no—
388Q16—816no—
389SD16—432no—
390M16—416no—
391C2×C8—28no—
392C2×C8—28no—
393C2×C8C22×D4116yes—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—416no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—84no—
404C4⋊C4—84no—
405C4⋊C4—84no—
406C4⋊C4—84no—
407C22⋊C4—216no—
408C22⋊C4—416no—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C22⋊C4—48no—
412C22⋊C4—416no—
413C42—24no—
414C42—24no—
415C42—42no—
416C42—42no—
417C23—22no—
418C23—21no—
419C23—21no—
420C23—24no—
421C23—22no—
422C23—22no—
423C23—24no—
424C23—28no—
425C23—24no—
426C23—44no—
427C23—44no—
428C23—48no—
429C23—416no—
430C23—44no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—84no—
435C23—84no—
436C23—88no—
437C23—84no—
438C23—84no—
439C23C2×#12714yes—
440C23C2×#12712yes—
441C23C2×#13714yes—
442C23C23×D411yescharacteristic
443Q8—28no—
444Q8—48no—
445Q8—44no—
446Q8—48no—
447Q8—48no—
448Q8—48no—
449Q8—88no—
450Q8—84no—
451D4—28no—
452D4—48no—
453D4—432no—
454D4—44no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—44no—
469C2×C4—48no—
470C2×C4—48no—
471C2×C4—432no—
472C2×C4—48no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—84no—
486C2×C4—84no—
487C2×C4—88no—
488C2×C4—88no—
489C2×C4—84no—
490C2×C4C2×#12714yes—
491C2×C4C2×#12712yes—
492C2×C4C2×#13718yes—
493C2×C4C23×D412yes—
494C2×C4C2×H(5,𝔽2)12yes—
495C2×C4C2×H(5,𝔽2)12yes—
496C8—216no—
497C22—22no—
498C22—24no—
499C22—24no—
500C22—22no—
501C22—24no—
502C22—24no—
503C22—24no—
504C22—44no—
505C22—44no—
506C22—44no—
507C22—48no—
508C22—88no—
509C22—816no—
510C22—84no—
511C22—82no—
512C22—82no—
513C22#315214yes—
514C22C22×#12711yescharacteristic
515C22#348411yescharacteristic
516C22#348411yescharacteristic
517C22C22×#13712yes—
518C22#358612yes—
519C4—24no—
520C4—22no—
521C4—28no—
522C4—44no—
523C4—44no—
524C4—82no—
525C4—84no—
526C4—82no—
527C4—84no—
528C4#348414yes—
529C2—24no—
530C2—44no—
531C2—84no—
532C2#3539011yescharacteristic
533C2C2×#315211yescharacteristic
534C2#6064511yescharacteristic
535C2#6157312yes—
536C2#6158612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
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
1. C2×#3195C2semidirect
3. #35011C2semidirect
5. #28362C2semidirect
6. #28313C2semidirect
7. #28302C2semidirect
8. #25795C2semidirect
9. #25730C2semidirect
10. #25722C2semidirect
11. #20524C2semidirect
19. C2×#497C22semidirect
20. C2×#494C22semidirect
22. C2×#483C22semidirect
23. C2×#470C22semidirect
28. C2×#451C22semidirect
36. #2502C22semidirect
37. #2434C22semidirect
46. #2096C22semidirect
47. #2094C22semidirect
48. #2016C22semidirect
49. #2016C22semidirect
50. #2015C22semidirect
51. #2013C22semidirect
52. #1971C22semidirect
53. #1968C22semidirect
54. #1959C22semidirect
55. #1956C22semidirect
56. #1753C22semidirect
378. C22×C4#137semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
C2×#30072
#350112
C2×#24322
#283622
#283132
#283022
#257952
#257302
#257222
#205242