Finite Group Databasebeta

← #6510819

Subgroups — #6510819

Order 512.

2378 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic, complemented
2C2×#3187C211yesmaximal, characteristic
3C22×#484C211yesmaximal, characteristic, complemented
4C22×#481C211yesmaximal, characteristic
5#35019C211yesmaximal, characteristic
6C2×#2502C211yesmaximal, characteristic, complemented
7#28347C214yesmaximal, complemented
8#28342C214yesmaximal, complemented
9#28315C214yesmaximal, complemented
10#27938C214yesmaximal, complemented
11#27933C214yesmaximal, complemented
12#27913C214yesmaximal, complemented
13#20526C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18#3189—28no—
19#3187—28no—
20C2×#484—28no—
21C2×#484C2214yes—
22C2×#484C2212yescomplemented
23C2×#484C2214yescomplemented
24C2×#484C2212yescomplemented
25C2×#481—28no—
26C2×#481C2212yes—
27C2×#481C2214yes—
28C2×#481C2214yes—
29C2×#483C2212yes—
30C2×#467C2214yes—
31C2×#465C2214yes—
32C2×#451C2212yescomplemented
33C2×#451C2212yescomplemented
34C2×#448C2212yes—
35C2×#450C2212yes—
36C22×#107C2212yescomplemented
37C22×#106C2212yes—
38C22×#103C2212yescomplemented
39C22×#102C2212yes—
40C22×C22⋊C8C2211yescharacteristic
41C22×C22⋊C8C2211yescharacteristic
42#2850C2214yes—
43#2639—22no—
44#2591—22no—
45#2561—22no—
46#2504C2212yes—
47#2503C2212yes—
48#2502—22no—
49#2502—22no—
50#2502C2214yes—
51#2502C2214yescomplemented
52#2436C2212yes—
53#2435C2212yes—
54C2×#398—22no—
55#2308—22no—
56C2×#391C2212yescomplemented
57C2×#385—22no—
58C2×#384C2212yescomplemented
59C2×C4×C22⋊C4C2211yescharacteristic
60#2097C2218yescomplemented
61#2096C2218yescomplemented
62#2093C2218yescomplemented
63#2015C2218yescomplemented
64#1973C2218yescomplemented
65#1971C2218yescomplemented
66#1959C2218yescomplemented
67#1847C2218yescomplemented
68#1752C2214yes—
69#1752C2214yescomplemented
70#1749C2214yescomplemented
71C23×Q8C2311yescharacteristic
72C23×Q8C2311yescharacteristic
73C23×D4—21no—
74C23×D4C2311yescharacteristic
75C24×C4—21no—
76C24×C4C2311yescharacteristic
77C22×Q16C2314yes—
78C22×SD16C2314yes—
79C23×C8C2312yes—
80C23×C8C2312yes—
81C2×#124C2314yes—
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8C2312yes—
91C2×C22⋊Q8C2312yes—
92C2×C22⋊Q8C2312yes—
93C2×C22⋊Q8C2312yes—
94C2×#121—21no—
95C2×#121—21no—
96C2×#121—24no—
97C2×#121—24no—
98C2×#121C2312yes—
99C2×#121C2312yes—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4D412yes—
118C22×C42C2312yes—
119#484—216no—
120#484—216no—
121#484—28no—
122#484—28no—
123#483—28no—
124#481—216no—
125#481—216no—
126#481—28no—
127#467—216no—
128#465—216no—
129#451—28no—
130#451—28no—
131#450—28no—
132#448—28no—
133C2×#107—28no—
134C2×#107C2318yes—
135C2×#107C2314yes—
136C2×#106—28no—
137C2×#106C2314yes—
138C2×#106C2318yes—
139C2×#103—28no—
140C2×#103C2314yes—
141C2×#103C2318yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102—28no—
145C2×#102C2318yes—
146C2×#102C2314yes—
147C2×C22⋊C8D418yes—
148C2×C22⋊C8D418yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×#97C2318yes—
154#400—24no—
155#398—28no—
156#398—28no—
157#397—24no—
158#394—24no—
159#393—24no—
160#391—24no—
161#391C2314yes—
162#391C2318yes—
163#390C2314yes—
164#388—28no—
165#386—28no—
166#386—24no—
167#386—24no—
168#385—28no—
169#385—28no—
170#384—24no—
171#384C2318yes—
172#382C2314yes—
173#381—24no—
174#379—24no—
175C4×C22⋊C4D412yes—
176C4×C22⋊C4D412yes—
177C4×C22⋊C4C2312yes—
178C4×C22⋊C4C2312yes—
179C2×#95—24no—
180C2×#95D412yes—
181#336C2318yes—
182C25—21no—
183C25C2×D411yescharacteristic
184C22×Q8—21no—
185C22×Q8—21no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—44no—
191C22×Q8—44no—
192C22×Q8C2412yes—
193C22×Q8C2412yes—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—21no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—48no—
202C22×D4C2412yes—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—22no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—84no—
214C23×C4C2×D414yes—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C2×Q16—28no—
224C2×Q16—48no—
225C2×SD16—28no—
226C2×SD16—48no—
227C22×C8—44no—
228C22×C8—44no—
229C22×C8C2×D418yes—
230C22×C8C2×D418yes—
231C22×C8C2414yes—
232C22×C8C2414yes—
233#124—48no—
234C22⋊Q8—24no—
235C22⋊Q8—24no—
236C22⋊Q8—24no—
237C22⋊Q8—24no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C22⋊Q8—24no—
242C22⋊Q8—44no—
243C22⋊Q8—44no—
244C22⋊Q8—44no—
245C22⋊Q8—44no—
246#121—24no—
247#121—24no—
248#121—24no—
249#121—24no—
250#121—44no—
251#121—44no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—216no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—88no—
281C2×C4⋊C4C2414yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4C2×D414yes—
298C2×C22⋊C4C2×D414yes—
299C2×C42—44no—
300C2×C42C2×D414yes—
301C2×C42C2414yes—
302#107—216no—
303#107—28no—
304#106—216no—
305#106—28no—
306#103—216no—
307#103—28no—
308#103—28no—
309#103—28no—
310#102—216no—
311#102—28no—
312C22⋊C8C2×D418yes—
313C22⋊C8C2×D418yes—
314C22⋊C8C2×D418yes—
315C22⋊C8C2×D418yes—
316#97—48no—
317#95—216no—
318#95—28no—
319#95—28no—
320#95C2×D414yes—
321#95C2×D414yes—
322C24—22no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24—44no—
327C24—82no—
328C24—82no—
329C24#12714yes—
330C24#13718yes—
331C24C22×D411yescharacteristic
332C24C22×D411yescharacteristic
333C24C22×D411yescharacteristic
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—28no—
339C2×Q8—28no—
340C2×Q8—44no—
341C2×Q8—48no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—82no—
345C2×Q8—84no—
346C2×Q8—84no—
347C2×Q8—84no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—28no—
351C2×D4—48no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—42no—
357C2×D4—84no—
358C2×D4—82no—
359C2×D4—816no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—48no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—82no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—82no—
393C22×C4—816no—
394C22×C4#12714yes—
395C22×C4#12714yes—
396C22×C4#12714yes—
397C22×C4C22×D418yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C2511yescharacteristic
408Q16—88no—
409SD16—88no—
410C2×C8—28no—
411C2×C8—28no—
412C2×C8—48no—
413C2×C8—48no—
414C2×C8C22×D418yes—
415C2×C8C22×D418yes—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—816no—
435C22⋊C4—28no—
436C22⋊C4—28no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C42—28no—
446C42—44no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—44no—
454C23—44no—
455C23—416no—
456C23—44no—
457C23—42no—
458C23—44no—
459C23—44no—
460C23—84no—
461C23—84no—
462C23—84no—
463C23—168no—
464C23#497116yes—
465C23C2×#12712yes—
466C23C2×#12712yes—
467C23C2×#12712yes—
468C23C2×#13714yes—
469C23C2×#13714yes—
470C23C2×#13714yes—
471C23C23×D411yescharacteristic
472Q8—44no—
473Q8—44no—
474Q8—44no—
475Q8—44no—
476Q8—84no—
477Q8—84no—
478D4—44no—
479D4—44no—
480D4—84no—
481D4—88no—
482D4—88no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—416no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—48no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—84no—
512C2×C4—88no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4—88no—
516C2×C4—88no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4C2×#12712yes—
520C2×C4C2×#12712yes—
521C2×C4C2×#12712yes—
522C2×C4C23×D411yescharacteristic
523C2×C4C23×D411yescharacteristic
524C2×C4#58514yes—
525C8—28no—
526C8—28no—
527C22—28no—
528C22—28no—
529C22—28no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—416no—
537C22—88no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—168no—
542C22C2×#49714yes—
543C22C2×#49714yes—
544C22C2×#49714yes—
545C22C22×#12711yescharacteristic
546C22#348512yes—
547C22C22×#13712yes—
548C22#358512yes—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—48no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4—82no—
557C4—82no—
558C4—84no—
559C4#348512yes—
560C4#348512yes—
561C2—24no—
562C2—24no—
563C2—24no—
564C2—162no—
565C2#3546311yescharacteristic
566C2C22×#49711yescharacteristic
567C2#6064411yescharacteristic
568C2#6170914yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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×#3189C2semidirect
3. C22×#484C2semidirect
6. C2×#2502C2semidirect
7. #28347C2semidirect
8. #28342C2semidirect
9. #28315C2semidirect
10. #27938C2semidirect
11. #27933C2semidirect
12. #27913C2semidirect
13. #20526C2semidirect
22. C2×#484C22semidirect
23. C2×#484C22semidirect
24. C2×#484C22semidirect
32. C2×#451C22semidirect
33. C2×#451C22semidirect
36. C22×#107C22semidirect
38. C22×#103C22semidirect
51. #2502C22semidirect
56. C2×#391C22semidirect
58. C2×#384C22semidirect
60. #2097C22semidirect
61. #2096C22semidirect
62. #2093C22semidirect
63. #2015C22semidirect
64. #1973C22semidirect
65. #1971C22semidirect
66. #1959C22semidirect
67. #1847C22semidirect
69. #1752C22semidirect
70. #1749C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
C2×#31872
C22×#4842
C22×#4812
#350192
C2×#25022
#283472
#283422
#283152
#279382
#279332
#279132
#205262