Finite Group Databasebeta

← #182004

Subgroups — #182004

Order 512.

1660 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2275C211yesmaximal, characteristic
2#20524C212yesmaximal, complemented
3C2×#1465C212yesmaximal
4#8153C212yesmaximal, complemented
5#8152C212yesmaximal, complemented
6C4×#324C212yesmaximal, complemented
7C4×#323C212yesmaximal, complemented
8#7909C212yesmaximal, complemented
9C22×C4×D4C411yescharacteristic
10C22×C4×D4C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#407C2214yes—
13#2275—28no—
14C2×C4×C4⋊C4C411yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×C4×C22⋊C4C411yescharacteristic
18C2×C4×C22⋊C4C411yescharacteristic
19C2×C4×C22⋊C4C411yescharacteristic
20C2×C4×C22⋊C4C2211yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×#376C411yescharacteristic
24C22×#95C411yescharacteristic
25C22×#95C2211yescharacteristic
26#1753—28no—
27#1749C2214yes—
28C4×C22⋊C8C2214yes—
29#1465C4116yescomplemented
30C2×#324C2214yes—
31C2×#323C2214yes—
32#1280C2214yes—
33#1280C2214yes—
34C23×D4C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4C2×C411yescharacteristic
38C24×C4C2×C411yescharacteristic
39C24×C4C2×C411yescharacteristic
40C24×C4C2311yescharacteristic
41C22×M16—24no—
42C2×C4×D4—24no—
43C2×C4×D4—24no—
44C2×C4×D4—24no—
45C2×C4×D4—24no—
46C2×C4×D4—24no—
47C2×C4×D4—24no—
48C2×C4×D4—24no—
49C2×C4×D4—24no—
50C2×C4×D4C2×C418yes—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2311yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69#407—28no—
70#407C2×C4116yescomplemented
71C2×C22⋊C8—24no—
72C2×C22⋊C8C2318yes—
73C2×#97—24no—
74C2×C4×C8—24no—
75C4×C4⋊C4—24no—
76C4×C4⋊C4—24no—
77C4×C4⋊C4—24no—
78C4×C22⋊C4—24no—
79C4×C22⋊C4—24no—
80C4×C22⋊C4—24no—
81C4×C22⋊C4—24no—
82C4×C22⋊C4—24no—
83C4×C22⋊C4D418yes—
84#376—24no—
85C2×#95—24no—
86C2×#95—24no—
87C2×#95C2×C411yescharacteristic
88C2×#95C2×C411yescharacteristic
89C2×#95C2×C411yescharacteristic
90C2×#95C2×C411yescharacteristic
91C2×#95C2×C411yescharacteristic
92C2×#95C2×C411yescharacteristic
93C2×#95C2×C411yescharacteristic
94C2×#95C2×C411yescharacteristic
95C2×#95C2311yescharacteristic
96C2×#95C2311yescharacteristic
97#336—24no—
98#336—24no—
99#324C2×C4116yescomplemented
100#323C2×C4116yescomplemented
101C25C22×C411yescharacteristic
102C25C22×C411yescharacteristic
103C22×D4—22no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4C4214yes—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—81no—
124C23×C4—81no—
125C23×C4—81no—
126C23×C4—81no—
127C23×C4—81no—
128C23×C4—81no—
129C23×C4—81no—
130C23×C4—81no—
131C23×C4—81no—
132C23×C4—81no—
133C23×C4—81no—
134C23×C4—81no—
135C23×C4C4214yes—
136C23×C4M1614yescomplemented
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C414yes—
150C23×C4C2×D414yes—
151C23×C4#4114yes—
152C23×C4C2411yescharacteristic
153C2×M16—216no—
154C2×M16—84no—
155C2×M16—84no—
156C22×C8—28no—
157C4×D4—232no—
158C4×D4—232no—
159C4×D4—216no—
160C4×D4—216no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4M1614yescomplemented
169C2×C4⋊C4C22×C414yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4C4214yes—
184C2×C22⋊C4C22⋊C414yes—
185C2×C22⋊C4C22⋊C414yes—
186C2×C22⋊C4M1614yescomplemented
187C2×C22⋊C4C22×C414yes—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—81no—
200C2×C42—81no—
201C2×C42—81no—
202C2×C42—81no—
203C2×C42—81no—
204C2×C42—81no—
205C2×C42C4214yes—
206C2×C42M1614yescomplemented
207C2×C42C2×D414yes—
208C2×C42C2×D414yes—
209C22⋊C8—216no—
210C22⋊C8—28no—
211C22⋊C8C22×C4116yes—
212#97—84no—
213C4×C8—84no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95C22⋊C414yes—
223#95C22⋊C414yes—
224C24—21no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—21no—
230C24—21no—
231C24—21no—
232C24—41no—
233C24—41no—
234C24—41no—
235C24—41no—
236C24—41no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24C2×C4212yes—
241C24C2×C4212yes—
242C24C2×C22⋊C412yes—
243C24#11712yes—
244C24C2×M1612yes—
245C24C2×M1612yes—
246C24C23×C411yescharacteristic
247C2×D4—28no—
248C2×D4—28no—
249C2×D4—28no—
250C2×D4—28no—
251C2×D4—28no—
252C2×D4—28no—
253C2×D4—28no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—28no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—84no—
312C22×C4—84no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4—84no—
320C22×C4—84no—
321C22×C4—84no—
322C22×C4—84no—
323C22×C4—84no—
324C22×C4—84no—
325C22×C4—84no—
326C22×C4—84no—
327C22×C4C2×C4212yes—
328C22×C4C2×C4212yes—
329C22×C4C2×C4212yes—
330C22×C4C2×C4212yes—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4#11712yes—
337C22×C4C4×D418yes—
338C22×C4C4×D418yes—
339C22×C4C2×M1612yes—
340C22×C4C2×M1612yes—
341C22×C4C2×M1612yes—
342C22×C4C2×M1612yes—
343C22×C4C2×M1612yes—
344C22×C4C2×M1612yes—
345C22×C4C2×M1612yes—
346C22×C4C2×M1612yes—
347C22×C4C2×M1612yes—
348C22×C4C2×M1612yes—
349C22×C4C23×C412yes—
350C22×C4C22×D412yes—
351C22×C4C2×#4112yes—
352M16—816no—
353C2×C8—216no—
354C2×C8—48no—
355C2×C8—48no—
356C4⋊C4—28no—
357C4⋊C4—28no—
358C4⋊C4—28no—
359C22⋊C4—28no—
360C22⋊C4—28no—
361C22⋊C4—28no—
362C22⋊C4—28no—
363C22⋊C4—28no—
364C22⋊C4—28no—
365C42—28no—
366C42—28no—
367C42—28no—
368C42—44no—
369C42—44no—
370C42—82no—
371C42—82no—
372C42—82no—
373C42—82no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23C4×C22⋊C414yes—
423C23C4×M1614yes—
424C23C22×C4211yescharacteristic
425C23C22×C22⋊C411yescharacteristic
426C23C2×#11711yescharacteristic
427C23C22×M1611yescharacteristic
428C23C22×M1611yescharacteristic
429C23C22×M1611yescharacteristic
430C23C22×M1611yescharacteristic
431D4—232no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—48no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—48no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—88no—
497C2×C4—88no—
498C2×C4—88no—
499C2×C4—88no—
500C2×C4—88no—
501C2×C4C4×C22⋊C414yes—
502C2×C4C4×M1614yes—
503C2×C4#40714yes—
504C2×C4#40714yes—
505C2×C4C2×C4×D414yes—
506C8—416no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—24no—
520C22—24no—
521C22—24no—
522C22—24no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22C2×C4×C22⋊C411yescharacteristic
542C22C2×C4×M1611yescharacteristic
543C22C2×C4×M1611yescharacteristic
544C22C2×#40711yescharacteristic
545C22C2×#40711yescharacteristic
546C22#288311yescharacteristic
547C22#288311yescharacteristic
548C22#288811yescharacteristic
549C22#288911yescharacteristic
550C22#292011yescharacteristic
551C22#292011yescharacteristic
552C4—24no—
553C4—24no—
554C4—24no—
555C4—24no—
556C4—24no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—42no—
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—44no—
570C2—44no—
571C2#1736711yescharacteristic
572C2#1736811yescharacteristic
573C2#1736911yescharacteristic
574C2#1737011yescharacteristic
575C2C4×#40711yescharacteristic
576C2#2039211yescharacteristic
577C2#2039711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20524C2semidirect
4. #8153C2semidirect
5. #8152C2semidirect
6. C4×#324C2semidirect
7. C4×#323C2semidirect
8. #7909C2semidirect
29. #1465C4semidirect
70. #407C2×C4semidirect
99. #324C2×C4semidirect
100. #323C2×C4semidirect
136. C23×C4M16semidirect
168. C2×C4⋊C4M16semidirect
186. C2×C22⋊C4M16semidirect
206. C2×C42M16semidirect

Maximal subgroups

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

subgroupindex
C2×#22752
#205242
C2×#14652
#81532
#81522
C4×#3242
C4×#3232
#79092