Finite Group Databasebeta

← #354457

Subgroups — #354457

Order 512.

2040 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#518C211yesmaximal, characteristic
2C2×#1867C212yesmaximal
3C2×C4×#99C211yesmaximal, characteristic
4#13784C211yesmaximal, characteristic
5#13782C212yesmaximal, complemented
6#10918C214yesmaximal, complemented
7#10745C212yesmaximal
8C4×#342C212yesmaximal, complemented
9C2×#518C2212yes—
10C22×C4×D4C2211yescharacteristic
11C22×#99C2211yescharacteristic
12C22×#99C2211yescharacteristic
13C2×#386C2212yes—
14C2×#385C2214yes—
15#2275—22no—
16#2275—22no—
17#2274—21no—
18#2273—21no—
19#2273C2212yes—
20C42×D4—24no—
21C4×#117—21no—
22C2×C4×C22⋊C4—21no—
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2212yes—
25C2×C4×C22⋊C4C2212yes—
26C22×#95C2211yescharacteristic
27#1867C4116yescomplemented
28C4×#99C418yescomplemented
29C2×#342C2214yes—
30C2×#342C2214yes—
31C2×#342C2212yes—
32#1437—21no—
33#1436—24no—
34#1436—21no—
35C4⋊#95—22no—
36#1434—21no—
37#1434—22no—
38C22⋊#95—21no—
39C22⋊#95C2212yes—
40C22⋊#95C2214yes—
41C23×D4C2311yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2312yes—
44C24×C4C2311yescharacteristic
45#518—24no—
46#518—28no—
47#518C2×C418yes—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—24no—
53C2×C4×D4C2×C414yes—
54C2×C4×D4Q814yes—
55C2×#117—22no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4D411yescharacteristic
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4D411yescharacteristic
61C22×C22⋊C4C2312yes—
62C22×C22⋊C4C2312yes—
63C22×C22⋊C4C2311yescharacteristic
64C22×C22⋊C4C2314yes—
65C22×C22⋊C4C2312yes—
66C22×C42—22no—
67C22×C42—22no—
68C22×C42—21no—
69C22×C42D411yescharacteristic
70C22×C42D414yes—
71C2×#99—24no—
72C2×#99C2×C414yes—
73C2×#99C2×C418yes—
74C2×#99C2×C414yes—
75#386—24no—
76#386C2×C418yes—
77#385—28no—
78#385C2×C4116yescomplemented
79C4×C4⋊C4—22no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—24no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—24no—
87C4×C22⋊C4—24no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4C2×C418yescomplemented
99#376—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—24no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—24no—
112C2×#95—22no—
113C2×#95—24no—
114C2×#95D412yes—
115C2×#95D412yes—
116C43—22no—
117#342—28no—
118#342C2×C4116yescomplemented
119#342C2×C418yescomplemented
120C25C2×D411yescharacteristic
121C25C2411yescharacteristic
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—24no—
125C22×D4C22×C412yes—
126C22×D4C22×C414yes—
127C22×D4C2×Q812yes—
128C22×D4#4114yes—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—82no—
142C23×C4—82no—
143C23×C4—82no—
144C23×C4—82no—
145C23×C4C4⋊C414yescomplemented
146C23×C4C22×C414yes—
147C23×C4C22×C412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D414yes—
152C23×C4C2×D414yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×Q812yes—
158C23×C4#4114yes—
159C4×D4—216no—
160C4×D4—216no—
161C4×D4—28no—
162C4×D4—28no—
163C4×D4—216no—
164C4×D4—216no—
165C4×D4—28no—
166C4×D4—28no—
167#117—28no—
168#117—28no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—216no—
174C2×C4⋊C4—84no—
175C2×C4⋊C4—84no—
176C2×C4⋊C4C4⋊C414yescomplemented
177C2×C22⋊C4—24no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4C4⋊C414yescomplemented
200C2×C22⋊C4C22×C414yes—
201C2×C22⋊C4C22×C414yes—
202C2×C22⋊C4C22×C412yes—
203C2×C22⋊C4C22×C418yes—
204C2×C22⋊C4C22×C414yes—
205C2×C22⋊C4C22×C418yes—
206C2×C22⋊C4C2×Q812yes—
207C2×C22⋊C4#4118yes—
208C2×C42—24no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—28no—
217C2×C42—24no—
218C2×C42—42no—
219C2×C42—84no—
220C2×C42—84no—
221C2×C42—84no—
222C2×C42—82no—
223C2×C42C4⋊C414yescomplemented
224#99—28no—
225#99C42116yescomplemented
226#95—22no—
227#95—22no—
228#95—24no—
229#95—28no—
230#95—216no—
231#95—28no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—41no—
237#95—42no—
238#95—41no—
239#95—42no—
240#95—82no—
241#95—84no—
242#95—84no—
243#95—82no—
244C24—21no—
245C24—21no—
246C24—22no—
247C24—42no—
248C24—41no—
249C24—41no—
250C24—41no—
251C24—42no—
252C24—41no—
253C24C2×C4⋊C412yes—
254C24C4×D414yes—
255C24C23×C411yescharacteristic
256C24C23×C412yes—
257C24C22×D411yescharacteristic
258C24C22×D412yes—
259C24C22×Q811yescharacteristic
260C24C2×#4112yes—
261C24C2×#4112yes—
262C24C2×#4114yes—
263C2×D4—24no—
264C2×D4—28no—
265C2×D4—24no—
266C2×D4—28no—
267C2×D4—24no—
268C2×D4—28no—
269C2×D4—28no—
270C2×D4—24no—
271C2×D4C2×C4214yes—
272C2×D4#11714yes—
273C2×D4C4×Q818yes—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—28no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—28no—
284C22×C4—28no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—28no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—41no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—84no—
334C22×C4—82no—
335C22×C4—84no—
336C22×C4—84no—
337C22×C4—84no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—84no—
343C22×C4—88no—
344C22×C4C2×C4⋊C412yes—
345C22×C4C2×C4⋊C412yes—
346C22×C4C2×C4⋊C412yes—
347C22×C4C2×C4⋊C412yes—
348C22×C4C2×C4⋊C412yes—
349C22×C4C4×D414yes—
350C22×C4C4×D418yes—
351C22×C4C4×D418yes—
352C22×C4C4×D414yes—
353C22×C4C4×D414yes—
354C22×C4#12011yescharacteristic
355C22×C4#12012yes—
356C22×C4#12011yescharacteristic
357C22×C4C22⋊Q814yes—
358C22×C4C22⋊Q814yes—
359C22×C4#12312yes—
360C22×C4#12312yes—
361C4⋊C4—28no—
362C4⋊C4—28no—
363C4⋊C4—28no—
364C4⋊C4—816no—
365C22⋊C4—216no—
366C22⋊C4—28no—
367C22⋊C4—24no—
368C22⋊C4—28no—
369C22⋊C4—24no—
370C22⋊C4—28no—
371C22⋊C4—24no—
372C22⋊C4—28no—
373C22⋊C4—28no—
374C22⋊C4—28no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—28no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—88no—
381C22⋊C4C2×C4218yescomplemented
382C42—216no—
383C42—24no—
384C42—24no—
385C42—48no—
386C42—84no—
387C42—82no—
388C42—82no—
389C42—88no—
390C42—84no—
391C42—84no—
392C23—24no—
393C23—22no—
394C23—22no—
395C23—21no—
396C23—22no—
397C23—21no—
398C23—22no—
399C23—24no—
400C23—22no—
401C23—24no—
402C23—41no—
403C23—42no—
404C23—41no—
405C23—42no—
406C23—44no—
407C23—42no—
408C23—41no—
409C23—42no—
410C23—41no—
411C23—42no—
412C23—44no—
413C23—42no—
414C23—44no—
415C23—42no—
416C23—42no—
417C23—44no—
418C23—42no—
419C23—41no—
420C23—41no—
421C23—81no—
422C23—81no—
423C23—82no—
424C23—82no—
425C23C4×C4⋊C414yes—
426C23C22×C4211yescharacteristic
427C23C22×C4⋊C411yescharacteristic
428C23C2×#11711yescharacteristic
429C23C2×C4×D412yes—
430C23C2×C4×D412yes—
431C23C2×C4×D414yes—
432C23C2×C4×Q812yes—
433C23C4×#4112yes—
434C23C4×#4114yes—
435C23C2×#12011yescharacteristic
436C23C2×C22⋊Q812yes—
437C23C2×#12311yescharacteristic
438C23#52514yes—
439C23#52512yes—
440C23#52914yes—
441C23#53312yes—
442D4—28no—
443D4—28no—
444D4—216no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—48no—
466C2×C4—44no—
467C2×C4—48no—
468C2×C4—44no—
469C2×C4—48no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—44no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—48no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—48no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—48no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—84no—
493C2×C4—81no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4—81no—
497C2×C4—88no—
498C2×C4—88no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—84no—
502C2×C4—82no—
503C2×C4—84no—
504C2×C4—82no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4C4×C4⋊C414yes—
512C2×C4#38512yes—
513C2×C4#38512yes—
514C2×C4#38514yes—
515C22—22no—
516C22—21no—
517C22—21no—
518C22—22no—
519C22—22no—
520C22—21no—
521C22—21no—
522C22—22no—
523C22—24no—
524C22—22no—
525C22—24no—
526C22—22no—
527C22—24no—
528C22—22no—
529C22—24no—
530C22—22no—
531C22—24no—
532C22—41no—
533C22—42no—
534C22—44no—
535C22—42no—
536C22—44no—
537C22—42no—
538C22—41no—
539C22—44no—
540C22—42no—
541C22—42no—
542C22—44no—
543C22—44no—
544C22—84no—
545C22—82no—
546C22C2×C4×C4⋊C411yescharacteristic
547C22C42×D412yes—
548C22C2×#38511yescharacteristic
549C22#229012yes—
550C22#229611yescharacteristic
551C22C4×#12012yes—
552C22C4×#12312yes—
553C22C4×C22⋊Q814yes—
554C22#302612yes—
555C22#302711yescharacteristic
556C22#302811yescharacteristic
557C4—22no—
558C4—22no—
559C4—24no—
560C4—24no—
561C4—48no—
562C4—42no—
563C4—42no—
564C4—48no—
565C4—48no—
566C4—84no—
567C4—84no—
568C4—84no—
569C2—22no—
570C2—22no—
571C2—22no—
572C2—21no—
573C2—21no—
574C2—24no—
575C2—44no—
576C2C4×#38511yescharacteristic
577C2#2088911yescharacteristic
578C2#2089011yescharacteristic
579C2#2148112yes—
580C2#2148212yes—
subgroupG/H
center, socleC23C2×#120
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #13782C2semidirect
6. #10918C2semidirect
8. C4×#342C2semidirect
27. #1867C4semidirect
28. C4×#99C4semidirect
78. #385C2×C4semidirect
98. C4×C22⋊C4C2×C4semidirect
118. #342C2×C4semidirect
119. #342C2×C4semidirect
145. C23×C4C4⋊C4semidirect
176. C2×C4⋊C4C4⋊C4semidirect
199. C2×C22⋊C4C4⋊C4semidirect
223. C2×C42C4⋊C4semidirect
225. #99C42semidirect
381. C22⋊C4C2×C42semidirect

Maximal subgroups

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

subgroupindex
C4×#5182
C2×#18672
C2×C4×#992
#137842
#137822
#109182
#107452
C4×#3422