Finite Group Databasebeta

← #360745

Subgroups — #360745

Order 512.

2042 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2C2×#1867C214yesmaximal
3C2×C4×#99C211yesmaximal, characteristic
4#13880C212yesmaximal, complemented
5#13872C211yesmaximal, characteristic
6#11344C214yesmaximal, complemented
7#10854C212yesmaximal, complemented
8C2×#518C2212yes—
9C22×C4×D4C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C22×#99C2212yes—
12#2349—24no—
13D4×C4⋊C4—22no—
14D4×C4⋊C4—22no—
15#2316—21no—
16#2315—21no—
17#2315C2212yes—
18C2×#386C2212yes—
19C2×#385C2218yes—
20#2285—21no—
21C2×#384—22no—
22C2×#380—21no—
23C2×#380—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2212yes—
26C2×C4×C22⋊C4C2212yes—
27#1867—216no—
28C4×#99C418yescomplemented
29C2×#342C2218yes—
30C2×#342C2212yes—
31#1444—24no—
32#1441—21no—
33#1439—21no—
34C4⋊#95—21no—
35C4⋊#95—22no—
36C22⋊#95C2212yes—
37C23×D4C2311yescharacteristic
38C24×C4D411yescharacteristic
39C24×C4C2312yes—
40C24×C4C2311yescharacteristic
41#518—28no—
42#518—44no—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—24no—
46C2×C4×D4—24no—
47C2×C4×D4—24no—
48C2×C4×D4C2×C414yes—
49C2×C4×D4Q814yes—
50C2×#117—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—28no—
59C22×C4⋊C4—44no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D412yes—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4D411yescharacteristic
66C22×C22⋊C4C2312yes—
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2311yescharacteristic
69C22×C22⋊C4C2314yes—
70C22×C42—22no—
71C22×C42D411yescharacteristic
72C2×#99C2×C418yes—
73C2×#99D418yes—
74C2×#99Q818yes—
75#389—24no—
76#389—24no—
77#386—28no—
78#385—24no—
79#385—24no—
80#385—24no—
81#385—24no—
82#385—216no—
83#385—48no—
84#384—21no—
85#384—21no—
86#384—48no—
87#382—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—22no—
96#380—24no—
97#380—44no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C22⋊C4—24no—
101C4×C22⋊C4C2×C418yescomplemented
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—28no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95D414yes—
110#342—216no—
111#342C2×C418yescomplemented
112C25C2×D411yescharacteristic
113C25C2411yescharacteristic
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4C22×C412yes—
119C22×D4C2×D414yes—
120C22×D4C2×Q812yes—
121C22×D4C2×Q814yes—
122C23×C4—24no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—44no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—82no—
139C23×C4C4⋊C414yescomplemented
140C23×C4C22×C412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D414yes—
145C23×C4C2×D414yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×Q812yes—
151C23×C4#4114yes—
152C4×D4—28no—
153C4×D4—28no—
154C4×D4—28no—
155C4×D4—28no—
156C4×D4—416no—
157C4×D4—416no—
158#117—44no—
159#117—44no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—48no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—416no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—416no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—88no—
191C2×C4⋊C4C4⋊C414yescomplemented
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4C4⋊C414yescomplemented
213C2×C22⋊C4C22×C412yes—
214C2×C22⋊C4C22×C418yes—
215C2×C22⋊C4C2×D418yes—
216C2×C22⋊C4C2×Q812yes—
217C2×C22⋊C4C2×Q818yes—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—28no—
223C2×C42—44no—
224C2×C42—44no—
225C2×C42C4⋊C414yescomplemented
226#99C4⋊C4132yescomplemented
227#95—24no—
228#95—24no—
229#95—28no—
230#95—24no—
231#95—24no—
232#95—416no—
233#95—44no—
234#95—42no—
235#95—44no—
236#95—82no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—41no—
242C24—41no—
243C24—41no—
244C24—41no—
245C24—41no—
246C24—41no—
247C24—41no—
248C24—41no—
249C24C2×C4⋊C412yes—
250C24#12712yes—
251C24C4⋊Q812yes—
252C24C23×C411yescharacteristic
253C24C22×D411yescharacteristic
254C24C22×D412yes—
255C24C22×D412yes—
256C24C22×Q811yescharacteristic
257C24C22×Q812yes—
258C24C2×#4112yes—
259C24H(5,𝔽2)12yes—
260C242−1+412yes—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—28no—
266C2×D4—28no—
267C2×D4—28no—
268C2×D4—28no—
269C2×D4C2×C4⋊C418yes—
270C2×D4C4⋊Q818yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—28no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—28no—
283C22×C4—28no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—28no—
293C22×C4—28no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—48no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—48no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—42no—
316C22×C4—48no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—48no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—84no—
335C22×C4—81no—
336C22×C4—81no—
337C22×C4—88no—
338C22×C4—88no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4C2×C4⋊C412yes—
344C22×C4C2×C4⋊C412yes—
345C22×C4C2×C4⋊C412yes—
346C22×C4C2×C4⋊C412yes—
347C22×C4C2×C4⋊C412yes—
348C22×C4C4×D414yes—
349C22×C4C4×D414yes—
350C22×C4#12011yescharacteristic
351C22×C4#12012yes—
352C22×C4#12011yescharacteristic
353C22×C4C22⋊Q814yes—
354C22×C4C22⋊Q814yes—
355C22×C4#12312yes—
356C22×C4#12312yes—
357C22×C4#12712yes—
358C22×C4C4⋊Q812yes—
359C4⋊C4—28no—
360C4⋊C4—48no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—84no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—832no—
375C22⋊C4—24no—
376C22⋊C4—28no—
377C22⋊C4—24no—
378C22⋊C4—28no—
379C22⋊C4—28no—
380C22⋊C4—28no—
381C22⋊C4—48no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—48no—
385C22⋊C4—48no—
386C22⋊C4—88no—
387C22⋊C4C2×C4⋊C4116yescomplemented
388C42—24no—
389C42—24no—
390C42—416no—
391C42—48no—
392C23—22no—
393C23—21no—
394C23—22no—
395C23—21no—
396C23—22no—
397C23—21no—
398C23—22no—
399C23—21no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—41no—
407C23—41no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—41no—
412C23—42no—
413C23—41no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23#38414yescomplemented
437C23C22×C4⋊C411yescharacteristic
438C23C22×C4⋊C412yes—
439C23C2×C4×D412yes—
440C23#52012yes—
441C23C2×#12011yescharacteristic
442C23C2×C22⋊Q812yes—
443C23C2×#12311yescharacteristic
444C23C2×#12711yescharacteristic
445C23C2×C4⋊Q811yescharacteristic
446C23C2×C4⋊Q812yes—
447C23#53511yescharacteristic
448C23#53611yescharacteristic
449C23#54412yes—
450C23D4214yes—
451C23#54714yes—
452C23D4×Q814yes—
453C23#55414yes—
454D4—216no—
455D4—216no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—28no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—216no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—48no—
478C2×C4—48no—
479C2×C4—48no—
480C2×C4—48no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—416no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—84no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—88no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—88no—
509C2×C4—88no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—164no—
518C2×C4—164no—
519C2×C4#38414yescomplemented
520C2×C4#38512yes—
521C2×C4#38514yes—
522C2×C4#38512yes—
523C2×C4B(4,𝔽2)12yes—
524C2×C4#45812yes—
525C22—22no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—24no—
534C22—21no—
535C22—21no—
536C22—24no—
537C22—21no—
538C22—21no—
539C22—24no—
540C22—21no—
541C22—21no—
542C22—24no—
543C22—21no—
544C22—21no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22C2×#38411yescharacteristic
563C22C2×#38511yescharacteristic
564C22#231711yescharacteristic
565C22D4×C4⋊C414yes—
566C22#243311yescharacteristic
567C22#243411yescharacteristic
568C22#244014yes—
569C22#250111yescharacteristic
570C22#250211yescharacteristic
571C22C2×B(4,𝔽2)11yescharacteristic
572C22C2×#45811yescharacteristic
573C22#302711yescharacteristic
574C22#302811yescharacteristic
575C4—24no—
576C4—22no—
577C4—24no—
578C4—416no—
579C4—42no—
580C4—42no—
581C4—84no—
582C4—84no—
583C4—84no—
584C4—162no—
585C4#186714yescomplemented
586C2—24no—
587C2—21no—
588C2—21no—
589C2—21no—
590C2—21no—
591C2—24no—
592C2—44no—
593C2#1388011yescharacteristic
594C2C2×#186711yescharacteristic
595C2#2089011yescharacteristic
596C2#2577111yescharacteristic
597C2#2577211yescharacteristic
598C2#2585211yescharacteristic
599C2#2585311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ 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
4. #13880C2semidirect
6. #11344C2semidirect
7. #10854C2semidirect
28. C4×#99C4semidirect
101. C4×C22⋊C4C2×C4semidirect
111. #342C2×C4semidirect
139. C23×C4C4⋊C4semidirect
191. C2×C4⋊C4C4⋊C4semidirect
212. C2×C22⋊C4C4⋊C4semidirect
225. C2×C42C4⋊C4semidirect
226. #99C4⋊C4semidirect
387. C22⋊C4C2×C4⋊C4semidirect
436. C23#384semidirect
519. C2×C4#384semidirect
585. C4#1867semidirect

Maximal subgroups

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

subgroupindex
#341812
C2×#18672
C2×C4×#992
#138802
#138722
#113442
#108542