Finite Group Databasebeta

← #360092

Subgroups — #360092

Order 512.

1972 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2322C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic, complemented
3C2×#1899C212yesmaximal
4C2×#1799C212yesmaximal
5#14339C211yesmaximal, characteristic, complemented
6#11203C212yesmaximal, complemented
7#11043C212yesmaximal
8#10846C212yesmaximal, complemented
9#10736C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117C2212yes—
13C23×C4⋊C4C2211yescharacteristic
14C22×#99C2214yes—
15#2322—24no—
16#2322—24no—
17#2316—28no—
18C2×#389—24no—
19C2×#389C412yescomplemented
20C2×#388C412yescomplemented
21C2×#387C412yescomplemented
22C2×#386C2211yescharacteristic
23C2×#385C412yescomplemented
24C2×#385C2211yescharacteristic
25C2×#381C2212yes—
26C2×#381C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×C4×C22⋊C4C2211yescharacteristic
31#1899—28no—
32#1799—28no—
33C2×#342C2214yes—
34C2×#342C2214yes—
35C2×#342C2214yes—
36#1441—24no—
37C22⋊#95C2212yes—
38C23×D4C2311yescharacteristic
39C24×C4C2×C411yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C2×#123—22no—
44C2×#123—22no—
45C2×#123—22no—
46C2×#123—22no—
47C2×#123D414yes—
48C2×#123Q814yes—
49C2×C4×D4—24no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—24no—
53C2×#117—28no—
54C2×#117—24no—
55C2×#117—24no—
56C2×#117—44no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4D414yes—
69C22×C4⋊C4D414yes—
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2311yescharacteristic
75C22×C22⋊C4C2311yescharacteristic
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2311yescharacteristic
78C22×C42—22no—
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C412yes—
81C2×#99—28no—
82C2×#99—28no—
83C2×#99—44no—
84#389—232no—
85#389—28no—
86#388—28no—
87#387—28no—
88#386—22no—
89#386—22no—
90#385—22no—
91#385—22no—
92#385—28no—
93#381—22no—
94#381—22no—
95#381—24no—
96#381—24no—
97#380—24no—
98#380—28no—
99#380—24no—
100#380—24no—
101#380D418yes—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—24no—
108C2×#95—24no—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114#342—28no—
115#342—28no—
116#342—44no—
117C25C22×C411yescharacteristic
118C25C2411yescharacteristic
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4C2×D412yes—
128C22×D4C2×Q812yes—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—24no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—82no—
140C23×C4—82no—
141C23×C4—82no—
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C412yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C412yes—
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C412yes—
148C23×C4C22×C412yes—
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D414yes—
159C23×C4C2×Q812yes—
160C23×C4#4112yes—
161C23×C4#4112yes—
162C23×C4#4114yes—
163#123—28no—
164#123—28no—
165#123—28no—
166#123—28no—
167C4×D4—44no—
168C4×D4—44no—
169C4×D4—44no—
170C4×D4—44no—
171#117—416no—
172#117—416no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—84no—
191C2×C4⋊C4—84no—
192C2×C4⋊C4—84no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4C22⋊C418yes—
195C2×C4⋊C4C22⋊C418yes—
196C2×C4⋊C4C4⋊C418yescomplemented
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4C4⋊C418yescomplemented
227C2×C22⋊C4C2×D418yes—
228C2×C22⋊C4C2×D412yes—
229C2×C22⋊C4C2×Q812yes—
230C2×C42—28no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—44no—
235C2×C42—44no—
236C2×C42—82no—
237#99—416no—
238#95—28no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—28no—
245#95—44no—
246#95—48no—
247#95—84no—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—22no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24—42no—
256C24C2×C4⋊C412yes—
257C24C23×C411yescharacteristic
258C24C22×D411yescharacteristic
259C24C22×D411yescharacteristic
260C24C22×D411yescharacteristic
261C24C22×D412yes—
262C24C22×Q811yescharacteristic
263C24C2×#4111yescharacteristic
264C24C2×#4111yescharacteristic
265C24C2×#4112yes—
266C24H(5,𝔽2)12yes—
267C242−1+412yes—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—41no—
277C2×D4—42no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—28no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—48no—
330C22×C4—48no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—48no—
339C22×C4—48no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4—84no—
348C22×C4—88no—
349C22×C4C2×C22⋊C414yes—
350C22×C4C2×C22⋊C414yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C4⋊C412yes—
354C22×C4C2×C4⋊C414yes—
355C22×C4C2×C4⋊C414yes—
356C22×C4#11714yes—
357C22×C4C4×D414yes—
358C22×C4C4×D414yes—
359C22×C4#12014yes—
360C22×C4#12114yes—
361C22×C4#12114yes—
362C22×C4#12114yes—
363C22×C4#12114yes—
364C22×C4C22⋊Q814yes—
365C22×C4C22⋊Q814yes—
366C22×C4#12314yes—
367C4⋊C4—216no—
368C4⋊C4—216no—
369C4⋊C4—216no—
370C4⋊C4—416no—
371C4⋊C4—44no—
372C4⋊C4—84no—
373C4⋊C4—88no—
374C4⋊C4—88no—
375C4⋊C4—88no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—216no—
381C22⋊C4—28no—
382C22⋊C4—28no—
383C22⋊C4—41no—
384C22⋊C4—42no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—42no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—84no—
397C42—416no—
398C42—44no—
399C42—84no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—24no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—41no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—41no—
425C23—82no—
426C23—84no—
427C23—82no—
428C23C22×C22⋊C411yescharacteristic
429C23C22×C4⋊C411yescharacteristic
430C23C2×#11711yescharacteristic
431C23C2×C4×D412yes—
432C23#52012yes—
433C23C2×#12011yescharacteristic
434C23C2×#12111yescharacteristic
435C23C2×#12111yescharacteristic
436C23C2×#12112yes—
437C23C2×C22⋊Q811yescharacteristic
438C23C2×C22⋊Q811yescharacteristic
439C23C2×#12311yescharacteristic
440C23#53511yescharacteristic
441C23#53611yescharacteristic
442C23#53712yes—
443C23D4212yes—
444C23#54612yes—
445C23#54712yes—
446C23#54812yes—
447C23D4×Q812yes—
448C23#55212yes—
449C23#55312yes—
450C23#55412yes—
451D4—42no—
452D4—42no—
453D4—44no—
454D4—44no—
455D4—42no—
456D4—42no—
457C2×C4—24no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—48no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—88no—
501C2×C4—84no—
502C2×C4—84no—
503C2×C4—88no—
504C2×C4—88no—
505C2×C4—84no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—84no—
509C2×C4—88no—
510C2×C4—88no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4#38018yes—
514C2×C4#38614yes—
515C2×C4#38614yes—
516C22—22no—
517C22—24no—
518C22—24no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—21no—
531C22—22no—
532C22—22no—
533C22—21no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—44no—
544C22—84no—
545C22—84no—
546C22—82no—
547C22—81no—
548C22—81no—
549C22C2×#38011yescharacteristic
550C22C2×#38611yescharacteristic
551C22#231111yescharacteristic
552C22D4×C4⋊C412yes—
553C22#235212yes—
554C22#240811yescharacteristic
555C22#240911yescharacteristic
556C22#241612yes—
557C22#241712yes—
558C22#248611yescharacteristic
559C22#248711yescharacteristic
560C22#288312yes—
561C22#288412yes—
562C4—28no—
563C4—24no—
564C4—24no—
565C4—42no—
566C4—44no—
567C4—44no—
568C4—88no—
569C4—82no—
570C4—81no—
571C4—81no—
572C4—164no—
573C2—21no—
574C2—22no—
575C2—22no—
576C2—22no—
577C2—21no—
578C2—22no—
579C2—21no—
580C2—21no—
581C2—82no—
582C2#1387211yescharacteristic
583C2#2063412yes—
584C2#2098411yescharacteristic
585C2#2098611yescharacteristic
586C2#2187712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2316C2semidirect
5. #14339C2semidirect
6. #11203C2semidirect
8. #10846C2semidirect
9. #10736C2semidirect
19. C2×#389C4semidirect
20. C2×#388C4semidirect
21. C2×#387C4semidirect
23. C2×#385C4semidirect
196. C2×C4⋊C4C4⋊C4semidirect
226. C2×C22⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23222
C2×#23162
C2×#18992
C2×#17992
#143392
#112032
#110432
#108462
#107362