Finite Group Databasebeta

← #432812

Subgroups — #432812

Order 512.

1508 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2452C211yesmaximal, characteristic
2C2×#2094C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4#15196C211yesmaximal, characteristic, complemented
5#11384C211yesmaximal, characteristic, complemented
6#11327C212yesmaximal, complemented
7#11220C212yesmaximal, complemented
8#11066C212yesmaximal
9#10925C211yesmaximal, characteristic
10#10874C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C22×C4×D4C2211yescharacteristic
15C22×#106C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2452—24no—
20#2452—24no—
21C2×#396—21no—
22C2×#393—21no—
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#389C2212yes—
28C2×#385—21no—
29C2×#384C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#2094—24no—
32#1990—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1444—21no—
42#1444—21no—
43#1441—21no—
44#1441—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8D414yes—
56C2×C22⋊Q8—28no—
57C2×C22⋊Q8—44no—
58C2×C22⋊Q8—44no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—24no—
62C2×C4×D4—42no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#106—24no—
85C2×#106—24no—
86C2×#106—42no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—42no—
90C2×#102—42no—
91C2×#102—42no—
92C2×#102Q818yescomplemented
93C2×C4×C8—22no—
94C2×C4×C8—42no—
95#396—28no—
96#393—28no—
97#390—22no—
98#390—22no—
99#390—22no—
100#390D414yes—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—42no—
106#389—42no—
107#385—28no—
108#384—22no—
109#384—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95D411yescharacteristic
120C2×#95D411yescharacteristic
121#340—24no—
122#340—24no—
123#340—24no—
124#340—42no—
125#340—42no—
126#336—24no—
127#336—22no—
128#336—42no—
129C25—21no—
130C22×Q8—24no—
131C22×Q8—44no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—82no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4C2×D412yes—
142C22×D4C2×Q814yes—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—44no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—82no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8C2×Q814yes—
173C22×C8#4114yes—
174C4⋊Q8—28no—
175C4⋊Q8—44no—
176C4⋊Q8—48no—
177C22⋊Q8—432no—
178C4×D4—48no—
179C4×D4—44no—
180C4×D4—44no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4SD1614yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×Q814yes—
225C2×C4⋊C4#4114yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—82no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251#106—48no—
252#102—48no—
253#102—48no—
254C4×C8—44no—
255#95—24no—
256#95—24no—
257#95—22no—
258#95—22no—
259#95—44no—
260#95—42no—
261#95—42no—
262#95—82no—
263#95—82no—
264C24—22no—
265C24—24no—
266C24—41no—
267C24—41no—
268C24—42no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—42no—
274C24—42no—
275C24C22×D411yescharacteristic
276C2×Q8—48no—
277C2×Q8—48no—
278C2×Q8—84no—
279C2×Q8—88no—
280C2×Q8—88no—
281C2×D4—24no—
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4C4⋊Q814yes—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—88no—
355C22×C4—84no—
356C22×C4—88no—
357C22×C4#12012yes—
358C22×C4#12012yes—
359C22×C4C2×SD1612yes—
360C22×C4C2×SD1612yes—
361C22×C4C2×SD1612yes—
362C22×C4#13512yes—
363C22×C4AGL(1,ℤ/8)12yes—
364C22×C4#13712yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×Q812yes—
368C22×C4C2×#4112yes—
369C22×C4C2×#4112yes—
370C22×C4H(5,𝔽2)12yes—
371C22×C4H(5,𝔽2)12yes—
372C22×C42−1+412yes—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—28no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C4⋊C4—88no—
410C4⋊C4#12314yes—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—84no—
416C42—22no—
417C42—22no—
418C42—24no—
419C42—41no—
420C42—41no—
421C42—42no—
422C42—42no—
423C42—42no—
424C23—24no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—44no—
437C23—81no—
438C23—82no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23C2×#12011yescharacteristic
448C23#53211yescharacteristic
449C23#53511yescharacteristic
450C23#54612yes—
451C23D4×Q812yes—
452C23C22×SD1611yescharacteristic
453C23C2×#13511yescharacteristic
454C23C2×AGL(1,ℤ/8)11yescharacteristic
455C23C2×#13711yescharacteristic
456C23#57612yes—
457C23#57712yes—
458Q8—816no—
459D4—28no—
460D4—44no—
461D4—44no—
462D4—44no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—816no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#45014yes—
526C2×C4#45214yes—
527C2×C4C2×#12311yescharacteristic
528C2×C4C2×C4⋊Q811yescharacteristic
529C2×C4#53312yes—
530C2×C4#53411yescharacteristic
531C2×C4#53611yescharacteristic
532C2×C4#53912yes—
533C2×C4#54212yes—
534C2×C4#54312yes—
535C2×C4#54612yes—
536C2×C4#54612yes—
537C2×C4D4×Q812yes—
538C2×C4#55012yes—
539C2×C4#55212yes—
540C2×C4#55412yes—
541C2×C4#55412yes—
542C2×C4#56212yes—
543C8—44no—
544C8—44no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—82no—
549C22—84no—
550C22—84no—
551C22—84no—
552C22—84no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22#240511yescharacteristic
557C22#243111yescharacteristic
558C22#243411yescharacteristic
559C22#255612yes—
560C22#269212yes—
561C22C2×#45011yescharacteristic
562C22C2×#45211yescharacteristic
563C22#300811yescharacteristic
564C22#301011yescharacteristic
565C22#316711yescharacteristic
566C22#317912yes—
567C22#318211yescharacteristic
568C22#320611yescharacteristic
569C22#320711yescharacteristic
570C22#322412yes—
571C22#329512yes—
572C22#330112yes—
573C22#331612yes—
574C22#331812yes—
575C22Q8×SD1612yes—
576C22#338012yes—
577C22#338412yes—
578C22#338512yes—
579C4—24no—
580C4—24no—
581C4—42no—
582C4—44no—
583C4—42no—
584C4—81no—
585C4—84no—
586C4—82no—
587C4—81no—
588C4—82no—
589C4—162no—
590C4#242312yes—
591C4#254912yes—
592C4#268712yes—
593C4#273212yes—
594C2—84no—
595C2#1546611yescharacteristic
596C2#2403311yescharacteristic
597C2#2403611yescharacteristic
598C2#2559711yescharacteristic
599C2#2561411yescharacteristic
600C2#2583411yescharacteristic
601C2#2583911yescharacteristic
602C2#2894912yes—
603C2#2896812yes—
604C2#3152212yes—
605C2#3152312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1990C2semidirect
4. #15196C2semidirect
5. #11384C2semidirect
6. #11327C2semidirect
7. #11220C2semidirect
10. #10874C2semidirect
92. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24522
C2×#20942
C2×#19902
#151962
#113842
#113272
#112202
#110662
#109252
#108742
#107882