Finite Group Databasebeta

← #432771

Subgroups — #432771

Order 512.

1522 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic, complemented
3C2×#1990C211yesmaximal, characteristic, complemented
4#15135C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11328C212yesmaximal, complemented
7#11207C212yesmaximal, complemented
8#11096C212yesmaximal
9#10921C211yesmaximal, characteristic
10#10880C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2440—24no—
19#2440—24no—
20C2×#399—21no—
21C2×#398—21no—
22C2×#395—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2211yescharacteristic
26C2×#385—21no—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32#2083—24no—
33#1990—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1445—21no—
44#1444—21no—
45#1439—21no—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C24×C4—21no—
49C24×C4C2311yescharacteristic
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—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—42no—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D411yescharacteristic
81C22×C22⋊C4D411yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#107—24no—
85C2×#107—42no—
86C2×#107—42no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8Q818yescomplemented
90C2×C4×C8—22no—
91C2×C4×C8—42no—
92#399—44no—
93#398—28no—
94#395—28no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—24no—
99#389—42no—
100#389D414yes—
101#385—28no—
102#384—22no—
103#384—22no—
104#384—22no—
105#384—22no—
106#380—22no—
107#380—22no—
108#380—24no—
109#380—42no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95D411yescharacteristic
120C2×#95D411yescharacteristic
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—42no—
127#340—42no—
128#336—22no—
129#336—42no—
130C25C2×D411yescharacteristic
131C22×Q8—24no—
132C22×Q8—44no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—82no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—44no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—82no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×Q814yes—
160C23×C4C2411yescharacteristic
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×C8—22no—
173C22×C8C2×Q814yes—
174C22×C8C2×Q814yes—
175C4⋊Q8—28no—
176C4⋊Q8—44no—
177C4⋊Q8—48no—
178C22⋊Q8—432no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
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—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4#4114yes—
232C2×C4⋊C4#4114yes—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4D814yes—
244C2×C22⋊C4Q1614yes—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253#107—48no—
254C22⋊C8—48no—
255C4×C8—44no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—22no—
260#95—22no—
261#95—44no—
262#95—44no—
263#95—42no—
264#95—42no—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24C4⋊Q812yes—
274C24C2×D812yes—
275C24C2×Q1612yes—
276C24C22×D411yescharacteristic
277C2×Q8—48no—
278C2×Q8—48no—
279C2×Q8—84no—
280C2×Q8—88no—
281C2×Q8—88no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—48no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—48no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—84no—
351C22×C4—88no—
352C22×C4—88no—
353C22×C4#12012yes—
354C22×C4#12012yes—
355C22×C4C4⋊Q812yes—
356C22×C4C2×D812yes—
357C22×C4C2×D812yes—
358C22×C4C2×Q1612yes—
359C22×C4C2×Q1612yes—
360C22×C4AGL(1,ℤ/8)12yes—
361C22×C4#13712yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×Q812yes—
365C22×C4C2×#4112yes—
366C22×C4C2×#4112yes—
367C22×C4H(5,𝔽2)12yes—
368C22×C4H(5,𝔽2)12yes—
369C22×C42−1+412yes—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—48no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—88no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C4⋊C4—84no—
412C4⋊C4#12314yes—
413C22⋊C4—28no—
414C22⋊C4—28no—
415C22⋊C4—48no—
416C22⋊C4—88no—
417C42—22no—
418C42—22no—
419C42—24no—
420C42—41no—
421C42—41no—
422C42—42no—
423C23—24no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—42no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—42no—
441C23—41no—
442C23—41no—
443C23—42no—
444C23—42no—
445C23#50014yes—
446C23C2×#12011yescharacteristic
447C23C2×C4⋊Q811yescharacteristic
448C23#53611yescharacteristic
449C23#54612yes—
450C23#54612yes—
451C23C22×D811yescharacteristic
452C23C22×Q1611yescharacteristic
453C23C2×AGL(1,ℤ/8)11yescharacteristic
454C23C2×#13711yescharacteristic
455C23#57712yes—
456C23#57712yes—
457Q8—816no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—816no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4#44714yes—
521C2×C4#45114yes—
522C2×C4C2×#12311yescharacteristic
523C2×C4#53211yescharacteristic
524C2×C4#53312yes—
525C2×C4#53411yescharacteristic
526C2×C4#53511yescharacteristic
527C2×C4#53912yes—
528C2×C4#54212yes—
529C2×C4#54312yes—
530C2×C4#54612yes—
531C2×C4D4×Q812yes—
532C2×C4D4×Q812yes—
533C2×C4#55012yes—
534C2×C4#55212yes—
535C2×C4#55412yes—
536C2×C4#55412yes—
537C2×C4#56212yes—
538C8—44no—
539C8—44no—
540C22—28no—
541C22—24no—
542C22—42no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22#240511yescharacteristic
552C22#243111yescharacteristic
553C22#243411yescharacteristic
554C22#254912yes—
555C22#268712yes—
556C22C2×#44711yescharacteristic
557C22C2×#45111yescharacteristic
558C22#300911yescharacteristic
559C22#301011yescharacteristic
560C22C2×#50011yescharacteristic
561C22#316312yes—
562C22#316511yescharacteristic
563C22#322611yescharacteristic
564C22#323011yescharacteristic
565C22#324412yes—
566C22#329412yes—
567C22#329412yes—
568C22#329912yes—
569C22#329912yes—
570C22#331012yes—
571C22#332012yes—
572C22#333112yes—
573C22#333212yes—
574C4—24no—
575C4—24no—
576C4—42no—
577C4—42no—
578C4—44no—
579C4—42no—
580C4—84no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—162no—
585C4#242312yes—
586C4#255612yes—
587C4#269212yes—
588C4#273212yes—
589C2—44no—
590C2#1546611yescharacteristic
591C2#2403211yescharacteristic
592C2#2404211yescharacteristic
593C2#2556711yescharacteristic
594C2#2557311yescharacteristic
595C2#2581311yescharacteristic
596C2#2582011yescharacteristic
597C2#2876912yes—
598C2#2878312yes—
599C2#3135012yes—
600C2#3135612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2083C2semidirect
3. C2×#1990C2semidirect
5. #11384C2semidirect
6. #11328C2semidirect
7. #11207C2semidirect
10. #10880C2semidirect
89. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
C2×#20832
C2×#19902
#151352
#113842
#113282
#112072
#110962
#109212
#108802
#107302