Finite Group Databasebeta

← #418663

Subgroups — #418663

Order 512.

1508 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic, complemented
2C2×#2441C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic, complemented
4C2×#1854C212yesmaximal
5#15125C211yesmaximal, characteristic
6#11385C211yesmaximal, characteristic, complemented
7#11324C212yesmaximal, complemented
8#11220C212yesmaximal, complemented
9#11096C212yesmaximal
10#10922C211yesmaximal, characteristic
11#10881C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C2×#501—28no—
16C2×#501—24no—
17C22×#107C2212yes—
18C22×#107C2212yes—
19C22×#106C2212yes—
20C22×C22⋊C8C2212yes—
21C22×#97C2211yescharacteristic
22#2441—24no—
23#2441—24no—
24C2×#399—21no—
25C2×#396—21no—
26C2×#393—21no—
27C2×#389—21no—
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#387—21no—
31C2×#384C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×C4×C22⋊C4C2211yescharacteristic
35#2057—24no—
36#1854—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44#1446—21no—
45#1442—21no—
46#1439—21no—
47C4⋊#95C2212yes—
48C4⋊#95C2211yescharacteristic
49C23×Q8—21no—
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—42no—
57C2×#125—24no—
58C2×#125—22no—
59C2×#125—42no—
60C2×#125D414yes—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—24no—
64C2×#117—42no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#501—232no—
87C2×#107—24no—
88C2×#107—24no—
89C2×#107—24no—
90C2×#107—24no—
91C2×#107—24no—
92C2×#107—42no—
93C2×#106—24no—
94C2×#106—24no—
95C2×#106D418yes—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8Q818yescomplemented
99C2×#97—24no—
100C2×#97—22no—
101#399—28no—
102#396—28no—
103#393—44no—
104#389—28no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389D414yes—
109#387—28no—
110#384—22no—
111#384—22no—
112#380—22no—
113#380—22no—
114#380—24no—
115#380—42no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95D411yescharacteristic
127C2×#95D411yescharacteristic
128#340—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—42no—
133#336—24no—
134#336—22no—
135C25C2×D411yescharacteristic
136C22×Q8—44no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—82no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
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×C8C2×D414yes—
172C22×C8C2×Q814yes—
173C22×C8C2×Q814yes—
174C22×C8#4114yes—
175C4⋊Q8—28no—
176C4⋊Q8—48no—
177C4⋊Q8—44no—
178#125—28no—
179#125—44no—
180#125—48no—
181#117—48no—
182#117—44no—
183#117—44no—
184C2×C4⋊C4—21no—
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—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2×D412yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4—82no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—44no—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251#107—216no—
252#107—48no—
253#106—216no—
254C22⋊C8—216no—
255#97—28no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—22no—
260#95—22no—
261#95—44no—
262#95—42no—
263#95—42no—
264#95—44no—
265#95—82no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—22no—
275C24—22no—
276C24C4⋊Q812yes—
277C24C22×D411yescharacteristic
278C2×Q8—48no—
279C2×Q8—84no—
280C2×Q8—84no—
281C2×Q8—88no—
282C2×Q8—88no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
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—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4#12012yes—
354C22×C4#12012yes—
355C22×C4C4⋊Q812yes—
356C22×C4AGL(1,ℤ/8)12yes—
357C22×C4#13712yes—
358C22×C4C22×D412yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×Q812yes—
362C22×C4C2×#4112yes—
363C22×C4H(5,𝔽2)12yes—
364C22×C42−1+412yes—
365C22×C42−1+412yes—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8C22⋊Q818yes—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—84no—
405C4⋊C4—84no—
406C4⋊C4#12014yes—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—48no—
410C22⋊C4—84no—
411C22⋊C4—88no—
412C42—22no—
413C42—22no—
414C42—24no—
415C42—48no—
416C42—41no—
417C42—41no—
418C42—42no—
419C23—22no—
420C23—24no—
421C23—22no—
422C23—22no—
423C23—24no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23C2×#12011yescharacteristic
443C23C2×C4⋊Q811yescharacteristic
444C23#53611yescharacteristic
445C23D4212yes—
446C23#54712yes—
447C23C2×AGL(1,ℤ/8)11yescharacteristic
448C23C2×#13711yescharacteristic
449C23#57511yescharacteristic
450C23#57511yescharacteristic
451C23#57612yes—
452C23#57812yes—
453Q8—816no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
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×C4C2×#12011yescharacteristic
518C2×C4C2×C22⋊Q812yes—
519C2×C4#53211yescharacteristic
520C2×C4#53411yescharacteristic
521C2×C4#53612yes—
522C2×C4#53611yescharacteristic
523C2×C4#54212yes—
524C2×C4#54412yes—
525C2×C4D4212yes—
526C2×C4#54812yes—
527C2×C4#54812yes—
528C2×C4D4×Q812yes—
529C2×C4D4×Q812yes—
530C2×C4D4×Q812yes—
531C2×C4#55412yes—
532C2×C4#56312yes—
533C8—28no—
534C8—44no—
535C22—42no—
536C22—44no—
537C22—42no—
538C22—42no—
539C22—44no—
540C22—81no—
541C22—81no—
542C22—82no—
543C22—82no—
544C22—81no—
545C22—81no—
546C22#240311yescharacteristic
547C22#243211yescharacteristic
548C22#243411yescharacteristic
549C22#244012yes—
550C22#267412yes—
551C22#300611yescharacteristic
552C22#300711yescharacteristic
553C22#300811yescharacteristic
554C22#301111yescharacteristic
555C22#316211yescharacteristic
556C22#316412yes—
557C22#316511yescharacteristic
558C22#323511yescharacteristic
559C22#323711yescharacteristic
560C22#324312yes—
561C22#326812yes—
562C22#326912yes—
563C22#327412yes—
564C22#327612yes—
565C22#334312yes—
566C22#334612yes—
567C22#334712yes—
568C22#334912yes—
569C4—42no—
570C4—44no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—82no—
576C4—82no—
577C4—84no—
578C4—82no—
579C4—162no—
580C4#240912yes—
581C4#245212yes—
582C4#267712yes—
583C4#272912yes—
584C2—82no—
585C2#1526511yescharacteristic
586C2#2386511yescharacteristic
587C2#2386911yescharacteristic
588C2#2571311yescharacteristic
589C2#2571611yescharacteristic
590C2#2581811yescharacteristic
591C2#2582111yescharacteristic
592C2#2606912yes—
593C2#2607712yes—
594C2#3079012yes—
595C2#3079612yes—
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
1. C22×#501C2semidirect
3. C2×#2057C2semidirect
6. #11385C2semidirect
7. #11324C2semidirect
8. #11220C2semidirect
11. #10881C2semidirect
98. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#24412
C2×#20572
C2×#18542
#151252
#113852
#113242
#112202
#110962
#109222
#108812