Finite Group Databasebeta

← #404130

Subgroups — #404130

Order 512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2345C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic, complemented
3C2×#1988C211yesmaximal, characteristic, complemented
4#15125C211yesmaximal, characteristic
5#11292C212yesmaximal, complemented
6#11228C211yesmaximal, characteristic, complemented
7#11095C212yesmaximal
8#11042C211yesmaximal, characteristic
9#10908C212yesmaximal, complemented
10#10807C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17C2×#393—21no—
18#2345—24no—
19#2345—24no—
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#387—21no—
23C2×#387—21no—
24C2×#380—21no—
25C2×#380C2211yescharacteristic
26C2×#380C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4—21no—
30C2×C4×C22⋊C4C2211yescharacteristic
31#2058—24no—
32#1988—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42#1446—21no—
43#1442—21no—
44#1439—21no—
45C4⋊#95C2212yes—
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—22no—
55C2×C4⋊Q8—42no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4Q814yescomplemented
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—42no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—42no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—42no—
88#393—44no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389D414yes—
93#387—28no—
94#387—28no—
95#380—28no—
96#380—22no—
97#380—22no—
98#380—24no—
99#380—42no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4D414yes—
104C4×C22⋊C4—28no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D411yescharacteristic
118C2×#95D411yescharacteristic
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—42no—
126#336—24no—
127#336—22no—
128C25C2×D411yescharacteristic
129C22×Q8—44no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—82no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—82no—
152C23×C4—82no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×Q812yes—
159C23×C4#4112yes—
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×C8—22no—
174C22×C8#4114yes—
175C4⋊Q8—28no—
176C4⋊Q8—48no—
177C4⋊Q8—44no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
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—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×Q812yes—
219C2×C4⋊C4C2×Q812yes—
220C2×C4⋊C4#4114yes—
221C2×C4⋊C4#4112yes—
222C2×C4⋊C4#4112yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4SD1614yes—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—82no—
244C2×C42—82no—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247#106—48no—
248C4⋊C8—44no—
249C22⋊C8—48no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—22no—
256#95—22no—
257#95—44no—
258#95—42no—
259#95—44no—
260#95—42no—
261#95—82no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24C22⋊Q812yes—
271C24C2×SD1612yes—
272C24#13712yes—
273C24C22×D411yescharacteristic
274C2×Q8—48no—
275C2×Q8—88no—
276C2×Q8—88no—
277C2×Q8—84no—
278C2×Q8—84no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—216no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
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—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
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—88no—
352C22×C4—88no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4#12012yes—
356C22×C4#12012yes—
357C22×C4C22⋊Q812yes—
358C22×C4C2×SD1612yes—
359C22×C4C2×SD1612yes—
360C22×C4#13712yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×Q811yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4112yes—
368C22×C4C2×#4112yes—
369C22×C4H(5,𝔽2)12yes—
370C22×C42−1+412yes—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C4⋊C4—216no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—48no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C4⋊C4C22⋊Q814yes—
410C4⋊C4C22⋊Q814yes—
411C22⋊C4—28no—
412C22⋊C4—44no—
413C22⋊C4—48no—
414C22⋊C4—88no—
415C42—22no—
416C42—22no—
417C42—24no—
418C42—41no—
419C42—41no—
420C42—42no—
421C42—84no—
422C42—84no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23—41no—
442C23—41no—
443C23—42no—
444C23—42no—
445C23#47514yes—
446C23C2×#12011yescharacteristic
447C23C2×C22⋊Q811yescharacteristic
448C23#52511yescharacteristic
449C23#54612yes—
450C23#54712yes—
451C23C22×SD1611yescharacteristic
452C23C2×#13711yescharacteristic
453C23C2×#13711yescharacteristic
454C23#57511yescharacteristic
455C23#57612yes—
456C23#57712yes—
457Q8—816no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—816no—
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—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4#44814yes—
524C2×C4C2×C22⋊Q811yescharacteristic
525C2×C4C2×C22⋊Q811yescharacteristic
526C2×C4#52511yescharacteristic
527C2×C4#52511yescharacteristic
528C2×C4#52511yescharacteristic
529C2×C4#52912yes—
530C2×C4#53211yescharacteristic
531C2×C4#53312yes—
532C2×C4#53511yescharacteristic
533C2×C4#53611yescharacteristic
534C2×C4#54212yes—
535C2×C4#54412yes—
536C2×C4#54612yes—
537C2×C4#54812yes—
538C2×C4#55112yes—
539C2×C4#55212yes—
540C2×C4#55512yes—
541C2×C4#55612yes—
542C8—44no—
543C8—82no—
544C22—28no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—44no—
550C22—42no—
551C22—42no—
552C22—44no—
553C22—42no—
554C22—44no—
555C22#240411yescharacteristic
556C22#243211yescharacteristic
557C22#243812yes—
558C22#247412yes—
559C22#284811yescharacteristic
560C22C2×#44811yescharacteristic
561C22#300711yescharacteristic
562C22#300811yescharacteristic
563C22#300911yescharacteristic
564C22C2×#47511yescharacteristic
565C22#307811yescharacteristic
566C22#308112yes—
567C22#310311yescharacteristic
568C22#311311yescharacteristic
569C22#311812yes—
570C22#329512yes—
571C22#331412yes—
572C22#332012yes—
573C22#332612yes—
574C22#333512yes—
575C22#334612yes—
576C22#334712yes—
577C22#335612yes—
578C4—24no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—44no—
583C4—44no—
584C4—81no—
585C4—82no—
586C4—84no—
587C4—82no—
588C4—162no—
589C4#239611yescharacteristic
590C4#243912yes—
591C4#247712yes—
592C4#259911yescharacteristic
593C4#261011yescharacteristic
594C4#262811yescharacteristic
595C2—44no—
596C2#1488711yescharacteristic
597C2#2396911yescharacteristic
598C2#2397411yescharacteristic
599C2#2570211yescharacteristic
600C2#2571311yescharacteristic
601C2#2599212yes—
602C2#2600812yes—
603C2#2733012yes—
604C2#2734512yes—
605C2#3295411yescharacteristic
606C2#3296211yescharacteristic
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×#2058C2semidirect
3. C2×#1988C2semidirect
5. #11292C2semidirect
6. #11228C2semidirect
9. #10908C2semidirect
10. #10807C2semidirect
65. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23452
C2×#20582
C2×#19882
#151252
#112922
#112282
#110952
#110422
#109082
#108072
#107302