Finite Group Databasebeta

← #431032

Subgroups — #431032

Order 512.

1632 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2452C211yesmaximal, characteristic
2C2×#2094C211yesmaximal, characteristic
3C2×#2084C211yesmaximal, characteristic, complemented
4C2×#2058C212yesmaximal, complemented
5C2×#1864C212yesmaximal
6#13910C211yesmaximal, characteristic, complemented
7#11346C211yesmaximal, characteristic, complemented
8#11327C212yesmaximal, complemented
9#11066C212yesmaximal
10#10931C211yesmaximal, characteristic
11#10884C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C22×C4×D4C2211yescharacteristic
15C22×#106C2212yes—
16C22×#106C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19#2452—24no—
20#2452—24no—
21C2×#396—21no—
22C2×#393—21no—
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#385—21no—
30C2×#384—21no—
31C2×#384C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#2094—24no—
35#2084—24no—
36#2058—28no—
37#1864—28no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46#1444—21no—
47#1444—21no—
48C4⋊#95—21no—
49C23×Q8—21no—
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—42no—
58C2×C22⋊Q8—28no—
59C2×C22⋊Q8—44no—
60C2×C22⋊Q8—44no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—24no—
64C2×C4×D4—42no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—42no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—42no—
99C2×#102Q818yescomplemented
100#396—28no—
101#393—28no—
102#390—22no—
103#390—22no—
104#390—22no—
105#390D414yes—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—24no—
110#389—24no—
111#389—42no—
112#389—42no—
113#389D414yes—
114#385—28no—
115#384—22no—
116#384—22no—
117#384—44no—
118C4×C4⋊C4—24no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C2×#95—21no—
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#340D418yes—
132#336—24no—
133#336—24no—
134#336—22no—
135#336—22no—
136C25—21no—
137C22×Q8—24no—
138C22×Q8—44no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—82no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C2×D412yes—
149C22×D4C2×Q814yes—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—44no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—82no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8C2×D414yes—
180C22×C8C2×Q814yes—
181C4⋊Q8—48no—
182C4⋊Q8—44no—
183C4⋊Q8—44no—
184C22⋊Q8—432no—
185C4×D4—48no—
186C4×D4—44no—
187C4×D4—44no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
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—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4SD1614yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×Q814yes—
236C2×C4⋊C4#4114yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—82no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265#106—48no—
266#106—48no—
267#102—216no—
268#102—48no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—22no—
273#95—22no—
274#95—44no—
275#95—44no—
276C24—22no—
277C24—24no—
278C24—41no—
279C24—41no—
280C24—42no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—42no—
286C24—42no—
287C24C22×D411yescharacteristic
288C2×Q8—48no—
289C2×Q8—48no—
290C2×Q8—88no—
291C2×Q8—84no—
292C2×Q8—88no—
293C2×D4—24no—
294C2×D4—22no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4C4⋊Q814yes—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—24no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—88no—
368C22×C4—84no—
369C22×C4—88no—
370C22×C4#12012yes—
371C22×C4#12012yes—
372C22×C4C2×SD1612yes—
373C22×C4C2×SD1612yes—
374C22×C4C2×SD1612yes—
375C22×C4AGL(1,ℤ/8)12yes—
376C22×C4AGL(1,ℤ/8)12yes—
377C22×C4C22×D412yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×Q812yes—
381C22×C4C2×#4112yes—
382C22×C4H(5,𝔽2)12yes—
383C22×C42−1+412yes—
384C22×C42−1+412yes—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—28no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—48no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—88no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4—88no—
422C4⋊C4C22⋊Q818yescomplemented
423C4⋊C4#12714yes—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—84no—
429C42—24no—
430C42—24no—
431C42—48no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—42no—
437C42—42no—
438C23—24no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23—44no—
445C23—42no—
446C23—44no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—81no—
452C23—81no—
453C23—82no—
454C23—81no—
455C23—81no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23C2×#12011yescharacteristic
462C23#52511yescharacteristic
463C23#53611yescharacteristic
464C23D4212yes—
465C23D4×Q812yes—
466C23C22×SD1611yescharacteristic
467C23C2×AGL(1,ℤ/8)11yescharacteristic
468C23C2×AGL(1,ℤ/8)11yescharacteristic
469C23#57511yescharacteristic
470C23#57712yes—
471C23#57712yes—
472Q8—816no—
473D4—28no—
474D4—44no—
475D4—44no—
476D4—44no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—816no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4#45014yes—
541C2×C4C2×C22⋊Q812yes—
542C2×C4C2×#12711yescharacteristic
543C2×C4C2×C4⋊Q811yescharacteristic
544C2×C4#53511yescharacteristic
545C2×C4#53611yescharacteristic
546C2×C4#53612yes—
547C2×C4#53912yes—
548C2×C4#54412yes—
549C2×C4D4212yes—
550C2×C4#54712yes—
551C2×C4#54712yes—
552C2×C4#54712yes—
553C2×C4D4×Q812yes—
554C2×C4D4×Q812yes—
555C2×C4#55412yes—
556C2×C4#56312yes—
557C8—44no—
558C8—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—84no—
563C22—82no—
564C22—84no—
565C22—82no—
566C22—82no—
567C22—84no—
568C22—82no—
569C22—84no—
570C22#240611yescharacteristic
571C22#243311yescharacteristic
572C22#243411yescharacteristic
573C22#244912yes—
574C22#267712yes—
575C22C2×#45011yescharacteristic
576C22#300611yescharacteristic
577C22#300911yescharacteristic
578C22#301011yescharacteristic
579C22#310311yescharacteristic
580C22#310911yescharacteristic
581C22#312512yes—
582C22#322711yescharacteristic
583C22#323411yescharacteristic
584C22#324412yes—
585C22#326512yes—
586C22#327112yes—
587C22#327212yes—
588C22D4×SD1612yes—
589C22Q8×SD1612yes—
590C22#338412yes—
591C22#338412yes—
592C22#338812yes—
593C4—24no—
594C4—42no—
595C4—48no—
596C4—42no—
597C4—42no—
598C4—44no—
599C4—42no—
600C4—82no—
601C4—81no—
602C4—82no—
603C4—162no—
604C4#244012yes—
605C4#254212yes—
606C4#268012yes—
607C4#272012yes—
608C2—84no—
609C2#1543011yescharacteristic
610C2#2406111yescharacteristic
611C2#2407411yescharacteristic
612C2#2575311yescharacteristic
613C2#2576811yescharacteristic
614C2#2583411yescharacteristic
615C2#2585111yescharacteristic
616C2#2637312yes—
617C2#2639212yes—
618C2#3095412yes—
619C2#3097512yes—
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×#2084C2semidirect
4. C2×#2058C2semidirect
6. #13910C2semidirect
7. #11346C2semidirect
8. #11327C2semidirect
11. #10884C2semidirect
99. C2×#102Q8semidirect
422. C4⋊C4C22⋊Q8semidirect

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×#20842
C2×#20582
C2×#18642
#139102
#113462
#113272
#110662
#109312
#108842