Finite Group Databasebeta

← #469981

Subgroups — #469981

Order 512.

1176 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2497C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#1948C211yesmaximal, characteristic, complemented
4#15150C211yesmaximal, characteristic
5#11323C212yesmaximal, complemented
6#11292C212yesmaximal, complemented
7#11228C211yesmaximal, characteristic, complemented
8#11095C212yesmaximal
9#11095C212yesmaximal
10#11042C211yesmaximal, characteristic
11#10807C211yesmaximal, characteristic, complemented
12C22×#106C2212yes—
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16#2497—24no—
17#2497—24no—
18C2×#400—21no—
19C2×#397—21no—
20C2×#397—21no—
21C2×#396—21no—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#388—21no—
25C2×#387—21no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#2084—24no—
34#1948—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4D411yescharacteristic
66C22×C22⋊C4D411yescharacteristic
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#106—24no—
71C2×#106—24no—
72C2×#106—42no—
73C2×#106—42no—
74C2×#106—42no—
75C2×#106—42no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—22no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—42no—
81#400—28no—
82#397—44no—
83#397—44no—
84#396—28no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389D414yes—
89#388—28no—
90#387—28no—
91#384—22no—
92#384—22no—
93#384—22no—
94#384—22no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—42no—
100#380—42no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4D414yes—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—42no—
122#340—42no—
123#336—24no—
124#336—22no—
125C25C2×D411yescharacteristic
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8#4114yes—
161C22×C8#4114yes—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
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—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4SD1614yes—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—82no—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232#106—48no—
233#106—48no—
234C4⋊C8—28no—
235C22⋊C8—48no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—22no—
242#95—22no—
243#95—44no—
244#95—42no—
245#95—42no—
246#95—44no—
247#95—44no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—22no—
252C24—22no—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—22no—
257C24—22no—
258C24C2×SD1612yes—
259C24C22×D411yescharacteristic
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
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#12012yes—
332C22×C4#12012yes—
333C22×C4C2×SD1612yes—
334C22×C4C2×SD1612yes—
335C22×C4AGL(1,ℤ/8)12yes—
336C22×C4#13712yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C2×#4112yes—
340C22×C4C2×#4112yes—
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C4H(5,𝔽2)11yescharacteristic
344C22×C42−1+412yes—
345C22×C42−1+412yes—
346C22×C42−1+411yescharacteristic
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—48no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—84no—
388C4⋊C4—84no—
389C22⋊C4—28no—
390C22⋊C4—44no—
391C22⋊C4—84no—
392C22⋊C4—84no—
393C22⋊C4—88no—
394C42—22no—
395C42—22no—
396C42—24no—
397C42—41no—
398C42—41no—
399C42—42no—
400C42—84no—
401C23—24no—
402C23—24no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—41no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23C2×#12011yescharacteristic
425C23#53511yescharacteristic
426C23#53711yescharacteristic
427C23#54712yes—
428C23#54712yes—
429C23C22×SD1611yescharacteristic
430C23C2×AGL(1,ℤ/8)11yescharacteristic
431C23C2×#13711yescharacteristic
432C23#57511yescharacteristic
433C23#57712yes—
434C23#57712yes—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4#44814yes—
500C2×C4#52511yescharacteristic
501C2×C4#53211yescharacteristic
502C2×C4#53511yescharacteristic
503C2×C4#53511yescharacteristic
504C2×C4#53611yescharacteristic
505C2×C4#53611yescharacteristic
506C2×C4#53711yescharacteristic
507C2×C4#53711yescharacteristic
508C2×C4#53912yes—
509C2×C4#54112yes—
510C2×C4#54112yes—
511C2×C4#54212yes—
512C2×C4#54712yes—
513C2×C4#54812yes—
514C2×C4#55112yes—
515C2×C4#55312yes—
516C2×C4#56312yes—
517C2×C4#56312yes—
518C8—44no—
519C8—44no—
520C22—44no—
521C22—42no—
522C22—42no—
523C22—44no—
524C22—42no—
525C22—81no—
526C22—81no—
527C22—82no—
528C22—81no—
529C22—82no—
530C22—81no—
531C22#240611yescharacteristic
532C22#243211yescharacteristic
533C22#261511yescharacteristic
534C22#271812yes—
535C22#272212yes—
536C22C2×#44811yescharacteristic
537C22#300611yescharacteristic
538C22#300911yescharacteristic
539C22#300911yescharacteristic
540C22#320811yescharacteristic
541C22#321611yescharacteristic
542C22#322112yes—
543C22#324611yescharacteristic
544C22#325411yescharacteristic
545C22#326012yes—
546C22#333512yes—
547C22#333512yes—
548C22#334412yes—
549C22#334512yes—
550C22#334812yes—
551C22#335012yes—
552C22#335412yes—
553C22#335612yes—
554C4—24no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—81no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4—82no—
563C4—162no—
564C4—162no—
565C4#260311yescharacteristic
566C4#263711yescharacteristic
567C4#266411yescharacteristic
568C4#272312yes—
569C4#272512yes—
570C4#282111yescharacteristic
571C2—82no—
572C2#1667311yescharacteristic
573C2#2409711yescharacteristic
574C2#2410411yescharacteristic
575C2#2570211yescharacteristic
576C2#2571411yescharacteristic
577C2#2966111yescharacteristic
578C2#2967311yescharacteristic
579C2#3228212yes—
580C2#3229712yes—
581C2#3240812yes—
582C2#3242212yes—
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×#2084C2semidirect
3. C2×#1948C2semidirect
5. #11323C2semidirect
6. #11292C2semidirect
7. #11228C2semidirect
11. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24972
C2×#20842
C2×#19482
#151502
#113232
#112922
#112282
#110952
#110422
#108072