Finite Group Databasebeta

← #470021

Subgroups — #470021

Order 512.

1166 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2513C211yesmaximal, characteristic
2C2×#2071C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic, complemented
4#15196C211yesmaximal, characteristic, complemented
5#11328C212yesmaximal, complemented
6#11268C212yesmaximal, complemented
7#11202C211yesmaximal, characteristic, complemented
8#11123C212yesmaximal
9#11076C211yesmaximal, characteristic
10#11066C212yesmaximal
11#10807C211yesmaximal, characteristic, complemented
12C22×#107C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2212yes—
16#2513—24no—
17#2513—24no—
18C2×#397—21no—
19C2×#397—21no—
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#388—21no—
26C2×#388—21no—
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#2071—24no—
32#1947—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1444—21no—
42#1444—21no—
43#1441—21no—
44#1441—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×D4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×#107—24no—
72C2×#107—42no—
73C2×#107—42no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—22no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—42no—
80C2×#102—42no—
81C2×#102—42no—
82#397—28no—
83#397—28no—
84#390—22no—
85#390—22no—
86#390—22no—
87#390—22no—
88#390—22no—
89#390D414yes—
90#389—22no—
91#389—22no—
92#389—42no—
93#389—42no—
94#388—28no—
95#388—28no—
96#384—22no—
97#384—22no—
98#382—22no—
99#382—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4D414yes—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D411yescharacteristic
114C2×#95D411yescharacteristic
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—42no—
120#336—24no—
121#336—22no—
122#336—42no—
123C25—21no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4C2×D412yes—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
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#4114yes—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
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—24no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4#4114yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—24no—
228C2×C42—82no—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231#107—48no—
232C4⋊C8—28no—
233#102—48no—
234#102—48no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—22no—
240#95—22no—
241#95—42no—
242#95—44no—
243#95—42no—
244#95—82no—
245#95—82no—
246C24—22no—
247C24—41no—
248C24—41no—
249C24—42no—
250C24—42no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24—42no—
256C24—42no—
257C24C22×D411yescharacteristic
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
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—42no—
332C22×C4#12012yes—
333C22×C4#12012yes—
334C22×C4#13512yes—
335C22×C4#13712yes—
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—42no—
351C2×C8—42no—
352C2×C8—42no—
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—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—44no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C4⋊C4—88no—
385C22⋊C4—48no—
386C22⋊C4—48no—
387C22⋊C4—84no—
388C22⋊C4—84no—
389C22⋊C4—84no—
390C22⋊C4—84no—
391C42—22no—
392C42—22no—
393C42—24no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—42no—
401C42—84no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—81no—
418C23—81no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23—81no—
423C23—81no—
424C23—82no—
425C23C2×#12011yescharacteristic
426C23#53511yescharacteristic
427C23#53711yescharacteristic
428C23#54712yes—
429C23#54812yes—
430C23C2×#13511yescharacteristic
431C23C2×#13711yescharacteristic
432C23C2×#13711yescharacteristic
433C23#57511yescharacteristic
434C23#57612yes—
435C23#57612yes—
436D4—44no—
437D4—44no—
438D4—81no—
439D4—81no—
440D4—81no—
441D4—81no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
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—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4#45214yes—
503C2×C4#52511yescharacteristic
504C2×C4#53211yescharacteristic
505C2×C4#53511yescharacteristic
506C2×C4#53511yescharacteristic
507C2×C4#53611yescharacteristic
508C2×C4#53611yescharacteristic
509C2×C4#53711yescharacteristic
510C2×C4#53711yescharacteristic
511C2×C4#53912yes—
512C2×C4#54112yes—
513C2×C4#54112yes—
514C2×C4#54212yes—
515C2×C4#54712yes—
516C2×C4#54712yes—
517C2×C4#55112yes—
518C2×C4#55312yes—
519C2×C4#56312yes—
520C2×C4#56312yes—
521C8—44no—
522C8—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—162no—
531C22—162no—
532C22—162no—
533C22—162no—
534C22#240611yescharacteristic
535C22#243211yescharacteristic
536C22#261511yescharacteristic
537C22#272312yes—
538C22#272512yes—
539C22C2×#45211yescharacteristic
540C22#300711yescharacteristic
541C22#300811yescharacteristic
542C22#300811yescharacteristic
543C22#321211yescharacteristic
544C22#321811yescharacteristic
545C22#322012yes—
546C22#324811yescharacteristic
547C22#325211yescharacteristic
548C22#325912yes—
549C22#333912yes—
550C22#334612yes—
551C22#334712yes—
552C22#335712yes—
553C22#336112yes—
554C22#337012yes—
555C22#337512yes—
556C22#337512yes—
557C4—24no—
558C4—42no—
559C4—42no—
560C4—81no—
561C4—81no—
562C4—82no—
563C4—82no—
564C4—84no—
565C4—81no—
566C4—82no—
567C4—162no—
568C4#260311yescharacteristic
569C4#263711yescharacteristic
570C4#266411yescharacteristic
571C4#271812yes—
572C4#272212yes—
573C4#282111yescharacteristic
574C2—162no—
575C2#1667311yescharacteristic
576C2#2409211yescharacteristic
577C2#2410511yescharacteristic
578C2#2564811yescharacteristic
579C2#2566011yescharacteristic
580C2#2966511yescharacteristic
581C2#2967011yescharacteristic
582C2#3244012yes—
583C2#3244212yes—
584C2#3248512yes—
585C2#3250312yes—
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×#1947C2semidirect
4. #15196C2semidirect
5. #11328C2semidirect
6. #11268C2semidirect
7. #11202C2semidirect
11. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25132
C2×#20712
C2×#19472
#151962
#113282
#112682
#112022
#111232
#110762
#110662
#108072