Finite Group Databasebeta

← #467139

Subgroups — #467139

Order 512.

1196 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2862C211yesmaximal, characteristic
2C2×#2101C212yesmaximal
3#14699C211yesmaximal, characteristic
4#11384C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic
7#11384C211yesmaximal, characteristic
8#11372C212yesmaximal
9#11268C212yesmaximal
10#11208C212yesmaximal
11#10795C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14#2862—28no—
15C2×#391—21no—
16C2×#391—21no—
17C2×#391—21no—
18C2×#391—21no—
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#387—21no—
22C2×#382C2212yes—
23C2×#382C2212yes—
24C2×C43C2211yescharacteristic
25#2101—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1436—21no—
39#1436—21no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44#1434—21no—
45C23×Q8—21no—
46C23×Q8—21no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8—42no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#391—28no—
75#391—28no—
76#391—28no—
77#391—28no—
78#389—22no—
79#389—24no—
80#389Q814yescomplemented
81#387—44no—
82#382—24no—
83#382—24no—
84#382—24no—
85#382D418yes—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C43—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—42no—
106#340—42no—
107#340—42no—
108#340—42no—
109#336—24no—
110#336—24no—
111#336—22no—
112#336—22no—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8—42no—
116C22×Q8—42no—
117C22×Q8—42no—
118C22×Q8—42no—
119C22×Q8—42no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—42no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C4⋊Q8—48no—
155C4⋊Q8—48no—
156C4⋊Q8—48no—
157C4⋊Q8—48no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2×Q812yes—
197C2×C4⋊C4C2×Q812yes—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—41no—
214C2×C42—41no—
215C2×C42—41no—
216C2×C42—41no—
217C2×C42C2×D414yes—
218C2×C42C2×Q812yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222#95—28no—
223#95—28no—
224#95—28no—
225#95—28no—
226#95—24no—
227#95—42no—
228#95—42no—
229#95—42no—
230#95—42no—
231#95—42no—
232#95—42no—
233#95—42no—
234#95—42no—
235#95—42no—
236C24C22×D411yescharacteristic
237C2×Q8—48no—
238C2×Q8—48no—
239C2×Q8—48no—
240C2×Q8—48no—
241C2×Q8—48no—
242C2×Q8—48no—
243C2×Q8—84no—
244C2×Q8—84no—
245C2×Q8—84no—
246C2×Q8—84no—
247C2×Q8—84no—
248C2×Q8—84no—
249C2×Q8—84no—
250C2×Q8—84no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
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—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4#12012yes—
326C22×C4#12012yes—
327C22×C4C22⋊Q812yes—
328C22×C4C22⋊Q812yes—
329C22×C4#12412yes—
330C22×C4#12412yes—
331C22×C4#13512yes—
332C22×C4#13512yes—
333C22×C4AGL(1,ℤ/8)12yes—
334C22×C4#13712yes—
335C22×C4C22×D412yes—
336C22×C4C22×Q811yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4H(5,𝔽2)12yes—
341C22×C42−1+412yes—
342C22×C42−1+411yescharacteristic
343C22×C42−1+411yescharacteristic
344C22×C42−1+412yes—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
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—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—84no—
379C42—24no—
380C42—24no—
381C42—24no—
382C42—24no—
383C42—24no—
384C42—24no—
385C42—44no—
386C42—44no—
387C42—44no—
388C42—44no—
389C23C2×#12011yescharacteristic
390C23C2×C22⋊Q811yescharacteristic
391C23C2×#12411yescharacteristic
392C23#53511yescharacteristic
393C23#53611yescharacteristic
394C23#53611yescharacteristic
395C23#53611yescharacteristic
396C23C2×#13511yescharacteristic
397C23C2×#13511yescharacteristic
398C23C2×AGL(1,ℤ/8)11yescharacteristic
399C23C2×#13711yescharacteristic
400C23#57511yescharacteristic
401C23#57511yescharacteristic
402C23#57511yescharacteristic
403C23#57511yescharacteristic
404Q8—816no—
405Q8—816no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4#48814yes—
489C2×C4#48914yes—
490C2×C4#53612yes—
491C2×C4#53612yes—
492C2×C4#54112yes—
493C2×C4#54112yes—
494C2×C4#54212yes—
495C2×C4#54412yes—
496C2×C4#54612yes—
497C2×C4#54612yes—
498C2×C4#54812yes—
499C2×C4D4×Q812yes—
500C2×C4#55611yescharacteristic
501C2×C4#55611yescharacteristic
502C2×C4#55711yescharacteristic
503C2×C4#55711yescharacteristic
504C2×C4#56312yes—
505C2×C4#56312yes—
506C8—82no—
507C8—82no—
508C22#240911yescharacteristic
509C22#243511yescharacteristic
510C22#256811yescharacteristic
511C22#261411yescharacteristic
512C22#262811yescharacteristic
513C22#264211yescharacteristic
514C22#282111yescharacteristic
515C22#300211yescharacteristic
516C22#300211yescharacteristic
517C22#300611yescharacteristic
518C22#300711yescharacteristic
519C22#307611yescharacteristic
520C22#307611yescharacteristic
521C22#307711yescharacteristic
522C22#307811yescharacteristic
523C22C2×#48811yescharacteristic
524C22C2×#48911yescharacteristic
525C22#314011yescharacteristic
526C22#314011yescharacteristic
527C22#321111yescharacteristic
528C22#321111yescharacteristic
529C22#321311yescharacteristic
530C22#321511yescharacteristic
531C22#322811yescharacteristic
532C22#322811yescharacteristic
533C22#323111yescharacteristic
534C22#323111yescharacteristic
535C22#323311yescharacteristic
536C22#323311yescharacteristic
537C22#323511yescharacteristic
538C22#323711yescharacteristic
539C22#324011yescharacteristic
540C22#324011yescharacteristic
541C22#324011yescharacteristic
542C22#324011yescharacteristic
543C4—24no—
544C4—24no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—84no—
550C4—84no—
551C4—82no—
552C4—82no—
553C4—161no—
554C4—161no—
555C4#271312yes—
556C4#271312yes—
557C4#271612yes—
558C4#271612yes—
559C2#1656011yescharacteristic
560C2#2430411yescharacteristic
561C2#2430611yescharacteristic
562C2#2585911yescharacteristic
563C2#2586411yescharacteristic
564C2#2922211yescharacteristic
565C2#2922411yescharacteristic
566C2#2964411yescharacteristic
567C2#2964511yescharacteristic
568C2#2998211yescharacteristic
569C2#2998511yescharacteristic
570C2#3022911yescharacteristic
571C2#3023411yescharacteristic
572C2#3284011yescharacteristic
573C2#3284211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
80. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28622
C2×#21012
#146992
#113842
#113722
#112682
#112082
#107952