Finite Group Databasebeta

← #403578

Subgroups — #403578

Order 512.

1390 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2C2×#2087C212yesmaximal
3#14699C211yesmaximal, characteristic
4#11384C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11324C211yesmaximal, characteristic
7#11267C211yesmaximal, characteristic
8#11228C212yesmaximal
9#10907C212yesmaximal
10#10885C212yesmaximal
11#10832C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×#107C2212yes—
15C22×C4⋊C8C2212yes—
16C4×C4⋊Q8—28no—
17C2×#391—21no—
18C2×#389—21no—
19C2×#389C2211yescharacteristic
20C2×#389C2212yes—
21C2×#389C2211yescharacteristic
22C2×#387—21no—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4C2212yes—
26C2×C43C2211yescharacteristic
27#2087—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38#1436—21no—
39#1436—21no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44#1434—21no—
45C23×Q8—21no—
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8Q814yes—
52C2×C4×Q8—28no—
53C2×C4×Q8—44no—
54C2×C4×Q8—44no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C42—22no—
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#107—24no—
76C2×#107—24no—
77C2×#107—42no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—42no—
81#391—28no—
82#389—28no—
83#389—24no—
84#389—22no—
85#389—24no—
86#389D418yes—
87#389Q814yescomplemented
88#387—44no—
89#384—24no—
90C4×C4⋊C4—28no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—24no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C43—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—42no—
113#336—22no—
114#336—22no—
115#336—24no—
116C22×Q8—24no—
117C22×Q8—24no—
118C22×Q8—24no—
119C22×Q8—42no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8—42no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2411yescharacteristic
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
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#4114yes—
150C4⋊Q8—216no—
151C4⋊Q8—216no—
152C4×Q8—432no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
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—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
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—21no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
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⋊C4C2×D414yes—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2×Q812yes—
196C2×C4⋊C4C2×Q812yes—
197C2×C4⋊C4C2×Q814yes—
198C2×C4⋊C4#4114yes—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—28no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—41no—
217C2×C42—41no—
218C2×C42—44no—
219C2×C42—44no—
220C2×C42Q1614yes—
221C2×C42C2×D414yes—
222C2×C42C2×Q812yes—
223C2×C42C2×Q812yes—
224C2×C42#4112yes—
225C2×C42#4112yes—
226#107—48no—
227C4⋊C8—48no—
228#95—28no—
229#95—28no—
230#95—24no—
231#95—42no—
232#95—42no—
233#95—42no—
234#95—42no—
235#95—42no—
236#95—42no—
237#95—42no—
238#95—42no—
239#95—42no—
240C24C22×D411yescharacteristic
241C2×Q8—216no—
242C2×Q8—48no—
243C2×Q8—48no—
244C2×Q8—48no—
245C2×Q8—48no—
246C2×Q8—48no—
247C2×Q8—48no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
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—24no—
266C22×C4—24no—
267C22×C4—24no—
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—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4#12012yes—
319C22×C4#12012yes—
320C22×C4C22⋊Q812yes—
321C22×C4C22⋊Q812yes—
322C22×C4C22⋊Q812yes—
323C22×C4C22⋊Q812yes—
324C22×C4C2×Q1612yes—
325C22×C4C2×Q1612yes—
326C22×C4C2×Q1612yes—
327C22×C4#13512yes—
328C22×C4AGL(1,ℤ/8)12yes—
329C22×C4AGL(1,ℤ/8)12yes—
330C22×C4AGL(1,ℤ/8)12yes—
331C22×C4#13712yes—
332C22×C4C22×D412yes—
333C22×C4C22×Q811yescharacteristic
334C22×C4C22×Q811yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4112yes—
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C42−1+411yescharacteristic
341C22×C42−1+412yes—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
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—
356C4⋊C4—216no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—416no—
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—84no—
377C4⋊C4C22⋊Q818yes—
378C42—28no—
379C42—24no—
380C42—24no—
381C42—24no—
382C42—24no—
383C42—24no—
384C42—24no—
385C42—28no—
386C42—44no—
387C42—44no—
388C42—416no—
389C23C2×#12011yescharacteristic
390C23C2×C22⋊Q811yescharacteristic
391C23C2×C22⋊Q811yescharacteristic
392C23#52511yescharacteristic
393C23#53211yescharacteristic
394C23#53611yescharacteristic
395C23#53711yescharacteristic
396C23C22×Q1611yescharacteristic
397C23C2×#13511yescharacteristic
398C23C2×AGL(1,ℤ/8)11yescharacteristic
399C23C2×AGL(1,ℤ/8)11yescharacteristic
400C23C2×AGL(1,ℤ/8)11yescharacteristic
401C23C2×#13711yescharacteristic
402C23#57511yescharacteristic
403C23#57511yescharacteristic
404Q8—432no—
405C2×C4—22no—
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—22no—
431C2×C4—28no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—82no—
468C2×C4—82no—
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#45114yes—
486C2×C4#47714yes—
487C2×C4#47714yes—
488C2×C4#47814yes—
489C2×C4C2×C22⋊Q812yes—
490C2×C4#52512yes—
491C2×C4#52912yes—
492C2×C4#53312yes—
493C2×C4#54112yes—
494C2×C4#54212yes—
495C2×C4#54612yes—
496C2×C4#54712yes—
497C2×C4#54812yes—
498C2×C4D4×Q812yes—
499C2×C4#55212yes—
500C2×C4#55712yes—
501C2×C4#55711yescharacteristic
502C2×C4#55711yescharacteristic
503C2×C4Q8211yescharacteristic
504C2×C4#55911yescharacteristic
505C8—44no—
506C8—82no—
507C22#239611yescharacteristic
508C22#240911yescharacteristic
509C22#243011yescharacteristic
510C22#256511yescharacteristic
511C22#258211yescharacteristic
512C22#263711yescharacteristic
513C22#284811yescharacteristic
514C22C2×#45111yescharacteristic
515C22#300211yescharacteristic
516C22#300311yescharacteristic
517C22#300611yescharacteristic
518C22C2×#47811yescharacteristic
519C22C2×#47711yescharacteristic
520C22C2×#47711yescharacteristic
521C22#307611yescharacteristic
522C22#307711yescharacteristic
523C22#307711yescharacteristic
524C22#307811yescharacteristic
525C22#307911yescharacteristic
526C22#310411yescharacteristic
527C22#310511yescharacteristic
528C22#311211yescharacteristic
529C22#311411yescharacteristic
530C22#317011yescharacteristic
531C22#317111yescharacteristic
532C22#318111yescharacteristic
533C22#318411yescharacteristic
534C22#322811yescharacteristic
535C22#323011yescharacteristic
536C22#323811yescharacteristic
537C22#324011yescharacteristic
538C22#324811yescharacteristic
539C22#324911yescharacteristic
540C22#325111yescharacteristic
541C22#325311yescharacteristic
542C4—24no—
543C4—24no—
544C4—24no—
545C4—24no—
546C4—42no—
547C4—48no—
548C4—42no—
549C4—84no—
550C4—82no—
551C4—82no—
552C4—161no—
553C4—161no—
554C4#245112yes—
555C4#245312yes—
556C4#245812yes—
557C4#246212yes—
558C2#1487811yescharacteristic
559C2#2377711yescharacteristic
560C2#2378111yescharacteristic
561C2#2430211yescharacteristic
562C2#2430411yescharacteristic
563C2#2549811yescharacteristic
564C2#2550211yescharacteristic
565C2#2917911yescharacteristic
566C2#2918211yescharacteristic
567C2#2928811yescharacteristic
568C2#2929011yescharacteristic
569C2#3014611yescharacteristic
570C2#3015111yescharacteristic
571C2#3295711yescharacteristic
572C2#3295911yescharacteristic
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
87. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×#20872
#146992
#113842
#113242
#112672
#112282
#109072
#108852
#108322