Finite Group Databasebeta

← #440893

Subgroups — #440893

Order 512.

1240 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2451C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic
3C2×#2072C211yesmaximal, characteristic
4#13911C211yesmaximal, characteristic
5#11384C212yesmaximal
6#11346C211yesmaximal, characteristic
7#11292C212yesmaximal
8#11190C212yesmaximal
9#10932C211yesmaximal, characteristic
10#10885C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16#2451—24no—
17#2451—24no—
18C2×#398—21no—
19C2×#395—21no—
20C2×#391—21no—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#387—21no—
26C2×#384C2211yescharacteristic
27C2×#382—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2099—24no—
31#2072—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1442—21no—
43#1442—21no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—42no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—24no—
58C2×C4×Q8—42no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C42—21no—
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—42no—
83#398—28no—
84#395—28no—
85#391—28no—
86#391—22no—
87#391—22no—
88#391—22no—
89#391D414yes—
90#389—22no—
91#389—22no—
92#389—22no—
93#389D414yes—
94#387—28no—
95#384—22no—
96#384—22no—
97#382—44no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D411yescharacteristic
113C2×#95D411yescharacteristic
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—42no—
120#336—22no—
121#336—22no—
122#336—42no—
123#336—42no—
124C22×Q8—24no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—44no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—82no—
135C22×Q8C2×D412yes—
136C22×Q8#4114yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
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—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C4⋊Q8—48no—
165C4⋊Q8—44no—
166C4⋊Q8—44no—
167C4×Q8—48no—
168C4×Q8—44no—
169C4×Q8—44no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4SD1614yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4#4114yes—
212C2×C4⋊C4#4114yes—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—44no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230#103—48no—
231#103—48no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—22no—
240#95—22no—
241#95—44no—
242#95—44no—
243#95—82no—
244#95—82no—
245C24C22×D411yescharacteristic
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—48no—
255C2×Q8—48no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—88no—
264C2×Q8—84no—
265C2×Q8—88no—
266C2×Q8#12414yes—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
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#12012yes—
327C22×C4#12012yes—
328C22×C4C2×SD1612yes—
329C22×C4C2×SD1612yes—
330C22×C4C2×SD1612yes—
331C22×C4#13712yes—
332C22×C4#13712yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4C2×#4112yes—
337C22×C4C2×#4112yes—
338C22×C4H(5,𝔽2)12yes—
339C22×C42−1+412yes—
340C22×C42−1+412yes—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
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—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—28no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—84no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C4⋊C4#12414yes—
379C42—24no—
380C42—24no—
381C42—48no—
382C42—48no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—42no—
388C42—42no—
389C23C2×#12011yescharacteristic
390C23#52511yescharacteristic
391C23#53611yescharacteristic
392C23#54612yes—
393C23#54812yes—
394C23C22×SD1611yescharacteristic
395C23C2×#13711yescharacteristic
396C23C2×#13711yescharacteristic
397C23#57511yescharacteristic
398C23#57612yes—
399C23#57812yes—
400Q8—28no—
401Q8—44no—
402Q8—44no—
403Q8—44no—
404Q8—816no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4#44814yes—
471C2×C4C2×#12411yescharacteristic
472C2×C4C2×#12411yescharacteristic
473C2×C4#52912yes—
474C2×C4#53511yescharacteristic
475C2×C4#53611yescharacteristic
476C2×C4#53912yes—
477C2×C4#54112yes—
478C2×C4#54212yes—
479C2×C4#54612yes—
480C2×C4#54712yes—
481C2×C4#54712yes—
482C2×C4#54812yes—
483C2×C4#55112yes—
484C2×C4#55212yes—
485C2×C4#55212yes—
486C2×C4#56312yes—
487C8—82no—
488C8—82no—
489C22#240611yescharacteristic
490C22#243511yescharacteristic
491C22#243611yescharacteristic
492C22#247712yes—
493C22#269012yes—
494C22C2×#44811yescharacteristic
495C22#300711yescharacteristic
496C22#300811yescharacteristic
497C22#301111yescharacteristic
498C22#310111yescharacteristic
499C22#311311yescharacteristic
500C22#312412yes—
501C22#322911yescharacteristic
502C22#323611yescharacteristic
503C22#324312yes—
504C22#329812yes—
505C22#331412yes—
506C22#331912yes—
507C22#332812yes—
508C22#336212yes—
509C22#337412yes—
510C22#337512yes—
511C22#337712yes—
512C4—24no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—44no—
518C4—44no—
519C4—82no—
520C4—84no—
521C4—81no—
522C4—82no—
523C4—162no—
524C4#248812yes—
525C4#248912yes—
526C4#269812yes—
527C4#272312yes—
528C2#1567011yescharacteristic
529C2#2406311yescharacteristic
530C2#2407311yescharacteristic
531C2#2588411yescharacteristic
532C2#2589711yescharacteristic
533C2#2593211yescharacteristic
534C2#2594311yescharacteristic
535C2#2742412yes—
536C2#2744112yes—
537C2#3145612yes—
538C2#3148512yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#24512
C2×#20992
C2×#20722
#139112
#113842
#113462
#112922
#111902
#109322
#108852
#107892