Finite Group Databasebeta

← #416398

Subgroups — #416398

Order 512.

1144 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1947C211yesmaximal, characteristic
2C2×#1834C211yesmaximal, characteristic
3C2×#1803C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#13926C211yesmaximal, characteristic
6#13911C211yesmaximal, characteristic
7#11379C212yesmaximal
8#11372C212yesmaximal
9#11124C212yesmaximal
10#11052C212yesmaximal
11#10834C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#391C2212yes—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1947—24no—
26#1834—24no—
27#1803—24no—
28C4×#103—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1442—22no—
38#1441—22no—
39C4⋊#95—22no—
40C4⋊#95—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×Q8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8Q814yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D412yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C42—21no—
65C22×C42—21no—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68C2×#107—24no—
69C2×#107—24no—
70C2×#107—42no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8D414yes—
74C2×#103—22no—
75C2×#103—22no—
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—42no—
83C2×C4×C8—22no—
84C2×C4×C8—22no—
85C2×C4×C8D414yes—
86#391—24no—
87#391—42no—
88#384—22no—
89#384—22no—
90#384—22no—
91#384—22no—
92#382—24no—
93#382—42no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C2×#95—24no—
99C2×#95—24no—
100C2×#95—21no—
101C2×#95—24no—
102C2×#95—24no—
103C2×#95D412yes—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—22no—
109#340—22no—
110#336—42no—
111#336—42no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8C2×Q812yes—
124C22×Q8#4112yes—
125C22×Q8#4112yes—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C2×D414yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2411yescharacteristic
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C22×C8C2×D414yes—
157C4×Q8—28no—
158C4×Q8—28no—
159C4×Q8—44no—
160C4×Q8—44no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
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—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
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—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C2×Q812yes—
203C2×C4⋊C4#4112yes—
204C2×C4⋊C4#4112yes—
205C2×C42—24no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42—44no—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×Q812yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222#107—48no—
223C4⋊C8—28no—
224#103—28no—
225#103—28no—
226#103—48no—
227C4×C8—28no—
228#95—24no—
229#95—48no—
230#95—48no—
231#95—42no—
232#95—84no—
233#95—84no—
234C24C22×D411yescharacteristic
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8#12514yes—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4C22⋊Q812yes—
305C22×C4C22⋊Q812yes—
306C22×C4#12312yes—
307C22×C4#12312yes—
308C22×C4#12712yes—
309C22×C4#12712yes—
310C22×C4#13514yes—
311C22×C4#13714yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×Q811yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4H(5,𝔽2)11yescharacteristic
323C22×C4H(5,𝔽2)11yescharacteristic
324C22×C42−1+411yescharacteristic
325C22×C42−1+411yescharacteristic
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—24no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—48no—
360C4⋊C4—48no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—48no—
378C42—84no—
379C42C22⋊Q814yes—
380C42C22⋊Q814yes—
381C42#12314yes—
382C42#12314yes—
383C23C2×C22⋊Q811yescharacteristic
384C23C2×#12311yescharacteristic
385C23C2×#12711yescharacteristic
386C23#54611yescharacteristic
387C23#54611yescharacteristic
388C23#54811yescharacteristic
389C23#54811yescharacteristic
390C23C2×#13512yes—
391C23C2×#13712yes—
392C23#57612yes—
393C23#57812yes—
394Q8—24no—
395Q8—24no—
396Q8—42no—
397Q8—42no—
398Q8—42no—
399Q8—42no—
400Q8—42no—
401Q8—42no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4#39914yes—
446C2×C4#39914yes—
447C2×C4#47918yes—
448C2×C4#48518yes—
449C2×C4#49514yes—
450C2×C4#49714yes—
451C2×C4C2×C22⋊Q811yescharacteristic
452C2×C4C2×C22⋊Q811yescharacteristic
453C2×C4C2×#12311yescharacteristic
454C2×C4C2×#12311yescharacteristic
455C2×C4#52511yescharacteristic
456C2×C4#52511yescharacteristic
457C2×C4C2×#12511yescharacteristic
458C2×C4#52911yescharacteristic
459C2×C4#52911yescharacteristic
460C2×C4#53311yescharacteristic
461C2×C4#53711yescharacteristic
462C2×C4#53711yescharacteristic
463C2×C4#54011yescharacteristic
464C2×C4#54111yescharacteristic
465C2×C4#54411yescharacteristic
466C2×C4#54411yescharacteristic
467C2×C4#54611yescharacteristic
468C2×C4#54611yescharacteristic
469C2×C4#54811yescharacteristic
470C2×C4#54811yescharacteristic
471C2×C4#55211yescharacteristic
472C2×C4#55211yescharacteristic
473C2×C4#55311yescharacteristic
474C2×C4#55311yescharacteristic
475C2×C4#55511yescharacteristic
476C2×C4#55511yescharacteristic
477C2×C4#55611yescharacteristic
478C2×C4#55611yescharacteristic
479C8—44no—
480C8—42no—
481C8—42no—
482C22C2×#39911yescharacteristic
483C22#242011yescharacteristic
484C22#242111yescharacteristic
485C22#243911yescharacteristic
486C22#244211yescharacteristic
487C22#254011yescharacteristic
488C22#254011yescharacteristic
489C22C2×#47912yes—
490C22#308312yes—
491C22C2×#48512yes—
492C22#309812yes—
493C22C2×#49511yescharacteristic
494C22C2×#49711yescharacteristic
495C22#315511yescharacteristic
496C22#315611yescharacteristic
497C22#330212yes—
498C22#330512yes—
499C22#332812yes—
500C22#333312yes—
501C22#336112yes—
502C22#336412yes—
503C22#337412yes—
504C22#337512yes—
505C4—24no—
506C4—44no—
507C4—84no—
508C4—82no—
509C4—82no—
510C4—82no—
511C4—162no—
512C4—162no—
513C4#209718yes—
514C4#242011yescharacteristic
515C4#242111yescharacteristic
516C4#243911yescharacteristic
517C4#244211yescharacteristic
518C4#248511yescharacteristic
519C4#248511yescharacteristic
520C4#262911yescharacteristic
521C4#285111yescharacteristic
522C2#1522311yescharacteristic
523C2C2×#209711yescharacteristic
524C2#2373711yescharacteristic
525C2#2497012yes—
526C2#2499512yes—
527C2#2602312yes—
528C2#2610912yes—
529C2#2839911yescharacteristic
530C2#2840311yescharacteristic
531C2#2841711yescharacteristic
532C2#2841811yescharacteristic
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×#19472
C2×#18342
C2×#18032
C2×C4×#1032
#139262
#139112
#113792
#113722
#111242
#110522
#108342