Finite Group Databasebeta

← #415004

Subgroups — #415004

Order 512.

1316 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2565C211yesmaximal, characteristic
2C2×#1991C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4C2×#1805C211yesmaximal, characteristic
5C2×C4×#103C211yesmaximal, characteristic
6#13954C211yesmaximal, characteristic
7#11208C212yesmaximal
8#11207C212yesmaximal
9#11067C212yesmaximal
10#11067C212yesmaximal
11#10834C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2565—24no—
19#2565—24no—
20C2×#398—22no—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#387—22no—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1991—24no—
32#1989—24no—
33#1805—24no—
34C4×#103—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1444—22no—
44C4⋊#95—22no—
45C23×Q8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×#125—22no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125D414yes—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8D414yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C42—21no—
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103Q818yes—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89#398—216no—
90#391—22no—
91#391—22no—
92#391—22no—
93#391—22no—
94#389—24no—
95#389—24no—
96#389—42no—
97#389—42no—
98#387—216no—
99#384—22no—
100#384—22no—
101#382—22no—
102#382—22no—
103#382—22no—
104#382—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4D414yes—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95D412yes—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#336—42no—
125#336—42no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8C2×D412yes—
135C22×Q8C2×Q814yes—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
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—22no—
160C22×C8C2×Q814yes—
161#125—28no—
162#125—28no—
163#125—44no—
164#125—44no—
165C4×Q8—28no—
166C4×Q8—28no—
167C4×Q8—44no—
168C4×Q8—44no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×Q814yes—
219C2×C4⋊C4#4114yes—
220C2×C42—24no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—82no—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42#4112yes—
239C4⋊C8—44no—
240#103—216no—
241#103—48no—
242C4×C8—44no—
243#95—28no—
244#95—28no—
245#95—28no—
246#95—24no—
247#95—24no—
248#95—44no—
249#95—44no—
250#95—42no—
251#95—42no—
252C24C22×D411yescharacteristic
253C2×Q8—22no—
254C2×Q8—24no—
255C2×Q8—22no—
256C2×Q8—24no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—24no—
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8C4⋊Q814yes—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
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#12012yes—
323C22×C4#12012yes—
324C22×C4#12112yes—
325C22×C4#12112yes—
326C22×C4#12412yes—
327C22×C4#12412yes—
328C22×C4#13514yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×Q812yes—
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4C2×#4111yescharacteristic
337C22×C4H(5,𝔽2)12yes—
338C22×C42−1+412yes—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
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—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—48no—
363C4⋊C4—416no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—48no—
371C4⋊C4—48no—
372C4⋊C4—48no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—88no—
378C4⋊C4—88no—
379C4⋊C4C22⋊Q818yes—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—42no—
385C42—42no—
386C42—48no—
387C42—84no—
388C42#12114yes—
389C42#12114yes—
390C42#12314yes—
391C42#12314yes—
392C23C2×#12011yescharacteristic
393C23C2×#12111yescharacteristic
394C23C2×#12411yescharacteristic
395C23#54612yes—
396C23D4×Q812yes—
397C23C2×#13512yes—
398C23#57512yes—
399C23#57712yes—
400C23#57812yes—
401Q8—24no—
402Q8—24no—
403Q8—42no—
404Q8—44no—
405Q8—42no—
406Q8—44no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—28no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4#39414yes—
464C2×C4#39414yes—
465C2×C4C2×#12111yescharacteristic
466C2×C4C2×#12111yescharacteristic
467C2×C4C2×C22⋊Q812yes—
468C2×C4C2×#12311yescharacteristic
469C2×C4C2×#12311yescharacteristic
470C2×C4C2×C4⋊Q811yescharacteristic
471C2×C4#53211yescharacteristic
472C2×C4#53312yes—
473C2×C4#53511yescharacteristic
474C2×C4#53611yescharacteristic
475C2×C4#53712yes—
476C2×C4#54212yes—
477C2×C4D4×Q812yes—
478C2×C4#55012yes—
479C2×C4#55212yes—
480C2×C4#55412yes—
481C2×C4#55412yes—
482C2×C4#55612yes—
483C8—44no—
484C8—82no—
485C22C2×#39411yescharacteristic
486C22#241712yes—
487C22#243111yescharacteristic
488C22#243411yescharacteristic
489C22#249212yes—
490C22#300212yes—
491C22#301312yes—
492C22#303512yes—
493C22#304211yescharacteristic
494C22#304411yescharacteristic
495C22#313111yescharacteristic
496C22#313511yescharacteristic
497C22#313712yes—
498C22#329612yes—
499C22#330012yes—
500C22#332312yes—
501C22#332712yes—
502C22#338112yes—
503C22#338312yes—
504C22#338812yes—
505C22#339012yes—
506C4—24no—
507C4—24no—
508C4—44no—
509C4—44no—
510C4—48no—
511C4—82no—
512C4—82no—
513C4—82no—
514C4—82no—
515C4—162no—
516C4#242312yes—
517C4#249911yescharacteristic
518C4#250211yescharacteristic
519C4#253812yes—
520C4#262011yescharacteristic
521C4#285711yescharacteristic
522C2#1518211yescharacteristic
523C2#2350411yescharacteristic
524C2#2351211yescharacteristic
525C2#2482112yes—
526C2#2482212yes—
527C2#2557612yes—
528C2#2584212yes—
529C2#2781812yes—
530C2#2782012yes—
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×#25652
C2×#19912
C2×#19892
C2×#18052
C2×C4×#1032
#139542
#112082
#112072
#110672
#108342