Finite Group Databasebeta

← #464875

Subgroups — #464875

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic
3C2×#2059C212yesmaximal
4C2×#2026C212yesmaximal
5C2×#1928C211yesmaximal, characteristic
6#15193C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic
8#11346C211yesmaximal, characteristic
9#11327C212yesmaximal
10#11067C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×#125C2212yes—
13C22×#107C2211yescharacteristic
14C22×#106C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18#2552—24no—
19#2552—24no—
20C2×#398—22no—
21C2×#391C2211yescharacteristic
22C2×#391C2212yes—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#384—21no—
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2096—24no—
31#2059—28no—
32#2026—28no—
33#1928—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43#1445—22no—
44#1445—21no—
45#1444—21no—
46C4⋊#95C2211yescharacteristic
47C23×Q8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C2×#125—24no—
52C2×#125—24no—
53C2×#125—24no—
54C2×#125—24no—
55C2×#125—42no—
56C2×#125—42no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C42—21no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×#107—24no—
74C2×#107—22no—
75C2×#106—24no—
76C2×#106—24no—
77C2×#106—42no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—22no—
83C2×#103—24no—
84C2×#103—42no—
85C2×#103Q814yes—
86#398—216no—
87#391—24no—
88#391—22no—
89#391—22no—
90#391—42no—
91#389—28no—
92#389—24no—
93#389—24no—
94#389—24no—
95#389—42no—
96#384—28no—
97#384—22no—
98#384D414yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4D414yes—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95D412yes—
113#340—24no—
114#340—22no—
115#340—24no—
116#340—24no—
117#340D418yes—
118#336—24no—
119#336—24no—
120#336—22no—
121#336—22no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8C2×Q812yes—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8C2×D414yes—
159C22×C8C2×Q812yes—
160#125—48no—
161#125—48no—
162#125—48no—
163#125—48no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
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—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×Q812yes—
211C2×C4⋊C4#4112yes—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—44no—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224#107—44no—
225#106—48no—
226#103—28no—
227#103—48no—
228#103—44no—
229#95—28no—
230#95—24no—
231#95—24no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—42no—
236#95—42no—
237#95—42no—
238#95—81no—
239C24C22×D411yescharacteristic
240C2×Q8—22no—
241C2×Q8—24no—
242C2×Q8—22no—
243C2×Q8—24no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4#12012yes—
313C22×C4#12012yes—
314C22×C4#13512yes—
315C22×C4#13712yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D412yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×Q811yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4H(5,𝔽2)12yes—
322C22×C42−1+412yes—
323C22×C42−1+411yescharacteristic
324C22×C42−1+411yescharacteristic
325C22×C42−1+412yes—
326C2×C8—22no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—42no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—44no—
353C4⋊C4—416no—
354C4⋊C4—416no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—44no—
360C4⋊C4—48no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—84no—
365C4⋊C4—88no—
366C4⋊C4—88no—
367C4⋊C4C22⋊Q814yes—
368C4⋊C4#12714yes—
369C42—22no—
370C42—22no—
371C42—24no—
372C42—48no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42—42no—
377C23C2×#12011yescharacteristic
378C23#53612yes—
379C23D4212yes—
380C23#54811yescharacteristic
381C23D4×Q811yescharacteristic
382C23C2×#13511yescharacteristic
383C23C2×#13711yescharacteristic
384C23#57512yes—
385C23#57711yescharacteristic
386C23#57712yes—
387C23#57811yescharacteristic
388Q8—42no—
389Q8—44no—
390Q8—42no—
391Q8—81no—
392Q8—81no—
393Q8—82no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4#45214yes—
452C2×C4C2×C22⋊Q811yescharacteristic
453C2×C4C2×#12711yescharacteristic
454C2×C4#53511yescharacteristic
455C2×C4#53611yescharacteristic
456C2×C4#53611yescharacteristic
457C2×C4#53611yescharacteristic
458C2×C4#53612yes—
459C2×C4#53612yes—
460C2×C4#53612yes—
461C2×C4#54212yes—
462C2×C4#54412yes—
463C2×C4D4212yes—
464C2×C4#54712yes—
465C2×C4#54811yescharacteristic
466C2×C4D4×Q811yescharacteristic
467C2×C4D4×Q812yes—
468C2×C4#56312yes—
469C2×C4#56312yes—
470C8—44no—
471C8—42no—
472C8—81no—
473C22#240911yescharacteristic
474C22#243311yescharacteristic
475C22#261411yescharacteristic
476C22#267412yes—
477C22#267712yes—
478C22C2×#45211yescharacteristic
479C22#300511yescharacteristic
480C22#300911yescharacteristic
481C22#301311yescharacteristic
482C22#323212yes—
483C22#323612yes—
484C22#324212yes—
485C22#324412yes—
486C22#326712yes—
487C22#326912yes—
488C22#327812yes—
489C22#328412yes—
490C22#336511yescharacteristic
491C22#337411yescharacteristic
492C22#337612yes—
493C22#338311yescharacteristic
494C22#338711yescharacteristic
495C22#338812yes—
496C4—24no—
497C4—44no—
498C4—44no—
499C4—42no—
500C4—84no—
501C4—82no—
502C4—82no—
503C4—84no—
504C4—82no—
505C4—161no—
506C4#254211yescharacteristic
507C4#262811yescharacteristic
508C4#265811yescharacteristic
509C4#267412yes—
510C4#267712yes—
511C4#282111yescharacteristic
512C2#1651111yescharacteristic
513C2#2433211yescharacteristic
514C2#2434311yescharacteristic
515C2#2575511yescharacteristic
516C2#2576411yescharacteristic
517C2#2964311yescharacteristic
518C2#2964811yescharacteristic
519C2#3081912yes—
520C2#3082612yes—
521C2#3096012yes—
522C2#3096812yes—
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×#25522
C2×#20962
C2×#20592
C2×#20262
C2×#19282
#151932
#113572
#113462
#113272
#110672
#108072