Finite Group Databasebeta

← #440873

Subgroups — #440873

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2565C211yesmaximal, characteristic
2C2×#2074C211yesmaximal, characteristic
3C2×#1835C211yesmaximal, characteristic
4#15235C211yesmaximal, characteristic
5#11385C212yesmaximal
6#11372C211yesmaximal, characteristic
7#11267C212yesmaximal
8#11190C212yesmaximal
9#10926C211yesmaximal, characteristic
10#10874C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×C4×C8C2211yescharacteristic
17#2565—24no—
18#2565—24no—
19C2×#398—22no—
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#387—22no—
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#2074—24no—
28#1835—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1443—22no—
39#1442—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#125—22no—
47C2×#125—22no—
48C2×#125—24no—
49C2×#125—42no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—24no—
53C2×C4×Q8—42no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
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×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#103—24no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—42no—
74C2×C4×C8—24no—
75C2×C4×C8—22no—
76#398—216no—
77#391—22no—
78#391—22no—
79#391—22no—
80#391D414yes—
81#387—216no—
82#384—22no—
83#384—22no—
84#382—22no—
85#382—22no—
86#382—22no—
87#382D414yes—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D412yes—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—42no—
104#340—42no—
105#340—42no—
106#336—24no—
107#336—22no—
108#336—24no—
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8C2×D412yes—
116C22×Q8#4114yes—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2411yescharacteristic
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8#4114yes—
142#125—48no—
143#125—48no—
144C4×Q8—48no—
145C4×Q8—48no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
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—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—24no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42#4114yes—
200#103—48no—
201#103—48no—
202C4×C8—28no—
203#95—28no—
204#95—28no—
205#95—28no—
206#95—24no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—82no—
211#95—82no—
212#95—82no—
213C24C22×D411yescharacteristic
214C2×Q8—22no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—24no—
219C2×Q8—24no—
220C2×Q8—24no—
221C2×Q8—24no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8#12414yes—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4#12014yes—
285C22×C4#12412yes—
286C22×C4#12412yes—
287C22×C4#13514yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C2×#4112yes—
291C22×C4C2×#4112yes—
292C22×C4C2×#4112yes—
293C22×C4H(5,𝔽2)12yes—
294C22×C42−1+412yes—
295C22×C42−1+412yes—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—48no—
314C4⋊C4—416no—
315C4⋊C4—48no—
316C4⋊C4—48no—
317C4⋊C4—48no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C42—22no—
330C42—22no—
331C42—24no—
332C42—24no—
333C42—24no—
334C42—48no—
335C42—41no—
336C42—41no—
337C42—42no—
338C23C2×#12011yescharacteristic
339C23C2×#12411yescharacteristic
340C23#53511yescharacteristic
341C23#54712yes—
342C23#54812yes—
343C23C2×#13512yes—
344C23#57512yes—
345C23#57712yes—
346C23#57812yes—
347Q8—28no—
348Q8—44no—
349Q8—44no—
350Q8—44no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4#52511yescharacteristic
410C2×C4C2×#12411yescharacteristic
411C2×C4#52912yes—
412C2×C4#53611yescharacteristic
413C2×C4#53611yescharacteristic
414C2×C4#53912yes—
415C2×C4#54112yes—
416C2×C4#54212yes—
417C2×C4#54612yes—
418C2×C4#54612yes—
419C2×C4#54712yes—
420C2×C4#54812yes—
421C2×C4#55112yes—
422C2×C4#55212yes—
423C2×C4#55212yes—
424C2×C4#56312yes—
425C8—44no—
426C8—82no—
427C22#240611yescharacteristic
428C22#243511yescharacteristic
429C22#243611yescharacteristic
430C22#248912yes—
431C22#272312yes—
432C22#300212yes—
433C22#301312yes—
434C22#313111yescharacteristic
435C22#313712yes—
436C22#314211yescharacteristic
437C22#321212yes—
438C22#322111yescharacteristic
439C22#322311yescharacteristic
440C22#333612yes—
441C22#333712yes—
442C22#334812yes—
443C22#334912yes—
444C22#336412yes—
445C22#336512yes—
446C22#337112yes—
447C22#337712yes—
448C4—42no—
449C4—42no—
450C4—44no—
451C4—44no—
452C4—44no—
453C4—82no—
454C4—82no—
455C4—82no—
456C4—162no—
457C4—162no—
458C4#247712yes—
459C4#248812yes—
460C4#269012yes—
461C4#269812yes—
462C2#1567011yescharacteristic
463C2#2409912yes—
464C2#2585811yescharacteristic
465C2#2586811yescharacteristic
466C2#2593712yes—
467C2#2773212yes—
468C2#2774512yes—
469C2#3243412yes—
470C2#3243612yes—
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×#20742
C2×#18352
#152352
#113852
#113722
#112672
#111902
#109262
#108742
#107892