Finite Group Databasebeta

← #409999

Subgroups — #409999

Order 512.

1880 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2C23×#107C211yesmaximal, characteristic
3C2×#2503C211yesmaximal, characteristic
4C2×#1892C211yesmaximal, characteristic
5C2×#1823C211yesmaximal, characteristic
6#13904C211yesmaximal, characteristic
7#11325C214yesmaximal, complemented
8#11124C214yesmaximal, complemented
9#10714C211yesmaximal, characteristic
10C24×C8C2211yescharacteristic
11C22×C22⋊Q8C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C2×#483—24no—
15C2×#483—24no—
16C2×#483—24no—
17C22×#107—24no—
18C22×#107—24no—
19C22×#107—24no—
20C22×#107C2214yes—
21C22×#106C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25#2503—24no—
26#2503—24no—
27C2×#400—22no—
28C2×#391C2212yes—
29C2×#382C2212yes—
30C2×#381—22no—
31C2×#381—22no—
32C2×#380C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#1892—24no—
35#1823—24no—
36C2×#340C2212yes—
37C2×#340C2214yes—
38C2×#336C2214yes—
39#1445—22no—
40C4⋊#95C2214yes—
41C23×Q8C2311yescharacteristic
42C24×C4—21no—
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C23×C8—22no—
46C23×C8—22no—
47C23×C8—28no—
48C23×C8D414yes—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8D414yes—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4D414yes—
71C22×C4⋊C4D414yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4D412yes—
79C22×C42C2312yes—
80#483—232no—
81C2×#107—232no—
82C2×#107—28no—
83C2×#107—28no—
84C2×#107—28no—
85C2×#106—24no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—28no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8D414yes—
96#400—216no—
97#391—24no—
98#391—24no—
99#382—24no—
100#382—24no—
101#381—216no—
102#381—48no—
103#380—22no—
104#380—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C2×#95—24no—
108C2×#95—24no—
109C2×#95—28no—
110C2×#95—21no—
111C2×#95D412yes—
112#340—28no—
113#340—24no—
114#336—28no—
115C25C2×D411yescharacteristic
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—24no—
122C22×Q8—24no—
123C22×Q8C2×D412yes—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—44no—
138C23×C4—44no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4C2×D414yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4#4112yes—
149C23×C4C2411yescharacteristic
150C22×C8—216no—
151C22×C8—216no—
152C22×C8—28no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—24no—
156C22×C8—216no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—24no—
160C22×C8—24no—
161C22×C8C2×D414yes—
162C22×C8C2×D414yes—
163C22×C8#4118yes—
164C22⋊Q8—28no—
165C22⋊Q8—28no—
166C22⋊Q8—44no—
167C22⋊Q8—44no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—22no—
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—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×D412yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—44no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—24no—
218C2×C42—24no—
219#107—232no—
220#106—216no—
221#103—216no—
222#103—48no—
223C22⋊C8—28no—
224#95—216no—
225#95—216no—
226#95—24no—
227#95—416no—
228#95—42no—
229C24—24no—
230C24—24no—
231C24#12012yes—
232C24#12112yes—
233C24#12712yes—
234C24#13514yes—
235C24AGL(1,ℤ/8)14yes—
236C24C22×D411yescharacteristic
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—28no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—44no—
249C2×Q8—44no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—28no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—28no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—48no—
284C22×C4—48no—
285C22×C4—416no—
286C22×C4—416no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—88no—
292C22×C4—88no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4—84no—
297C22×C4#12014yes—
298C22×C4#12014yes—
299C22×C4#12012yes—
300C22×C4#12114yes—
301C22×C4#12114yes—
302C22×C4#12112yes—
303C22×C4#12712yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4114yes—
311C22×C42−1+414yes—
312C2×C8—232no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—28no—
316C2×C8—28no—
317C2×C8—24no—
318C2×C8—28no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—44no—
322C2×C8#12318yes—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—48no—
336C4⋊C4—48no—
337C4⋊C4—48no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C22⋊C4—48no—
342C22⋊C4—48no—
343C22⋊C4—84no—
344C42—42no—
345C42—42no—
346C42—44no—
347C23—216no—
348C23—216no—
349C23—24no—
350C23—24no—
351C23—24no—
352C23—24no—
353C23—24no—
354C23—24no—
355C23#39214yes—
356C23#44918yes—
357C23#46318yes—
358C23#49514yes—
359C23#49614yes—
360C23C2×#12011yescharacteristic
361C23C2×#12111yescharacteristic
362C23C2×#12711yescharacteristic
363C23#54714yes—
364C23C2×#13512yes—
365C23C2×AGL(1,ℤ/8)12yes—
366C23#57814yes—
367Q8—44no—
368Q8—44no—
369Q8—81no—
370Q8—81no—
371Q8—82no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—28no—
378C2×C4—28no—
379C2×C4—28no—
380C2×C4—48no—
381C2×C4—48no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—84no—
400C2×C4—82no—
401C2×C4—84no—
402C2×C4—816no—
403C2×C4—816no—
404C2×C4—82no—
405C2×C4—88no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4#39214yes—
410C2×C4C2×#12011yescharacteristic
411C2×C4C2×#12011yescharacteristic
412C2×C4C2×#12111yescharacteristic
413C2×C4C2×#12111yescharacteristic
414C2×C4C2×#12312yes—
415C2×C4#52512yes—
416C2×C4#52512yes—
417C2×C4#53612yes—
418C2×C4#53612yes—
419C2×C4#53612yes—
420C2×C4#54714yes—
421C2×C4#54714yes—
422C2×C4#55214yes—
423C8—28no—
424C8—44no—
425C22—216no—
426C22—28no—
427C22—216no—
428C22—28no—
429C22—24no—
430C22—24no—
431C22#201218yes—
432C22C2×#39211yescharacteristic
433C22#240612yes—
434C22#241312yes—
435C22#250112yes—
436C22C2×#44912yes—
437C22#301112yes—
438C22C2×#46312yes—
439C22#304412yes—
440C22C2×#49511yescharacteristic
441C22C2×#49611yescharacteristic
442C22#315612yes—
443C22#333618yes—
444C22#334318yes—
445C4—24no—
446C4—84no—
447C4—84no—
448C4—84no—
449C4—82no—
450C4—82no—
451C4—162no—
452C4#240612yes—
453C4#241312yes—
454C4#248612yes—
455C4#261411yescharacteristic
456C4#284911yescharacteristic
457C2—28no—
458C2#1506911yescharacteristic
459C2C2×#201211yescharacteristic
460C2#2342511yescharacteristic
461C2#2409414yes—
462C2#2457714yes—
463C2#2793112yes—
464C2#2793912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
7. #11325C2semidirect
8. #11124C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
C23×#1072
C2×#25032
C2×#18922
C2×#18232
#139042
#113252
#111242
#107142