Finite Group Databasebeta

← #413038

Subgroups — #413038

Order 512.

1908 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2858C211yesmaximal, characteristic
2C2×#2042C214yesmaximal, complemented
3C2×#1961C211yesmaximal, characteristic
4C2×#1961C211yesmaximal, characteristic
5C2×#1960C211yesmaximal, characteristic, complemented
6C2×#1960C211yesmaximal, characteristic, complemented
7#14707C211yesmaximal, characteristic, complemented
8#11009C214yesmaximal, complemented
9#10693C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×#127C2211yescharacteristic
12C22×#124C2212yes—
13C22×#124C2212yes—
14C22×#103C2214yes—
15C22×#103C2214yes—
16C22×#102C2214yes—
17C22×#102C2214yes—
18C22×#97C2211yescharacteristic
19C22×#97C2211yescharacteristic
20#2858—24no—
21#2858—24no—
22C2×#390—24no—
23C2×#389C2212yes—
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C43C2211yescharacteristic
27#2042—216no—
28#1961—24no—
29#1961—24no—
30#1960—24no—
31#1960—24no—
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1436—24no—
35C4⋊#95C2212yes—
36C23×Q8C2312yes—
37C23×D4—22no—
38C23×D4C2312yes—
39C23×C8C2312yes—
40C23×C8C2312yes—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8D414yes—
47C2×#127—22no—
48C2×#127—22no—
49C2×#127—22no—
50C2×#127—22no—
51C2×#127—22no—
52C2×#127D414yes—
53C2×#124—24no—
54C2×#124—24no—
55C2×#124—24no—
56C2×#124—24no—
57C2×#124—28no—
58C2×#124—28no—
59C2×#124—44no—
60C2×#124—44no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—24no—
66C22×C22⋊C4—24no—
67C22×C42D414yes—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×#103—28no—
72C2×#103—28no—
73C2×#103—28no—
74C2×#103—28no—
75C2×#103—44no—
76C2×#103—44no—
77C2×#102—28no—
78C2×#102—28no—
79C2×#102—28no—
80C2×#102—28no—
81C2×#102—44no—
82C2×#102—44no—
83C2×#97—22no—
84C2×#97—22no—
85C2×#97—22no—
86C2×#97—22no—
87C2×#97—22no—
88C2×#97—22no—
89#390—232no—
90#389—28no—
91#382—22no—
92#382—22no—
93#382—22no—
94#382D414yes—
95C2×#95—22no—
96C2×#95—24no—
97C2×#95—22no—
98C2×#95—22no—
99C43—22no—
100C43—22no—
101#336—24no—
102#336D418yescomplemented
103C25—22no—
104C22×Q8—22no—
105C22×Q8—24no—
106C22×Q8—22no—
107C22×Q8—24no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—24no—
112C22×Q8—24no—
113C22×Q8C2×D414yes—
114C22×D4—28no—
115C22×D4—24no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—24no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—24no—
123C22×D4—24no—
124C22×D4—48no—
125C22×D4—48no—
126C22×D4—44no—
127C22×D4—44no—
128C22×D4C2×D414yes—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2411yescharacteristic
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8C2×D414yes—
149C22×C8C2×D414yes—
150C4⋊Q8—28no—
151C4⋊Q8—28no—
152C4⋊Q8—44no—
153C4⋊Q8—44no—
154#127—28no—
155#127—28no—
156#127—44no—
157#127—44no—
158#124—416no—
159#124—416no—
160#124—416no—
161#124—416no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—48no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4C2×D414yes—
175C2×C22⋊C4—216no—
176C2×C22⋊C4—216no—
177C2×C22⋊C4—48no—
178C2×C22⋊C4—48no—
179C2×C22⋊C4—48no—
180C2×C22⋊C4—48no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—28no—
188C2×C42—44no—
189C2×C42—44no—
190C2×C42C2×D414yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42#4114yes—
195C2×C42#4112yes—
196C2×C42#4112yes—
197C2×C42#4112yes—
198#103—416no—
199#103—416no—
200#102—416no—
201#102—416no—
202#97—44no—
203#97—44no—
204#95—28no—
205#95—48no—
206#95—44no—
207#95—44no—
208C24—24no—
209C24—42no—
210C24—42no—
211C24—42no—
212C24—42no—
213C24—42no—
214C24—42no—
215C24—48no—
216C24—48no—
217C24C22×D411yescharacteristic
218C2×Q8—24no—
219C2×Q8—24no—
220C2×Q8—28no—
221C2×Q8—28no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—44no—
225C2×Q8—44no—
226C2×Q8—44no—
227C2×Q8—44no—
228C2×Q8—44no—
229C2×Q8—44no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—44no—
233C2×Q8—44no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—28no—
237C2×D4—28no—
238C2×D4—432no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—416no—
252C2×D4—88no—
253C2×D4—88no—
254C2×D4—816no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
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—44no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4#12014yes—
283C22×C4#12114yes—
284C22×C4#12414yes—
285C22×C4#12414yes—
286C22×C4#12714yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D412yes—
291C22×C4C2×#4111yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4112yes—
295C22×C4H(5,𝔽2)12yes—
296C22×C4H(5,𝔽2)12yes—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C4⋊C4—28no—
312C4⋊C4—44no—
313C4⋊C4—416no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—88no—
317C22⋊C4—432no—
318C22⋊C4—432no—
319C42—24no—
320C42—24no—
321C42—24no—
322C42—24no—
323C42—416no—
324C42#12114yes—
325C42#12114yes—
326C42#12314yes—
327C42#12314yes—
328C23—44no—
329C23—44no—
330C23—44no—
331C23—416no—
332C23—82no—
333C23—84no—
334C23—84no—
335C23—84no—
336C23—84no—
337C23—84no—
338C23—82no—
339C23—84no—
340C23—88no—
341C23—88no—
342C23—88no—
343C23—82no—
344C23—82no—
345C23C2×#12011yescharacteristic
346C23C2×#12111yescharacteristic
347C23C2×#12411yescharacteristic
348C23C2×#12411yescharacteristic
349C23C2×#12711yescharacteristic
350C23#53511yescharacteristic
351C23#53511yescharacteristic
352C23#57514yes—
353C23#57514yes—
354Q8—48no—
355Q8—48no—
356Q8—84no—
357Q8—84no—
358D4—48no—
359D4—48no—
360D4—832no—
361D4—84no—
362D4—84no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—28no—
366C2×C4—28no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—48no—
381C2×C4—44no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—84no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—84no—
393C2×C4—84no—
394C2×C4#39418yes—
395C2×C4C2×#12111yescharacteristic
396C2×C4C2×#12111yescharacteristic
397C2×C4C2×#12311yescharacteristic
398C2×C4C2×#12311yescharacteristic
399C2×C4#53212yes—
400C2×C4#53212yes—
401C2×C4#54012yes—
402C2×C4#54012yes—
403C2×C4D4212yes—
404C2×C4D4212yes—
405C2×C4D4212yes—
406C2×C4#54612yes—
407C2×C4#54612yes—
408C2×C4#55012yes—
409C2×C4#55012yes—
410C2×C4#55912yes—
411C8—44no—
412C8—44no—
413C22—88no—
414C22—84no—
415C22—88no—
416C22—88no—
417C22—164no—
418C22—164no—
419C22—164no—
420C22—162no—
421C22—162no—
422C22C2×#39411yescharacteristic
423C22#243311yescharacteristic
424C22#243511yescharacteristic
425C22#254111yescharacteristic
426C22#254811yescharacteristic
427C22#264111yescharacteristic
428C22#285811yescharacteristic
429C22#300514yes—
430C22#304014yes—
431C22#313512yes—
432C22#314012yes—
433C22#314012yes—
434C22#314212yes—
435C22#315012yes—
436C22#315312yes—
437C22#321914yes—
438C22#321914yes—
439C4—28no—
440C4—44no—
441C4—44no—
442C4—84no—
443C4—84no—
444C4—162no—
445C4#242812yes—
446C4#242812yes—
447C4#244512yes—
448C4#244512yes—
449C2—164no—
450C2#1514011yescharacteristic
451C2#2350612yes—
452C2#2572712yes—
453C2#2586212yes—
454C2#2844612yes—
455C2#2875712yes—
456C2#3022512yes—
457C2#3323311yescharacteristic
458C2#3323711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2042C2semidirect
5. C2×#1960C2semidirect
6. C2×#1960C2semidirect
7. #14707C2semidirect
8. #11009C2semidirect
9. #10693C2semidirect
102. #336D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28582
C2×#20422
C2×#19612
C2×#19602
#147072
#110092
#106932