Finite Group Databasebeta

← #394836

Subgroups — #394836

Order 512.

1788 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35224C211yesmaximal, characteristic
2C22×#382C211yesmaximal, characteristic
3C2×#2061C212yesmaximal, complemented
4#20844C212yesmaximal
5#11339C214yesmaximal, complemented
6#11211C212yesmaximal, complemented
7#10727C211yesmaximal, characteristic
8#10719C212yesmaximal
9C2×#544C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C2×#428C2214yes—
14C2×#407C2212yes—
15#2848—22no—
16#2627—22no—
17#2442—24no—
18#2440—24no—
19#2317C2212yes—
20#2316C2211yescharacteristic
21C2×#382—28no—
22C2×#382—22no—
23C2×#382C2212yes—
24C2×#382C2212yes—
25#2280C2212yes—
26C22×#95—22no—
27#2061—28no—
28#1853—24no—
29#1852—24no—
30C2×#343C2212yes—
31C2×#343C2212yes—
32#1736C2214yes—
33C2×#339C2214yes—
34C2×#339C2214yes—
35C24×C4—21no—
36C24×C4—21no—
37C24×C4D412yes—
38C24×C4C2311yescharacteristic
39C22×M16C2314yes—
40#544—24no—
41#544—42no—
42C2×C4⋊Q8—22no—
43C2×#125—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—24no—
46C2×#117C2312yes—
47C2×#117C2312yes—
48C22×C4⋊C4—24no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4—42no—
54C22×C4⋊C4—42no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C42—21no—
60C22×C42—22no—
61C22×C42—24no—
62C22×C42—24no—
63C22×C42Q814yescomplemented
64C22×C42C2312yes—
65#428—28no—
66#428—44no—
67C2×#107—24no—
68C2×#106—24no—
69#407—24no—
70#407Q818yescomplemented
71C2×C22⋊C8—24no—
72#400—24no—
73#398—28no—
74#395—24no—
75#389—24no—
76#389—24no—
77#385—28no—
78#385—24no—
79#384—22no—
80#382—216no—
81#382—22no—
82#382—24no—
83#382—24no—
84#382—24no—
85#382D418yes—
86#381—22no—
87#380—24no—
88#380—22no—
89#380D414yes—
90#380D414yes—
91#379—22no—
92C4×C22⋊C4—24no—
93C2×#95—216no—
94C2×#95—24no—
95C2×#95—24no—
96#343—24no—
97#343—42no—
98#340—28no—
99#339—28no—
100#339—28no—
101C25C2×D411yescharacteristic
102C22×Q8—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—24no—
108C23×C4—24no—
109C23×C4—24no—
110C23×C4—22no—
111C23×C4—24no—
112C23×C4—22no—
113C23×C4—24no—
114C23×C4—24no—
115C23×C4—42no—
116C23×C4—44no—
117C23×C4—44no—
118C23×C4—48no—
119C23×C4—42no—
120C23×C4—48no—
121C23×C4—42no—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D414yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×Q814yes—
127C23×C4C2×Q812yes—
128C23×C4#4112yes—
129C23×C4C2411yescharacteristic
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—24no—
133C2×M16—24no—
134C2×M16—24no—
135C2×M16C2×Q818yes—
136C22×C8—24no—
137C4⋊Q8—48no—
138#125—48no—
139C22⋊Q8—44no—
140C22⋊Q8—44no—
141C22⋊Q8—88no—
142#117—24no—
143#117—24no—
144#117—24no—
145#117—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—416no—
162C2×C4⋊C4—416no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4C2×D414yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D414yes—
170C2×C4⋊C4C2×D414yes—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—216no—
178C2×C42—24no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—216no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42C2×D414yes—
186C2×C42C2×Q814yes—
187C2×C42#4114yes—
188#107—48no—
189#106—48no—
190C22⋊C8—216no—
191#95—216no—
192#95—28no—
193#95—24no—
194#95—216no—
195#95—24no—
196#95—24no—
197C24—21no—
198C24—21no—
199C24—21no—
200C24—21no—
201C24—24no—
202C24—24no—
203C24#12012yes—
204C24C22⋊Q812yes—
205C24C4⋊Q812yes—
206C24C22×D411yescharacteristic
207C2×Q8—44no—
208C2×Q8—84no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—28no—
220C22×C4—28no—
221C22×C4—216no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—48no—
235C22×C4—48no—
236C22×C4—48no—
237C22×C4—416no—
238C22×C4—416no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—48no—
244C22×C4—416no—
245C22×C4—416no—
246C22×C4—48no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—48no—
250C22×C4—84no—
251C22×C4—88no—
252C22×C4—88no—
253C22×C4—84no—
254C22×C4—84no—
255C22×C4#12012yes—
256C22×C4#12014yes—
257C22×C4C22⋊Q818yes—
258C22×C4C22⋊Q812yes—
259C22×C4C4⋊Q818yes—
260C22×C4C4⋊Q812yes—
261C22×C4C22×D412yes—
262C22×C4C22×D412yes—
263C22×C4C22×Q811yescharacteristic
264C22×C4C22×Q812yes—
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4112yes—
267C22×C4H(5,𝔽2)12yes—
268C22×C42−1+412yes—
269M16—28no—
270M16—28no—
271M16—44no—
272M16—44no—
273C2×C8—28no—
274C2×C8—44no—
275C2×C8—44no—
276C4⋊C4—28no—
277C4⋊C4—24no—
278C4⋊C4—28no—
279C4⋊C4—24no—
280C4⋊C4—28no—
281C4⋊C4—48no—
282C4⋊C4—48no—
283C4⋊C4—44no—
284C4⋊C4—48no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—84no—
288C4⋊C4—88no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—48no—
294C22⋊C4—48no—
295C42—216no—
296C42—24no—
297C42—24no—
298C42—24no—
299C42—44no—
300C42—44no—
301C42C22⋊Q818yescomplemented
302C23—24no—
303C23—24no—
304C23—28no—
305C23—24no—
306C23—24no—
307C23—28no—
308C23—28no—
309C23—28no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23#39514yes—
321C23#45414yes—
322C23#45514yes—
323C23C2×#12011yescharacteristic
324C23C2×#12011yescharacteristic
325C23C2×C22⋊Q811yescharacteristic
326C23#52511yescharacteristic
327C23C2×C4⋊Q811yescharacteristic
328C23#53411yescharacteristic
329C23#53611yescharacteristic
330Q8—164no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—216no—
338C2×C4—216no—
339C2×C4—28no—
340C2×C4—24no—
341C2×C4—432no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—48no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—82no—
361C2×C4—816no—
362C2×C4—84no—
363C2×C4—816no—
364C2×C4—84no—
365C2×C4—82no—
366C2×C4#39514yes—
367C2×C4C2×C22⋊Q812yes—
368C2×C4C2×C22⋊Q812yes—
369C2×C4#52512yes—
370C2×C4C2×C4⋊Q812yes—
371C2×C4#53912yes—
372C2×C4#54412yes—
373C2×C4D4214yes—
374C2×C4#54714yes—
375C2×C4D4×Q814yes—
376C2×C4#55414yes—
377C8—48no—
378C22—28no—
379C22—28no—
380C22—28no—
381C22—28no—
382C22—22no—
383C22—22no—
384C22—24no—
385C22—22no—
386C22—22no—
387C22—24no—
388C22—24no—
389C22—21no—
390C22—21no—
391C22—24no—
392C22—21no—
393C22—21no—
394C22—24no—
395C22—24no—
396C22#206118yescomplemented
397C22C2×#39511yescharacteristic
398C22#240311yescharacteristic
399C22#240611yescharacteristic
400C22#243411yescharacteristic
401C22#258311yescharacteristic
402C22#262711yescharacteristic
403C22#284811yescharacteristic
404C22C2×#45512yes—
405C22C2×#45412yes—
406C4—24no—
407C4—48no—
408C4—44no—
409C4—84no—
410C4—82no—
411C4—82no—
412C4—82no—
413C4—84no—
414C4#244014yes—
415C4#244914yes—
416C2—21no—
417C2—28no—
418C2—21no—
419C2—21no—
420C2—21no—
421C2#1472011yescharacteristic
422C2C2×#206112yes—
423C2#2390711yescharacteristic
424C2#2391011yescharacteristic
425C2#2407712yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2061C2semidirect
5. #11339C2semidirect
6. #11211C2semidirect
63. C22×C42Q8semidirect
70. #407Q8semidirect
301. C42C22⋊Q8semidirect
396. C22#2061semidirect

Maximal subgroups

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

subgroupindex
#352242
C22×#3822
C2×#20612
#208442
#113392
#112112
#107272
#107192