Finite Group Databasebeta

← #379863

Subgroups — #379863

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3#20911C212yesmaximal
4C2×#1821C211yesmaximal, characteristic
5#11211C214yesmaximal, complemented
6#11055C214yesmaximal, complemented
7#10733C212yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C22×#117C2212yes—
10C23×C4⋊C4C2211yescharacteristic
11C2×#419C2214yes—
12C2×#407C2212yes—
13#2503—24no—
14#2503C418yescomplemented
15C2×#400—22no—
16#2316—24no—
17#2316—22no—
18#2316C2214yes—
19C2×#391C2212yes—
20C2×#389—24no—
21C2×#382C2212yes—
22C2×#381—22no—
23C2×#380—22no—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1873—24no—
26#1872—24no—
27#1821—24no—
28C2×#343C2214yes—
29#1736C2214yes—
30C2×#339C2218yes—
31C23×Q8C2311yescharacteristic
32C24×C4D412yes—
33C24×C4C2311yescharacteristic
34C22×M16C2314yes—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8C2×C414yes—
40C2×#117—22no—
41C2×#117—22no—
42C2×#117—48no—
43C2×#117D414yes—
44C2×#117C2314yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4C2×C414yes—
51C22×C4⋊C4D412yes—
52C22×C4⋊C4D414yes—
53C22×C4⋊C4D414yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C42—22no—
60C22×C42C2312yes—
61#419—48no—
62C2×#103—28no—
63#407—24no—
64#407—24no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67#400—216no—
68#391—24no—
69#391C2×C418yescomplemented
70#389—216no—
71#389—216no—
72#382—24no—
73#382C2×C418yescomplemented
74#381—216no—
75#380—28no—
76#380—28no—
77C4×C22⋊C4—22no—
78C4×C22⋊C4C2×C414yes—
79C2×#95—24no—
80C2×#95—21no—
81C2×#95—24no—
82C2×#95—24no—
83#343—28no—
84#340—28no—
85#339—48no—
86C25C2×D411yescharacteristic
87C22×Q8—21no—
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—24no—
91C22×Q8—24no—
92C22×Q8C22×C412yes—
93C23×C4—24no—
94C23×C4—24no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—48no—
103C23×C4C22⋊C418yes—
104C23×C4C22×C412yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D414yes—
110C23×C4#4114yes—
111C23×C4C2411yescharacteristic
112C2×M16—24no—
113C2×M16—24no—
114C2×M16—28no—
115C2×M16—28no—
116C22×C8—24no—
117C22⋊Q8—28no—
118C22⋊Q8—28no—
119C22⋊Q8—44no—
120C22⋊Q8—44no—
121#117—28no—
122#117—28no—
123#117—48no—
124#117—48no—
125#117—48no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—216no—
140C2×C4⋊C4—48no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—88no—
144C2×C4⋊C4C22⋊C418yes—
145C2×C4⋊C4C22×C412yes—
146C2×C4⋊C4C22×C414yes—
147C2×C4⋊C4C2×D418yes—
148C2×C4⋊C4C2×D414yes—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—48no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—42no—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—24no—
161C2×C42—24no—
162C2×C42—22no—
163C2×C42—48no—
164C2×C42C22×C414yes—
165#103—416no—
166C22⋊C8—44no—
167C22⋊C8—44no—
168#95—216no—
169#95—24no—
170#95—216no—
171#95—216no—
172C24—22no—
173C24—22no—
174C24—24no—
175C24—42no—
176C24C2×C22⋊C412yes—
177C24#12011yescharacteristic
178C24#12011yescharacteristic
179C24#12312yes—
180C24C22×D411yescharacteristic
181C2×Q8—24no—
182C2×Q8—28no—
183C2×Q8—42no—
184C2×Q8—44no—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×Q8—48no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—28no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—28no—
196C22×C4—216no—
197C22×C4—24no—
198C22×C4—28no—
199C22×C4—24no—
200C22×C4—28no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—28no—
205C22×C4—48no—
206C22×C4—416no—
207C22×C4—48no—
208C22×C4—48no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—48no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—48no—
222C22×C4—88no—
223C22×C4—84no—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C418yes—
226C22×C4C4×D414yes—
227C22×C4C4×D414yes—
228C22×C4#12012yes—
229C22×C4#12114yes—
230C22×C4#12114yes—
231C22×C4#12312yes—
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D412yes—
234C22×C4C22×D411yescharacteristic
235C22×C4C2×#4112yes—
236C22×C4H(5,𝔽2)14yes—
237C22×C42−1+414yes—
238M16—48no—
239M16—48no—
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—44no—
243C4⋊C4—28no—
244C4⋊C4—28no—
245C4⋊C4—216no—
246C4⋊C4—24no—
247C4⋊C4—28no—
248C4⋊C4—28no—
249C4⋊C4—48no—
250C4⋊C4—42no—
251C4⋊C4—48no—
252C4⋊C4—816no—
253C22⋊C4—48no—
254C22⋊C4—48no—
255C22⋊C4—48no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C42—28no—
259C42—48no—
260C42—48no—
261C42—44no—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—24no—
268C23—21no—
269C23—21no—
270C23—24no—
271C23—24no—
272C23—24no—
273C23—22no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—48no—
278C23—48no—
279C23#38612yes—
280C23#38612yes—
281C23#45514yes—
282C23C22×C22⋊C411yescharacteristic
283C23C2×#12011yescharacteristic
284C23C2×#12311yescharacteristic
285C23#53512yes—
286C23#53612yes—
287Q8—48no—
288Q8—84no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—28no—
294C2×C4—28no—
295C2×C4—28no—
296C2×C4—28no—
297C2×C4—28no—
298C2×C4—416no—
299C2×C4—48no—
300C2×C4—48no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—48no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—84no—
312C2×C4—88no—
313C2×C4—84no—
314C2×C4—84no—
315C2×C4—816no—
316C2×C4—82no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4#38614yes—
322C2×C4C2×C4×D412yes—
323C2×C4#52014yes—
324C2×C4C2×#12112yes—
325C2×C4#53414yes—
326C2×C4#53614yes—
327C2×C4#53714yes—
328C2×C4#53918yes—
329C8—84no—
330C22—22no—
331C22—24no—
332C22—24no—
333C22—21no—
334C22—21no—
335C22—24no—
336C22—24no—
337C22—21no—
338C22—21no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—48no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22#188514yes—
348C22C2×#38611yescharacteristic
349C22#231112yes—
350C22#261311yescharacteristic
351C22#261411yescharacteristic
352C22#262312yes—
353C22C2×#45512yes—
354C22#302012yes—
355C4—24no—
356C4—48no—
357C4—84no—
358C4—82no—
359C4—88no—
360C4—82no—
361C4#232214yes—
362C4#261814yes—
363C2—22no—
364C2—22no—
365C2—24no—
366C2—42no—
367C2#1425611yescharacteristic
368C2C2×#188511yescharacteristic
369C2#2095411yescharacteristic
370C2#2961812yes—
371C2#2965212yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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
5. #11211C2semidirect
6. #11055C2semidirect
14. #2503C4semidirect
69. #391C2×C4semidirect
73. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#23162
#209112
C2×#18212
#112112
#110552
#107332