Finite Group Databasebeta

← #239285

Subgroups — #239285

Order 512.

858 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 858 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60663C211yesmaximal, characteristic
2#21350C211yesmaximal, characteristic
3#17718C212yesmaximal
4#17545C212yesmaximal
5#10548C218yesmaximal, complemented
6#7862C211yesmaximal, characteristic
7C2×#533C2212yes—
8#3166—24no—
9#3165—22no—
10#3165—22no—
11#3162C2214yes—
12#3160—22no—
13#2920C2212yes—
14C2×#431C2211yescharacteristic
15#2911—22no—
16C2×C4×M16C2211yescharacteristic
17C4×#117C2211yescharacteristic
18#1940C2214yes—
19#1851—22no—
20C4×C4⋊C8—22no—
21#1554—24no—
22#1548—22no—
23#1546—22no—
24#1542C2218yes—
25#1508—24no—
26#1506—22no—
27#1506—22no—
28#1503C2218yes—
29#1275C2214yes—
30C22×M16D412yes—
31#533—22no—
32#533—22no—
33#533—22no—
34#533—22no—
35#533—22no—
36#533—22no—
37#533D414yes—
38#533C2314yes—
39C2×C4⋊Q8D412yes—
40C2×C4⋊Q8D412yes—
41C2×#125—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C4×Q8—22no—
45C2×C4×Q8—22no—
46C2×#117—21no—
47C2×#117—22no—
48C22×C42C2311yescharacteristic
49#501—28no—
50#501—28no—
51#500—22no—
52#500—22no—
53#499—28no—
54#498—24no—
55#432—22no—
56#432—22no—
57#431—22no—
58#431—22no—
59#431—22no—
60#431C2314yes—
61#428—28no—
62#428—28no—
63#427—28no—
64#423—22no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8C2312yes—
67#407—22no—
68C2×C22⋊C8—21no—
69C4×M16—22no—
70C4×M16—22no—
71C4×M16—22no—
72C2×#97C2312yes—
73C2×C4×C8—21no—
74C2×C4×C8D412yes—
75C4×C4⋊C4—21no—
76C4×C4⋊C4C2312yes—
77C4×C22⋊C4—21no—
78C4×C22⋊C4D412yes—
79#376D411yescharacteristic
80#376D411yescharacteristic
81C43—21no—
82#336—24no—
83#333—28no—
84#332—28no—
85#328—28no—
86#328—28no—
87C22×Q8—22no—
88C22×Q8—22no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4#4111yescharacteristic
92C2×M16—24no—
93C2×M16—28no—
94C2×M16—42no—
95C22×C8—22no—
96C22×C8C2×D414yes—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8—22no—
102C4⋊Q8—22no—
103C4⋊Q8C2×D414yes—
104C4⋊Q8C2×D414yes—
105#125—24no—
106#125—24no—
107#125—42no—
108#125—42no—
109C22⋊Q8—24no—
110C22⋊Q8—24no—
111C22⋊Q8—24no—
112C22⋊Q8—24no—
113C22⋊Q8—42no—
114C22⋊Q8—42no—
115C22⋊Q8—42no—
116C22⋊Q8—42no—
117C4×Q8—24no—
118C4×Q8—24no—
119C4×Q8—24no—
120C4×Q8—24no—
121C4×Q8—42no—
122C4×Q8—42no—
123C4×Q8—42no—
124C4×Q8—42no—
125#117—24no—
126#117—24no—
127#117—44no—
128#117—42no—
129#117—42no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—42no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—42no—
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C2×C42C2×D411yescharacteristic
148C2×C42#4112yes—
149C2×C42#4112yes—
150C2×C42#4111yescharacteristic
151C2×C42C2411yescharacteristic
152#107—28no—
153#107—28no—
154#106—216no—
155C4⋊C8—24no—
156C4⋊C8—24no—
157C4⋊C8—48no—
158C4⋊C8—42no—
159C22⋊C8—24no—
160C22⋊C8—42no—
161C22⋊C8—42no—
162#97—24no—
163#97—42no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—24no—
167C4×C8—41no—
168C4×C8—41no—
169#95—21no—
170#95—21no—
171#95C2×D412yes—
172#95C2×D412yes—
173C24#12111yescharacteristic
174C2×Q8—24no—
175C2×Q8—24no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—84no—
179C2×Q8—84no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4#12012yes—
188C22×C4#12112yes—
189C22×C4#12111yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12112yes—
193C22×C4#12312yes—
194C22×C4#12312yes—
195C22×C4#12412yes—
196C22×C4C22×D411yescharacteristic
197C22×C4C22×D411yescharacteristic
198C22×C4C2×#4111yescharacteristic
199M16—216no—
200M16—84no—
201C2×C8—24no—
202C2×C8—24no—
203C2×C8—42no—
204C2×C8—44no—
205C2×C8—44no—
206C2×C8#13718yes—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C22⋊C4—44no—
223C42—24no—
224C42—44no—
225C42—42no—
226C42—44no—
227C42—44no—
228C42—44no—
229C42#12011yescharacteristic
230C42#12011yescharacteristic
231C42#12112yes—
232C42#12112yes—
233C42#12312yes—
234C42#12312yes—
235C42#12411yescharacteristic
236C42#12411yescharacteristic
237C42C22×D412yes—
238C42C2×#4112yes—
239C23—22no—
240C23—41no—
241C23—41no—
242C23#39412yes—
243C23C2×#12111yescharacteristic
244Q8—84no—
245Q8—84no—
246Q8—84no—
247Q8—84no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—84no—
268C2×C4—84no—
269C2×C4#39412yes—
270C2×C4#39411yescharacteristic
271C2×C4#39411yescharacteristic
272C2×C4#39411yescharacteristic
273C2×C4#39411yescharacteristic
274C2×C4C2×#12011yescharacteristic
275C2×C4C2×#12112yes—
276C2×C4C2×#12312yes—
277C2×C4C2×#12411yescharacteristic
278C2×C4C2×#13714yes—
279C8—28no—
280C8—84no—
281C8—82no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—82no—
288C22C2×#39411yescharacteristic
289C22#303912yes—
290C4—22no—
291C4—22no—
292C4—21no—
293C4—22no—
294C4—21no—
295C4—44no—
296C4—82no—
297C4—84no—
298C4—84no—
299C4#205012yes—
300C4#205312yes—
301C4#300412yes—
302C4#313912yes—
303C2—22no—
304C2—42no—
305C2#2350711yescharacteristic
306C2C2×#205011yescharacteristic
307C2C2×#205311yescharacteristic
subgroupG/H
centerC2×C4C2×#124
commutator, Frattini, cosocleC2×C42C24
socleC22C2×#394
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10548C2semidirect

Maximal subgroups

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

subgroupindex
#606632
#213502
#177182
#175452
#105482
#78622