Finite Group Databasebeta

← #241237

Subgroups — #241237

Order 512.

862 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 202 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
1C2×#2844C211yesmaximal, characteristic
2#21986C212yesmaximal
3#20801C211yesmaximal, characteristic
4#10584C214yesmaximal, complemented
5#10583C214yesmaximal, complemented
6#8033C212yesmaximal
7#8030C211yesmaximal, characteristic
8C2×#432C2212yes—
9C2×#428—21no—
10C2×#428—21no—
11C2×C4×M16C2211yescharacteristic
12#2844—22no—
13#2844—24no—
14#2844C2214yes—
15C2×#400—21no—
16C2×#400—22no—
17C2×#395—21no—
18C2×#395—22no—
19#2285C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#382C2212yes—
22#1949—24no—
23#1853—22no—
24#1852—22no—
25#1847C2212yes—
26#1847C2212yes—
27#1847C2214yes—
28#1846—22no—
29#1836C2214yes—
30#1295C2214yes—
31#1295C2218yes—
32C23×Q8—21no—
33C22×M16—21no—
34C2×#117—21no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4D411yescharacteristic
38C22×C4⋊C4D411yescharacteristic
39C22×C4⋊C4D412yes—
40C22×C4⋊C4D412yes—
41C22×C42C2311yescharacteristic
42#432—24no—
43#432Q818yescomplemented
44#428—44no—
45#428—44no—
46C2×#107—22no—
47C2×#107—22no—
48C2×#107—24no—
49C2×#106—22no—
50C2×#106—22no—
51C2×#106—24no—
52C2×C4⋊C8—22no—
53C2×C22⋊C8—22no—
54C4×M16—22no—
55C4×M16Q814yescomplemented
56C2×#97C2314yes—
57C2×#97C2312yes—
58C2×C4×C8—21no—
59#400—28no—
60#400—24no—
61#400—28no—
62#400—24no—
63#395—28no—
64#395—24no—
65#395—28no—
66#395—24no—
67#384C2311yescharacteristic
68#384C2311yescharacteristic
69#382—22no—
70#382—22no—
71#382—22no—
72#382—21no—
73#382D412yes—
74#382D414yes—
75#382C2314yes—
76#382C2312yes—
77#380—22no—
78C4×C4⋊C4—22no—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85#340—24no—
86#340—24no—
87C22×Q8—24no—
88C22×Q8—84no—
89C22×Q8—82no—
90C23×C4—21no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—21no—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D412yes—
96C2×M16—24no—
97C2×M16—24no—
98C2×M16—24no—
99C22×C8—22no—
100C22×C8—24no—
101#117—42no—
102#117—42no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—82no—
116C2×C4⋊C4—82no—
117C2×C4⋊C4—84no—
118C2×C4⋊C4—82no—
119C2×C4⋊C4C2×D414yes—
120C2×C4⋊C4C2×D414yes—
121C2×C4⋊C4C2×D412yes—
122C2×C4⋊C4C2×D412yes—
123C2×C22⋊C4—41no—
124C2×C42—41no—
125C2×C42C2×D412yes—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×Q812yes—
128C2×C42C2×Q814yes—
129C2×C42#4112yes—
130C2×C42C2411yescharacteristic
131#107—44no—
132#107—44no—
133#107—84no—
134#106—44no—
135#106—44no—
136#106—84no—
137C4⋊C8—28no—
138C22⋊C8—28no—
139#97—22no—
140#97—24no—
141#97C2×Q814yescomplemented
142#97C2×Q818yescomplemented
143C4×C8—24no—
144#95—28no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95—24no—
152#95—24no—
153#95—42no—
154C24#12011yescharacteristic
155C2×Q8—84no—
156C2×Q8—88no—
157C2×Q8—816no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4#12012yes—
179C22×C4#12011yescharacteristic
180C22×C4C22⋊Q812yes—
181C22×C4C22⋊Q814yes—
182C22×C4C4⋊Q812yes—
183C22×C4C4⋊Q814yes—
184C22×C4C22×D411yescharacteristic
185C22×C4C22×D412yes—
186M16—216no—
187C2×C8—24no—
188C2×C8—24no—
189C2×C8—24no—
190C2×C8—28no—
191C2×C8—44no—
192C2×C8—44no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—44no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—44no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C4⋊C4—82no—
206C4⋊C4—82no—
207C4⋊C4—84no—
208C4⋊C4—88no—
209C22⋊C4—82no—
210C42—82no—
211C42C22⋊Q812yes—
212C42C22⋊Q814yes—
213C42C4⋊Q814yes—
214C42C4⋊Q812yes—
215C42C22×Q811yescharacteristic
216C42C22×Q812yes—
217C42C2×#4111yescharacteristic
218C23—22no—
219C23—22no—
220C23#39512yes—
221C23#45312yes—
222C23#45612yes—
223C23C2×#12011yescharacteristic
224Q8—816no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—84no—
248C2×C4—88no—
249C2×C4—84no—
250C2×C4—88no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4#39512yes—
256C2×C4#39514yes—
257C2×C4C2×C22⋊Q811yescharacteristic
258C2×C4C2×C22⋊Q812yes—
259C2×C4C2×C4⋊Q811yescharacteristic
260C2×C4C2×C4⋊Q812yes—
261C8—28no—
262C8—48no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—24no—
267C22#206014yes—
268C22C2×#39511yescharacteristic
269C22C2×#45311yescharacteristic
270C22C2×#45611yescharacteristic
271C22#301912yes—
272C22#302012yes—
273C4—21no—
274C4—21no—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—24no—
279C4—82no—
280C4—84no—
281C4—82no—
282C4—84no—
283C4—161no—
284C4—161no—
285C2—24no—
286C2#1899211yescharacteristic
287C2#1899411yescharacteristic
288C2#1900712yes—
289C2C2×#206011yescharacteristic
290C2#2356112yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10584C2semidirect
5. #10583C2semidirect
43. #432Q8semidirect
55. C4×M16Q8semidirect
141. #97C2×Q8semidirect
142. #97C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28442
#219862
#208012
#105842
#105832
#80332
#80302