Finite Group Databasebeta

← #241320

Subgroups — #241320

Order 512.

800 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 218 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#37642C211yesmaximal, characteristic
2#21964C212yesmaximal
3#20760C211yesmaximal, characteristic
4#10578C214yesmaximal, complemented
5#10575C214yesmaximal, complemented
6#8041C211yesmaximal, characteristic
7#7897C212yesmaximal
8C2×#423C2212yes—
9C22×C4×C8C2211yescharacteristic
10#2844—22no—
11#2628—24no—
12#2628—22no—
13#2571—22no—
14#2571C2212yes—
15#2551—22no—
16#2551C2212yes—
17#2285C2211yescharacteristic
18#2285C2212yes—
19#2285C2211yescharacteristic
20#1948—22no—
21#1948C2214yes—
22#1947—22no—
23#1947C2214yes—
24#1835—22no—
25#1834C2212yes—
26#1834C2212yes—
27#1809—24no—
28#1735—22no—
29#1296C2212yes—
30#1296C2212yes—
31#1279C2218yes—
32C22×M16—22no—
33C23×C8—21no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8—24no—
36C2×#117—22no—
37C2×#117—21no—
38C2×#117D412yes—
39C22×C42C2311yescharacteristic
40C2×#107—24no—
41C2×#106—24no—
42#423—24no—
43#423Q818yescomplemented
44C2×C4⋊C8—22no—
45C2×C4⋊C8C2314yes—
46C2×C4×C8—21no—
47C2×C4×C8—22no—
48C2×C4×C8Q814yescomplemented
49C2×C4×C8C2312yes—
50#400—28no—
51#400—24no—
52#399—24no—
53#395—28no—
54#395—24no—
55#393—24no—
56#389—24no—
57#389—24no—
58#389—24no—
59#384D412yes—
60#384D412yes—
61#384C2312yes—
62#384C2311yescharacteristic
63#384C2311yescharacteristic
64#382—22no—
65#382—24no—
66#382—22no—
67#382—21no—
68#382D412yes—
69#380—22no—
70#380—22no—
71#380—22no—
72#380D414yes—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4C2314yes—
77#340—24no—
78#340—24no—
79#340—24no—
80#340—24no—
81C22×Q8—24no—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D412yes—
84C2×M16—28no—
85C2×M16—82no—
86C2×M16—82no—
87C22×C8—24no—
88C22×C8—22no—
89C22×C8—24no—
90C22×C8—42no—
91C22×C8—42no—
92C4⋊Q8—84no—
93C4⋊Q8—84no—
94C4⋊Q8—84no—
95#117—44no—
96#117—44no—
97#117—82no—
98#117—82no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4C2×D414yes—
111C2×C4⋊C4C2×D414yes—
112C2×C4⋊C4C2×D412yes—
113C2×C4⋊C4C2×D412yes—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—41no—
117C2×C42—22no—
118C2×C42—24no—
119C2×C42—41no—
120C2×C42C2×D412yes—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×Q812yes—
123C2×C42C2×Q814yes—
124C2×C42#4112yes—
125C2×C42C2411yescharacteristic
126#107—84no—
127#106—84no—
128C4⋊C8—28no—
129C4⋊C8—24no—
130C4⋊C8C2×Q818yes—
131C4×C8—24no—
132C4×C8—22no—
133C4×C8C2×Q814yes—
134#95—24no—
135#95—24no—
136#95—28no—
137#95—24no—
138#95—24no—
139#95—24no—
140#95—42no—
141C24#12011yescharacteristic
142C2×Q8—88no—
143C2×Q8—84no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—42no—
155C22×C4#12012yes—
156C22×C4#12011yescharacteristic
157C22×C4C22⋊Q812yes—
158C22×C4C22⋊Q814yes—
159C22×C4C4⋊Q812yes—
160C22×C4C4⋊Q814yes—
161C22×C4C22×D411yescharacteristic
162C22×C4C22×D412yes—
163M16—88no—
164C2×C8—24no—
165C2×C8—28no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—48no—
169C2×C8—44no—
170C2×C8—44no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—44no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—82no—
189C4⋊C4—82no—
190C4⋊C4—88no—
191C4⋊C4—84no—
192C22⋊C4—82no—
193C22⋊C4—84no—
194C22⋊C4—82no—
195C42—44no—
196C42—44no—
197C42—44no—
198C42—82no—
199C42C22⋊Q812yes—
200C42C22⋊Q814yes—
201C42C4⋊Q812yes—
202C42C4⋊Q814yes—
203C42C22×Q811yescharacteristic
204C42C22×Q812yes—
205C42C2×#4111yescharacteristic
206C23—22no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23#39512yes—
211C23C2×#12011yescharacteristic
212Q8—168no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4#39514yes—
237C2×C4#39512yes—
238C2×C4#45412yes—
239C2×C4#45512yes—
240C2×C4C2×C22⋊Q811yescharacteristic
241C2×C4C2×C22⋊Q812yes—
242C2×C4C2×C4⋊Q811yescharacteristic
243C2×C4C2×C4⋊Q812yes—
244C2×C4#57612yes—
245C2×C4#57712yes—
246C2×C4#57712yes—
247C2×C4#57812yes—
248C8—44no—
249C8—48no—
250C22—24no—
251C22—22no—
252C22—41no—
253C22—41no—
254C22—42no—
255C22C2×#39511yescharacteristic
256C22#301212yes—
257C22#301312yes—
258C22C2×#45511yescharacteristic
259C22C2×#45411yescharacteristic
260C4—24no—
261C4—21no—
262C4—21no—
263C4—84no—
264C4—82no—
265C4—82no—
266C4—84no—
267C4—162no—
268C4#206114yes—
269C4#308112yes—
270C4#308212yes—
271C4#316312yes—
272C4#316412yes—
273C2—42no—
274C2#1869111yescharacteristic
275C2#1869311yescharacteristic
276C2#1869712yes—
277C2#2355712yes—
278C2C2×#206111yescharacteristic
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. #10578C2semidirect
5. #10575C2semidirect
43. #423Q8semidirect
48. C2×C4×C8Q8semidirect

Maximal subgroups

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

subgroupindex
#376422
#219642
#207602
#105782
#105752
#80412
#78972