Finite Group Databasebeta

← #122001

Subgroups — #122001

Order 512.

658 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 142 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#35344C211yesmaximal, characteristic
2C4×H(3,ℤ/4)C211yesmaximal, characteristic
3#8074C212yesmaximal
4#7940C212yesmaximal
5#7856C211yesmaximal, characteristic
6#2920C2212yes—
7C2×C4×M16C2211yescharacteristic
8#2864C412yescomplemented
9#2862C412yescomplemented
10#2628—22no—
11#2607—24no—
12#2605—24no—
13#2331—22no—
14#2283C2212yes—
15C4×#117C2211yescharacteristic
16C4×#117C2211yescharacteristic
17C2×H(3,ℤ/4)—22no—
18#1300—24no—
19#1284—24no—
20#1274—22no—
21C22×M16—21no—
22C2×C4⋊Q8—22no—
23C2×#117—21no—
24C2×#117D412yes—
25C22×C42C2311yescharacteristic
26#432—22no—
27#432—22no—
28#431—22no—
29#431Q814yescomplemented
30#423—22no—
31#407—22no—
32C4×M16—22no—
33C4×M16—22no—
34C4×M16—22no—
35C2×#97Q812yescomplemented
36C2×C4×C8—21no—
37#400—24no—
38#399—24no—
39#395—24no—
40#393—24no—
41#391—28no—
42#391—28no—
43#389—24no—
44#387—28no—
45#387—28no—
46#383—22no—
47#382—22no—
48#382C2×C414yes—
49#381D414yes—
50#380—22no—
51C4×C4⋊C4—22no—
52C4×C4⋊C4—22no—
53C4×C4⋊C4—22no—
54C4×C22⋊C4—22no—
55C4×C22⋊C4—22no—
56C4×C22⋊C4—22no—
57#376—21no—
58#376D412yes—
59#376D414yes—
60C43C2×C411yescharacteristic
61C43C2×C411yescharacteristic
62C43Q812yescomplemented
63H(3,ℤ/4)—28no—
64C22×Q8—24no—
65C23×C4C2×D411yescharacteristic
66C23×C4C2×D412yes—
67C2×M16—24no—
68C2×M16—42no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—42no—
72C22×C8—22no—
73C4⋊Q8—44no—
74C4⋊Q8—44no—
75#117—24no—
76#117—24no—
77#117—42no—
78#117—42no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—28no—
83C2×C4⋊C4—41no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—41no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—24no—
91C2×C42—24no—
92C2×C42C22⋊C412yes—
93C2×C42C22⋊C412yes—
94C2×C42C22×C411yescharacteristic
95C2×C42C2×D411yescharacteristic
96C2×C42C2×D412yes—
97C2×C42C2×Q811yescharacteristic
98C2×C42C2×Q812yes—
99C2×C42C2×Q811yescharacteristic
100C2×C42#4111yescharacteristic
101C2×C42#4112yes—
102C2×C42#4111yescharacteristic
103C4⋊C8—24no—
104C4⋊C8—42no—
105C22⋊C8—24no—
106C22⋊C8—42no—
107#97—24no—
108#97—42no—
109C4×C8—24no—
110C4×C8—42no—
111#95—24no—
112#95—24no—
113#95—28no—
114#95—24no—
115#95—24no—
116#95—24no—
117#95—24no—
118#95—42no—
119#95C22⋊C418yes—
120C24#12011yescharacteristic
121C2×Q8—48no—
122C2×Q8—88no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—28no—
127C22×C4—22no—
128C22×C4—42no—
129C22×C4—82no—
130C22×C4C2×C22⋊C412yes—
131C22×C4C2×C22⋊C411yescharacteristic
132C22×C4#12011yescharacteristic
133C22×C4#12012yes—
134C22×C4C22⋊Q811yescharacteristic
135C22×C4C22⋊Q811yescharacteristic
136C22×C4C22⋊Q812yes—
137C22×C4C22⋊Q812yes—
138C22×C4#12411yescharacteristic
139C22×C4#12412yes—
140C22×C4C4⋊Q811yescharacteristic
141C22×C4C4⋊Q812yes—
142M16—44no—
143M16—44no—
144M16—84no—
145C2×C8—24no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—44no—
149C4⋊C4—44no—
150C4⋊C4—48no—
151C4⋊C4—44no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C4⋊C4—88no—
155C22⋊C4—44no—
156C22⋊C4—44no—
157C22⋊C4—84no—
158C42—22no—
159C42—24no—
160C42—22no—
161C42—22no—
162C42—42no—
163C42—82no—
164C42—82no—
165C42—84no—
166C42—84no—
167C42#10414yes—
168C42C4×Q811yescharacteristic
169C42C4×Q812yes—
170C42C4×Q811yescharacteristic
171C42C22⋊Q812yes—
172C42C22⋊Q812yes—
173C42C22⋊Q812yes—
174C42#12412yes—
175C42#12411yescharacteristic
176C42C4⋊Q812yes—
177C42C4⋊Q811yescharacteristic
178C23—22no—
179C23—42no—
180C23#37911yescharacteristic
181C23#39311yescharacteristic
182C23#39511yescharacteristic
183Q8—168no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—24no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—48no—
202C2×C4—48no—
203C2×C4—42no—
204C2×C4—48no—
205C2×C4—84no—
206C2×C4#39112yes—
207C2×C4#39112yes—
208C2×C4#39111yescharacteristic
209C2×C4#39111yescharacteristic
210C2×C4#39311yescharacteristic
211C2×C4#39312yes—
212C2×C4#39511yescharacteristic
213C2×C4#39512yes—
214C2×C4C2×#10412yes—
215C2×C4#42112yes—
216C8—44no—
217C8—84no—
218C22—22no—
219C22—21no—
220C22—21no—
221C22—22no—
222C22—22no—
223C22—82no—
224C22#144611yescharacteristic
225C22#178912yes—
226C4—21no—
227C4—21no—
228C4—22no—
229C4—22no—
230C4—22no—
231C4—22no—
232C4—44no—
233C4—82no—
234C4—88no—
235C4—84no—
236C4#198612yes—
237C4#199411yescharacteristic
238C4#199511yescharacteristic
239C2—22no—
240C2—41no—
241C2—41no—
242C2#1138611yescharacteristic
243C2#1138912yes—
subgroupG/H
center, socleC22#1446
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #2864C4semidirect
9. #2862C4semidirect
29. #431Q8semidirect
35. C2×#97Q8semidirect
62. C43Q8semidirect

Maximal subgroups

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

subgroupindex
#353442
C4×H(3,ℤ/4)2
#80742
#79402
#78562