Finite Group Databasebeta

← #120289

Subgroups — #120289

Order 512.

524 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 140 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#21969C211yesmaximal, characteristic
2#20793C211yesmaximal, characteristic
3#8097C212yesmaximal
4#8004C212yesmaximal
5C2×#1292C211yesmaximal, characteristic
6#2920C2212yes—
7C2×#424—22no—
8C2×#423C2211yescharacteristic
9C2×#423C2212yes—
10C22×C4⋊C8C2211yescharacteristic
11C4×#117C2211yescharacteristic
12#1950—24no—
13#1843C414yes—
14#1842—22no—
15#1810—22no—
16#1303—22no—
17#1303—22no—
18#1292—24no—
19#1292—22no—
20C22×M16—21no—
21C22×M16—22no—
22C23×C8D412yes—
23C2×#117—21no—
24C22×C42C2311yescharacteristic
25#432—22no—
26#432—22no—
27#431—22no—
28#431—22no—
29C2×#108—24no—
30#424—48no—
31#423—22no—
32#423—22no—
33#423—22no—
34#423—22no—
35#423—22no—
36#423—22no—
37#423D414yes—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—28no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8C2×C412yes—
44C2×C4⋊C8C2×C412yes—
45C2×C4⋊C8D412yes—
46C2×C4⋊C8Q814yes—
47#407—22no—
48C4×M16—22no—
49C4×C4⋊C4—21no—
50C4×C4⋊C4D411yescharacteristic
51C4×C4⋊C4D411yescharacteristic
52C4×C22⋊C4—21no—
53C4×C22⋊C4Q812yes—
54#376C2×C412yes—
55C43—21no—
56#341—24no—
57C23×C4C2×D411yescharacteristic
58C23×C4C2×Q811yescharacteristic
59C23×C4#4111yescharacteristic
60C2×M16—24no—
61C2×M16—44no—
62C2×M16—41no—
63C2×M16—42no—
64C2×M16—41no—
65C2×M16—42no—
66C2×M16—82no—
67C2×M16—82no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—24no—
72C22×C8—44no—
73C22×C8C22⋊C414yes—
74#117—24no—
75#117—24no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C42—22no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—22no—
83C2×C42C22×C411yescharacteristic
84C2×C42C2×D412yes—
85C2×C42C2×D411yescharacteristic
86C2×C42C2×D411yescharacteristic
87C2×C42C2×Q812yes—
88C2×C42#4112yes—
89C2×C42#4112yes—
90C2×C42#4111yescharacteristic
91#108—84no—
92C4⋊C8—24no—
93C4⋊C8—24no—
94C4⋊C8—24no—
95C4⋊C8—24no—
96C4⋊C8—24no—
97C4⋊C8—24no—
98C4⋊C8—24no—
99C4⋊C8—416no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102C4⋊C8—42no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105#97—42no—
106C4×C8—42no—
107#95—22no—
108#95C4⋊C412yes—
109#95C4⋊C412yes—
110C24C22⋊Q811yescharacteristic
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—81no—
119C22×C4C2×C22⋊C411yescharacteristic
120C22×C4C2×C4⋊C411yescharacteristic
121C22×C4#11711yescharacteristic
122C22×C4#12011yescharacteristic
123C22×C4#12111yescharacteristic
124C22×C4#12111yescharacteristic
125C22×C4C22⋊Q812yes—
126C22×C4C22⋊Q811yescharacteristic
127C22×C4#12311yescharacteristic
128C22×C4#12411yescharacteristic
129C22×C4#12512yes—
130C22×C4#12612yes—
131C22×C4C4⋊Q811yescharacteristic
132C22×C4C4⋊Q812yes—
133M16—44no—
134M16—44no—
135M16—82no—
136M16—84no—
137M16—82no—
138M16—82no—
139M16—82no—
140C2×C8—24no—
141C2×C8—44no—
142C2×C8—44no—
143C2×C8—44no—
144C2×C8—42no—
145C2×C8—44no—
146C2×C8—44no—
147C2×C8—82no—
148C2×C8—84no—
149C4⋊C4—24no—
150C4⋊C4—24no—
151C22⋊C4—28no—
152C42—24no—
153C42—24no—
154C42—42no—
155C42—41no—
156C42—41no—
157C42—82no—
158C42C4×D412yes—
159C42C4×Q812yes—
160C42#12112yes—
161C42#12112yes—
162C42C22⋊Q812yes—
163C42C22⋊Q812yes—
164C42#12312yes—
165C42#12512yes—
166C23—22no—
167C23—21no—
168C23—21no—
169C23—41no—
170C23#38011yescharacteristic
171C23#39511yescharacteristic
172C23#40011yescharacteristic
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4#38611yescharacteristic
197C2×C4#38712yes—
198C2×C4#38912yes—
199C2×C4#39111yescharacteristic
200C2×C4#39411yescharacteristic
201C2×C4#39612yes—
202C2×C4#39812yes—
203C2×C4#39811yescharacteristic
204C2×C4#45312yes—
205C2×C4#45612yes—
206C2×C4#47112yes—
207C2×C4#47211yescharacteristic
208C2×C4#47311yescharacteristic
209C8—44no—
210C8—84no—
211C8—82no—
212C8—84no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—42no—
221C22—42no—
222C22#144111yescharacteristic
223C22#181511yescharacteristic
224C22#181611yescharacteristic
225C22#206012yes—
226C4—21no—
227C4—21no—
228C4—22no—
229C4—22no—
230C4—22no—
231C4—44no—
232C4—44no—
233C4#188611yescharacteristic
234C4#188711yescharacteristic
235C4#203912yes—
236C4#206312yes—
237C4#206412yes—
238C2—21no—
239C2—21no—
240C2—22no—
241C2—41no—
242C2#1121011yescharacteristic
243C2#1123511yescharacteristic
244C2#1124211yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#219692
#207932
#80972
#80042
C2×#12922