Finite Group Databasebeta

← #108742

Subgroups — #108742

Order 512.

453 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 148 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#20482C211yesmaximal, characteristic
2#8063C212yesmaximal
3#8006C211yesmaximal, characteristic
4#8001C211yesmaximal, characteristic
5#7960C212yesmaximal
6#2920C2212yes—
7C2×#431C2211yescharacteristic
8C2×#423C2212yes—
9C2×C4×M16C2211yescharacteristic
10#2283C2211yescharacteristic
11#1770—21no—
12C8×C4⋊C4—21no—
13#1753—22no—
14#1746C412yes—
15#1726C412yes—
16#1299C418yescomplemented
17#1293—22no—
18#1292—21no—
19#1292—21no—
20#1286—22no—
21#1286—22no—
22C22×M16—21no—
23C22×M16—22no—
24C22×C42C2311yescharacteristic
25#432—22no—
26#432—22no—
27#431—22no—
28#431—22no—
29#431—22no—
30#431C2×C414yes—
31#423—22no—
32#423—22no—
33#423—22no—
34#423—22no—
35#423C2×C414yes—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8D411yescharacteristic
39C2×C4⋊C8D411yescharacteristic
40#407—22no—
41C2×C22⋊C8D412yes—
42C4×M16—22no—
43C4×M16—22no—
44C4×M16C2×C414yes—
45C2×#97—22no—
46C2×#97C2×C411yescharacteristic
47C2×#97C2×C411yescharacteristic
48C2×C4×C8—21no—
49C2×C4×C8—22no—
50C2×C4×C8C2×C412yes—
51#383—21no—
52#382D411yescharacteristic
53#382D411yescharacteristic
54#381Q812yes—
55#380—21no—
56C4×C4⋊C4—21no—
57C4×C22⋊C4—21no—
58#376C2×C412yes—
59#336—22no—
60#336—22no—
61C23×C4C2×D411yescharacteristic
62C23×C4C2×Q811yescharacteristic
63C23×C4#4111yescharacteristic
64C2×M16—24no—
65C2×M16—24no—
66C2×M16—42no—
67C2×M16—42no—
68C2×M16—42no—
69C2×M16—42no—
70C2×M16—82no—
71C2×M16—82no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—42no—
75C22×C8C22⋊C412yes—
76C22×C8C22⋊C412yes—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C22⋊C4—22no—
80C2×C42—22no—
81C2×C42C22×C411yescharacteristic
82C2×C42C22×C412yes—
83C2×C42C2×D411yescharacteristic
84C2×C42C2×D411yescharacteristic
85C2×C42#4112yes—
86C2×C42#4111yescharacteristic
87C2×C42#4112yes—
88C2×C42#4112yes—
89C4⋊C8—24no—
90C4⋊C8—24no—
91C4⋊C8—24no—
92C4⋊C8—22no—
93C4⋊C8—22no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C4⋊C8—42no—
97C22⋊C8—24no—
98C22⋊C8—24no—
99C22⋊C8—42no—
100#97—21no—
101#97—21no—
102#97—44no—
103#97—42no—
104#97C4212yes—
105#97C4212yes—
106C4×C8—22no—
107C4×C8—22no—
108C4×C8—24no—
109C4×C8—24no—
110C4×C8—44no—
111#95—22no—
112#95—21no—
113#95—21no—
114#95—22no—
115#95C4⋊C412yes—
116#95C4⋊C412yes—
117C24C22⋊Q811yescharacteristic
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—81no—
126C22×C4C2×C22⋊C411yescharacteristic
127C22×C4C2×C4⋊C411yescharacteristic
128C22×C4#11711yescharacteristic
129C22×C4#11712yes—
130C22×C4C4×D412yes—
131C22×C4C4×Q812yes—
132C22×C4#12111yescharacteristic
133C22×C4#12111yescharacteristic
134C22×C4C22⋊Q811yescharacteristic
135C22×C4#12312yes—
136C22×C4#12512yes—
137C22×C4#12612yes—
138M16—48no—
139M16—82no—
140M16—82no—
141M16—82no—
142M16—82no—
143M16—84no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—44no—
155C2×C8—44no—
156C2×C8—84no—
157C4⋊C4—42no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C22⋊C4—82no—
161C42—42no—
162C42—42no—
163C42C2×C4211yescharacteristic
164C42#11711yescharacteristic
165C42#11712yes—
166C42#11711yescharacteristic
167C42#11711yescharacteristic
168C42C4×D412yes—
169C42C4×D412yes—
170C42#12312yes—
171C42#12412yes—
172C42#12612yes—
173C23—21no—
174C23—21no—
175C23—41no—
176C23—41no—
177C23#38011yescharacteristic
178C23#38712yes—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4#37611yescharacteristic
202C2×C4C4×C22⋊C411yescharacteristic
203C2×C4C4×C4⋊C411yescharacteristic
204C2×C4#38111yescharacteristic
205C2×C4#38211yescharacteristic
206C2×C4#38311yescharacteristic
207C2×C4#38712yes—
208C2×C4#38812yes—
209C2×C4#38812yes—
210C2×C4#47112yes—
211C2×C4#47211yescharacteristic
212C2×C4#47311yescharacteristic
213C8—24no—
214C8—42no—
215C8—42no—
216C8—42no—
217C8—84no—
218C8—84no—
219C22—21no—
220C22—21no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—81no—
225C22—81no—
226C22#143411yescharacteristic
227C22#181511yescharacteristic
228C22#181611yescharacteristic
229C4—21no—
230C4—21no—
231C4—22no—
232C4—21no—
233C4—22no—
234C4—21no—
235C4—21no—
236C4—82no—
237C4—82no—
238C4#182711yescharacteristic
239C4#182711yescharacteristic
240C4#193012yes—
241C4#193112yes—
242C2—21no—
243C2—21no—
244C2—42no—
245C2#1081211yescharacteristic
246C2#1082411yescharacteristic
247C2#1082711yescharacteristic
subgroupG/H
center, socleC22#1434
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C22⋊Q8
LCSG ▷ 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
16. #1299C4semidirect

Maximal subgroups

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

subgroupindex
#204822
#80632
#80062
#80012
#79602