Finite Group Databasebeta

← #121583

Subgroups — #121583

Order 512.

572 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 150 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#17807C212yesmaximal
3#10832C211yesmaximal, characteristic
4#8002C211yesmaximal, characteristic
5#7867C212yesmaximal
6#2920C2212yes—
7#2920C2212yes—
8C22×C4⋊C8C2211yescharacteristic
9C22×C4⋊C8C2211yescharacteristic
10C2×C43C2211yescharacteristic
11#1950—22no—
12#1950—22no—
13#1810—22no—
14C2×#336—22no—
15#1573—24no—
16#1571—22no—
17#1570—22no—
18#1565—24no—
19#1292C414yescomplemented
20#1276—24no—
21C23×C8—21no—
22C23×C8D412yes—
23C22×C42—21no—
24C22×C42—21no—
25C22×C42Q812yes—
26C22×C42C2311yescharacteristic
27#432—22no—
28#432—22no—
29#432—22no—
30#432Q814yes—
31#431—22no—
32#431—22no—
33#431—22no—
34#431D414yes—
35C2×#108—22no—
36C2×#108—22no—
37#423C2×C414yes—
38#423D412yes—
39#423D412yes—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8C2×C412yes—
46C2×C4⋊C8D412yes—
47C2×C4⋊C8Q812yes—
48#407—22no—
49#407—22no—
50C4×M16—22no—
51C4×M16—22no—
52C43—22no—
53C43—22no—
54#341—24no—
55#336—44no—
56#335—28no—
57#334—28no—
58C23×C4—22no—
59C23×C4C2×D411yescharacteristic
60C23×C4C2×Q811yescharacteristic
61C23×C4#4111yescharacteristic
62C2×M16—24no—
63C2×M16—22no—
64C2×M16—22no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—42no—
69C22×C8—42no—
70C22×C8—42no—
71C22×C8—42no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8C4⋊C414yes—
75C2×C42—24no—
76C2×C42—22no—
77C2×C42—24no—
78C2×C42—22no—
79C2×C42—44no—
80C2×C42—42no—
81C2×C42—41no—
82C2×C42—41no—
83C2×C42—42no—
84C2×C42—41no—
85C2×C42—41no—
86C2×C42C22×C411yescharacteristic
87C2×C42C2×D412yes—
88C2×C42C2×D412yes—
89C2×C42C2×D411yescharacteristic
90C2×C42C2×Q812yes—
91C2×C42C2×Q811yescharacteristic
92C2×C42#4112yes—
93C2×C42#4111yescharacteristic
94#108—82no—
95#108—82no—
96C4⋊C8—24no—
97C4⋊C8—24no—
98C4⋊C8—22no—
99C4⋊C8—24no—
100C4⋊C8—22no—
101C4⋊C8—22no—
102C4⋊C8—22no—
103C4⋊C8—24no—
104C4⋊C8—42no—
105C4⋊C8C4⋊C418yes—
106C22⋊C8—24no—
107C22⋊C8—24no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110#97—24no—
111#97—42no—
112C4×C8—24no—
113C4×C8—42no—
114C24C22⋊Q811yescharacteristic
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—24no—
122C22×C4—24no—
123C22×C4—24no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4C2×C4⋊C411yescharacteristic
130C22×C4C4×D411yescharacteristic
131C22×C4C4×Q811yescharacteristic
132C22×C4#12011yescharacteristic
133C22×C4#12111yescharacteristic
134C22×C4C22⋊Q812yes—
135C22×C4C22⋊Q812yes—
136C22×C4#12312yes—
137C22×C4#12311yescharacteristic
138C22×C4#12511yescharacteristic
139C22×C4#12711yescharacteristic
140C22×C4C4⋊Q811yescharacteristic
141C22×C4C4⋊Q812yes—
142C22×C4C4⋊Q811yescharacteristic
143M16—44no—
144M16—44no—
145M16—42no—
146M16—42no—
147M16—42no—
148M16—42no—
149C2×C8—28no—
150C2×C8—24no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—44no—
154C2×C8—44no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—82no—
159C2×C8—82no—
160C2×C8—81no—
161C2×C8—81no—
162C42—28no—
163C42—44no—
164C42—42no—
165C42—42no—
166C42—48no—
167C42—42no—
168C42—42no—
169C42C2×C4⋊C412yes—
170C42C4×D412yes—
171C42#12012yes—
172C42#12112yes—
173C42C22⋊Q812yes—
174C42C22⋊Q812yes—
175C42#12512yes—
176C42C4⋊Q812yes—
177C23—21no—
178C23—21no—
179C23—22no—
180C23—21no—
181C23—21no—
182C23#38911yescharacteristic
183C23#39511yescharacteristic
184C23#39911yescharacteristic
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—24no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—28no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4#38411yescharacteristic
211C2×C4#38511yescharacteristic
212C2×C4#38512yes—
213C2×C4#38912yes—
214C2×C4#39211yescharacteristic
215C2×C4#39512yes—
216C2×C4#39812yes—
217C2×C4#39811yescharacteristic
218C2×C4#44312yes—
219C2×C4#44412yes—
220C2×C4#45311yescharacteristic
221C2×C4#45411yescharacteristic
222C2×C4#45511yescharacteristic
223C2×C4#45611yescharacteristic
224C8—44no—
225C8—48no—
226C8—82no—
227C8—82no—
228C8—82no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22#144411yescharacteristic
240C22#194511yescharacteristic
241C22#194611yescharacteristic
242C22#206011yescharacteristic
243C22#206111yescharacteristic
244C22#209112yes—
245C4—22no—
246C4—22no—
247C4—22no—
248C4—21no—
249C4—21no—
250C4—48no—
251C4#185611yescharacteristic
252C4#185711yescharacteristic
253C4#186614yes—
254C4#200611yescharacteristic
255C4#200711yescharacteristic
256C2—22no—
257C2—21no—
258C2—21no—
259C2—21no—
260C2—21no—
261C2#1132911yescharacteristic
262C2#1133811yescharacteristic
263C2#1134211yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1444
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #1292C4semidirect

Maximal subgroups

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

subgroupindex
#219692
#178072
#108322
#80022
#78672