Finite Group Databasebeta

← #122801

Subgroups — #122801

Order 512.

438 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 66 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#24268C211yesmaximal, characteristic
2#11488C211yesmaximal, characteristic
3#8611C211yesmaximal, characteristic
4#7086C214yesmaximal, complemented
5C23×Q16C411yescharacteristic
6C2×#467—21no—
7C2×C8×D4C2211yescharacteristic
8#2409—21no—
9#2163—21no—
10#2056C412yes—
11#2001C2212yes—
12#1969—21no—
13#1810—21no—
14#1808C411yescharacteristic
15#1787—21no—
16#1465—21no—
17#1314—21no—
18#1314C2212yes—
19#1293C2212yes—
20C23×Q8—21no—
21C22×Q16—21no—
22C22×Q16—41no—
23C22×Q16—41no—
24C22×Q16—44no—
25C22×Q16C2×C412yes—
26C23×C8C2×C411yescharacteristic
27C23×C8D411yescharacteristic
28C2×C22⋊Q8—22no—
29C2×C4×D4D411yescharacteristic
30#467—44no—
31C8×D4—41no—
32C8×D4—41no—
33C2×#108—41no—
34C2×#107—21no—
35C2×C4⋊C8C2×C411yescharacteristic
36C2×#103—22no—
37C2×#103—22no—
38#407—22no—
39C2×C22⋊C8D411yescharacteristic
40C2×C22⋊C8C2311yescharacteristic
41C2×C4×C8D411yescharacteristic
42#395—41no—
43#393—41no—
44#391—22no—
45#386—41no—
46#385—41no—
47#380—21no—
48C4⋊C16—41no—
49C22⋊C16—22no—
50#346—41no—
51#341—41no—
52#340C2×C412yes—
53#324—41no—
54#323—41no—
55C22×Q8—22no—
56C22×Q8—42no—
57C22×Q8—42no—
58C22×Q8—41no—
59C22×Q8—44no—
60C22×Q8—82no—
61C22×D4SD1611yescharacteristic
62C23×C4—41no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4C2×D411yescharacteristic
65C2×Q16—24no—
66C2×Q16—42no—
67C2×Q16—42no—
68C2×Q16—48no—
69C2×Q16—44no—
70C2×Q16—44no—
71C2×Q16—44no—
72C2×Q16—41no—
73C2×Q16—41no—
74C2×M16—42no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—42no—
80C22×C8—42no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22×C411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C22⋊Q8—44no—
86C22⋊Q8—82no—
87C4×D4—41no—
88C4×D4—41no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4D811yescharacteristic
92C2×C22⋊C4—41no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4D811yescharacteristic
95C2×C22⋊C4SD1611yescharacteristic
96C2×C42C22⋊C411yescharacteristic
97C2×C16—42no—
98#108—82no—
99#107—42no—
100C4⋊C8—41no—
101#103—44no—
102#103—82no—
103C22⋊C8—41no—
104C22⋊C8—41no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107C4×C8—21no—
108C4×C8—21no—
109#95—41no—
110#95—42no—
111#95—42no—
112C24#10211yescharacteristic
113C24C2×SD1611yescharacteristic
114C2×Q8—24no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—48no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—82no—
122C2×Q8—84no—
123C2×Q8—88no—
124C2×Q8—84no—
125C2×D4—22no—
126C2×D4—41no—
127C2×D4—41no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—84no—
139C22×C4—82no—
140C22×C4#10211yescharacteristic
141C22×C4#10211yescharacteristic
142C22×C4#10211yescharacteristic
143C22×C4C2×C22⋊C411yescharacteristic
144C22×C4C2×D811yescharacteristic
145Q16—44no—
146Q16—44no—
147Q16—44no—
148Q16—44no—
149M16—82no—
150M16—82no—
151C2×C8—22no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—81no—
162C2×C8—82no—
163C2×C8#9911yescharacteristic
164C2×C8#9911yescharacteristic
165C4⋊C4—41no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—84no—
169C22⋊C4—41no—
170C22⋊C4—41no—
171C22⋊C4—84no—
172C22⋊C4—82no—
173C42—21no—
174C42—21no—
175C16—82no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23#36212yes—
186C23C2×#10211yescharacteristic
187Q8—44no—
188Q8—44no—
189Q8—82no—
190Q8—88no—
191Q8—82no—
192D4—44no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—44no—
210C2×C4—81no—
211C2×C4—82no—
212C2×C4—88no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4C2×#9911yescharacteristic
218C2×C4#42111yescharacteristic
219C8—21no—
220C8—41no—
221C8—41no—
222C8—41no—
223C8—162no—
224C8AΣL(1,𝔽16)12yes—
225C22—21no—
226C22—21no—
227C22—22no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—81no—
237C22#151911yescharacteristic
238C22#214311yescharacteristic
239C22C2×#36211yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—41no—
244C4—41no—
245C4—84no—
246C4—162no—
247C4C2×AΣL(1,𝔽16)11yescharacteristic
248C4#212811yescharacteristic
249C2—21no—
250C2—21no—
251C2—42no—
252C2—81no—
253C2#1187111yescharacteristic
254C2#1187311yescharacteristic
255C2#1189611yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC2×C22⋊C8C23
prime radicalC22×D4SD16
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7086C2semidirect

Maximal subgroups

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

subgroupindex
#242682
#114882
#86112
#70862