Finite Group Databasebeta

← #122807

Subgroups — #122807

Order 512.

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

normal subgroups only

This group has 296 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#28707C211yesmaximal, characteristic
2#11465C211yesmaximal, characteristic
3#8611C211yesmaximal, characteristic
4#7087C212yesmaximal, complemented
5#7085C212yesmaximal, complemented
6C2×#499C411yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2562—21no—
9C2×C22⋊C16—21no—
10#2076C411yescharacteristic
11#2076C2211yescharacteristic
12#2075C411yescharacteristic
13#2075C2211yescharacteristic
14#1989—21no—
15#1810—21no—
16#1807—21no—
17#1807—21no—
18C4×#106C411yescharacteristic
19#1465—21no—
20#1322—21no—
21#1314C2212yes—
22#1293C2212yes—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×#125—21no—
26C2×C4×D4D411yescharacteristic
27C22×C16—41no—
28#499—41no—
29#499—41no—
30C8×D4—41no—
31C8×D4—41no—
32C2×#108—41no—
33C2×#107C2×C411yescharacteristic
34C2×#107C2×C411yescharacteristic
35C2×#106—21no—
36C2×#106—21no—
37C2×C4⋊C8D411yescharacteristic
38#407—22no—
39C2×C22⋊C8C2×C411yescharacteristic
40C2×C22⋊C8C2311yescharacteristic
41C2×C4×C8C2×C411yescharacteristic
42#400—41no—
43#394—41no—
44#388—41no—
45#388—41no—
46#380—21no—
47#380—21no—
48#380—21no—
49#380—21no—
50C4×C4⋊C4—21no—
51C22⋊C16—22no—
52C22⋊C16—41no—
53#341—41no—
54#340—21no—
55#340—21no—
56#340—21no—
57#340—21no—
58#340C2×C411yescharacteristic
59#340C2×C411yescharacteristic
60#324—41no—
61#323—41no—
62C22×D4SD1611yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C2×D411yescharacteristic
65C2×M16—42no—
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—42no—
71C22×C8—42no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22×C411yescharacteristic
75C22×C8C2×D411yescharacteristic
76#125—42no—
77#125—42no—
78C4×D4—41no—
79C4×D4—41no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4D811yescharacteristic
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4D811yescharacteristic
92C2×C22⋊C4SD1611yescharacteristic
93C2×C42—41no—
94C2×C42C22⋊C411yescharacteristic
95C2×C16—42no—
96C2×C16—82no—
97#108—82no—
98#107—41no—
99#107—41no—
100#106—42no—
101#106—41no—
102#106—41no—
103C4⋊C8—41no—
104C22⋊C8—41no—
105C22⋊C8—41no—
106C22⋊C8—42no—
107C22⋊C8—42no—
108C4×C8—21no—
109C4×C8—21no—
110#95—41no—
111#95—41no—
112#95—41no—
113#95—41no—
114#95—41no—
115C24#10211yescharacteristic
116C24C2×SD1611yescharacteristic
117C2×D4—22no—
118C2×D4—41no—
119C2×D4—41no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4#10211yescharacteristic
133C22×C4#10211yescharacteristic
134C22×C4#10211yescharacteristic
135C22×C4C2×C22⋊C411yescharacteristic
136C22×C4C2×D811yescharacteristic
137M16—82no—
138M16—82no—
139C2×C8—22no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8—82no—
151C2×C8#9911yescharacteristic
152C2×C8#9911yescharacteristic
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—82no—
163C4⋊C4—82no—
164C22⋊C4—41no—
165C22⋊C4—41no—
166C22⋊C4—82no—
167C22⋊C4—82no—
168C22⋊C4—161no—
169C22⋊C4—161no—
170C42—21no—
171C42—21no—
172C42—161no—
173C16—82no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23#35912yes—
184C23C2×#10211yescharacteristic
185D4—44no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4C2×#9911yescharacteristic
216C2×C4#42111yescharacteristic
217C8—21no—
218C8—41no—
219C8—41no—
220C8—41no—
221C8—162no—
222C8#35412yes—
223C22—21no—
224C22—21no—
225C22—22no—
226C22—42no—
227C22—42no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—81no—
234C22—81no—
235C22#151911yescharacteristic
236C22C2×#35911yescharacteristic
237C22#214811yescharacteristic
238C4—21no—
239C4—41no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—161no—
244C4—161no—
245C4—161no—
246C4—161no—
247C4#212311yescharacteristic
248C4C2×#35411yescharacteristic
249C2—21no—
250C2—21no—
251C2—42no—
252C2—81no—
253C2#1187211yescharacteristic
254C2#1187811yescharacteristic
255C2#1189311yescharacteristic
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. #7087C2semidirect
5. #7085C2semidirect

Maximal subgroups

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

subgroupindex
#287072
#114652
#86112
#70872
#70852