Finite Group Databasebeta

← #142160

Subgroups — #142160

Order 512.

416 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 106 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#33208C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3C4×#341C211yesmaximal, characteristic
4#7259C212yesmaximal
5#7259C212yesmaximal
6C2×#499C411yescharacteristic
7C2×C4×M16—21no—
8C22×C4×C8C2211yescharacteristic
9#2851—21no—
10C2×C4⋊C16—21no—
11C2×C22⋊C16C2212yes—
12#1991—21no—
13#1942C411yescharacteristic
14#1942C411yescharacteristic
15#1941—21no—
16#1846C411yescharacteristic
17#1808C2211yescharacteristic
18#1808C2211yescharacteristic
19#1807—21no—
20C2×#341—21no—
21C2×#341C2212yes—
22#1374—21no—
23C22×M16—22no—
24C23×C8D411yescharacteristic
25C23×C8C2311yescharacteristic
26C2×#125—21no—
27C22×C42D411yescharacteristic
28C22×C16—22no—
29#499—42no—
30C2×#107C2×C411yescharacteristic
31C2×#107C2×C411yescharacteristic
32C2×#107C2×C411yescharacteristic
33C2×#107C2×C411yescharacteristic
34C2×#106—21no—
35C2×#106—21no—
36C4×M16—24no—
37C4×M16—42no—
38C2×#97—41no—
39C2×C4×C8—42no—
40C2×C4×C8—41no—
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45#389—21no—
46#389—21no—
47#387—41no—
48#387—41no—
49#385—41no—
50#385—41no—
51#382—21no—
52#380—21no—
53#380—21no—
54C4⋊C16—42no—
55C22⋊C16—24no—
56#341—24no—
57#341—22no—
58#341Q814yes—
59#340—21no—
60#340—21no—
61#340—21no—
62#340—21no—
63C23×C4SD1611yescharacteristic
64C23×C4SD1611yescharacteristic
65C23×C4C2×D411yescharacteristic
66C2×M16—24no—
67C2×M16—24no—
68C2×M16—24no—
69C2×M16—42no—
70C2×M16—42no—
71C2×M16—42no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8—41no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8—42no—
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22×C411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C22×C8C2×Q812yes—
83C22×C8#4112yes—
84#125—42no—
85#125—42no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C42—22no—
97C2×C42—22no—
98C2×C42C22⋊C411yescharacteristic
99C2×C42C22⋊C411yescharacteristic
100C2×C42D811yescharacteristic
101C2×C42D811yescharacteristic
102C2×C16—42no—
103C2×C16—42no—
104C2×C16—42no—
105#107—41no—
106#107—41no—
107#107—41no—
108#107—41no—
109#106—42no—
110#106—42no—
111#97—44no—
112C4×C8—22no—
113C4×C8—22no—
114C4×C8—41no—
115C4×C8—41no—
116C4×C8—44no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—41no—
123C24C2×SD1611yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4#10211yescharacteristic
139C22×C4#10211yescharacteristic
140C22×C4#10211yescharacteristic
141C22×C4#10211yescharacteristic
142C22×C4C2×C22⋊C411yescharacteristic
143C22×C4C22⋊Q812yes—
144C22×C4C2×D811yescharacteristic
145C22×C4#13712yes—
146M16—28no—
147M16—44no—
148M16—44no—
149M16—44no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—44no—
162C2×C8—44no—
163C2×C8—44no—
164C2×C8—44no—
165C2×C8#10412yes—
166C2×C8C4×Q812yes—
167C2×C8#12411yescharacteristic
168C2×C8C4⋊Q811yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—82no—
178C4⋊C4—82no—
179C4⋊C4—82no—
180C4⋊C4—82no—
181C22⋊C4—82no—
182C22⋊C4—82no—
183C42—22no—
184C42—22no—
185C42—21no—
186C42—21no—
187C42#9912yes—
188C42#10412yes—
189C16—82no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23C2×#10211yescharacteristic
197C23#47512yes—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4#32712yes—
228C2×C4#32712yes—
229C2×C4#35912yes—
230C2×C4#39111yescharacteristic
231C2×C4C2×#9911yescharacteristic
232C2×C4#41011yescharacteristic
233C2×C4#41911yescharacteristic
234C2×C4C2×#10411yescharacteristic
235C2×C4C2×#10411yescharacteristic
236C2×C4#47412yes—
237C8—21no—
238C8—22no—
239C8—21no—
240C8—22no—
241C8—48no—
242C8—41no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22C2×#32711yescharacteristic
251C22#150211yescharacteristic
252C22#198811yescharacteristic
253C22C2×#35911yescharacteristic
254C22#214311yescharacteristic
255C22#214811yescharacteristic
256C22#214811yescharacteristic
257C4—22no—
258C4—22no—
259C4—22no—
260C4—22no—
261C4—41no—
262C4—161no—
263C4—161no—
264C4—161no—
265C4—161no—
266C4#150611yescharacteristic
267C4#150811yescharacteristic
268C4#199411yescharacteristic
269C4#199611yescharacteristic
270C2—42no—
271C2#990911yescharacteristic
272C2#1170911yescharacteristic
273C2#1171011yescharacteristic
274C2#1174911yescharacteristic
275C2#1175011yescharacteristic
276C2#1327411yescharacteristic
277C2#1328011yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332082
#127832
C4×#3412
#72592