Finite Group Databasebeta

← #142192

Subgroups — #142192

Order 512.

368 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 98 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
3#10822C211yesmaximal, characteristic
4#7259C212yesmaximal
5#7259C212yesmaximal
6C2×#499C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#2851—21no—
9C2×C4⋊C16—21no—
10C2×C22⋊C16C2212yes—
11#1991—21no—
12#1942C411yescharacteristic
13#1942C411yescharacteristic
14#1941—21no—
15#1846C411yescharacteristic
16#1808C2211yescharacteristic
17#1808C2211yescharacteristic
18#1807—21no—
19C2×#341—21no—
20C2×#341C2212yes—
21#1728—21no—
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—
36C2×C4×C8—42no—
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41#389—21no—
42#389—21no—
43#387—41no—
44#387—41no—
45#385—41no—
46#385—41no—
47#382—21no—
48#380—21no—
49#380—21no—
50C4⋊C16—42no—
51C22⋊C16—24no—
52#341—22no—
53#341—42no—
54#341Q814yes—
55#340—21no—
56#340—21no—
57#340—21no—
58#340—21no—
59#336—41no—
60#336—41no—
61C23×C4SD1611yescharacteristic
62C23×C4SD1611yescharacteristic
63C23×C4C2×D411yescharacteristic
64C2×M16—24no—
65C2×M16—42no—
66C2×M16—42no—
67C2×M16—42no—
68C2×M16—42no—
69C2×M16—42no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8—41no—
75C22×C8—42no—
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22×C411yescharacteristic
79C22×C8C2×D411yescharacteristic
80C22×C8C2×Q812yes—
81C22×C8#4112yes—
82#125—42no—
83#125—42no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C42—41no—
95C2×C42—41no—
96C2×C42C22⋊C411yescharacteristic
97C2×C42C22⋊C411yescharacteristic
98C2×C42D811yescharacteristic
99C2×C42D811yescharacteristic
100C2×C16—42no—
101C2×C16—42no—
102C2×C16—42no—
103#107—41no—
104#107—41no—
105#107—41no—
106#107—41no—
107#106—42no—
108#106—42no—
109C4×C8—22no—
110C4×C8—22no—
111C4×C8—41no—
112C4×C8—41no—
113#95—41no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119C24C2×SD1611yescharacteristic
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4#10211yescharacteristic
135C22×C4#10211yescharacteristic
136C22×C4#10211yescharacteristic
137C22×C4#10211yescharacteristic
138C22×C4C2×C22⋊C411yescharacteristic
139C22×C4C22⋊Q812yes—
140C22×C4C2×D811yescharacteristic
141C22×C4#13712yes—
142M16—44no—
143M16—44no—
144M16—82no—
145M16—82no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—44no—
158C2×C8—44no—
159C2×C8—82no—
160C2×C8—82no—
161C2×C8#9912yes—
162C2×C8C4×Q812yes—
163C2×C8#12411yescharacteristic
164C2×C8C4⋊Q811yescharacteristic
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C4⋊C4—82no—
176C4⋊C4—82no—
177C22⋊C4—82no—
178C22⋊C4—82no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—41no—
184C42—41no—
185C16—82no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23C2×#10211yescharacteristic
193C23#47512yes—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—81no—
214C2×C4—81no—
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#35912yes—
226C2×C4#39111yescharacteristic
227C2×C4C2×#9911yescharacteristic
228C2×C4#41011yescharacteristic
229C2×C4#41911yescharacteristic
230C2×C4#42111yescharacteristic
231C2×C4#42111yescharacteristic
232C2×C4#47412yes—
233C8—21no—
234C8—22no—
235C8—21no—
236C8—22no—
237C8—41no—
238C8—84no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22#151711yescharacteristic
247C22#151911yescharacteristic
248C22#198811yescharacteristic
249C22C2×#35911yescharacteristic
250C22#214311yescharacteristic
251C22#214811yescharacteristic
252C22#214811yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—161no—
259C4—161no—
260C4—161no—
261C4—161no—
262C4#151111yescharacteristic
263C4#151511yescharacteristic
264C4#199511yescharacteristic
265C4#199611yescharacteristic
266C2—42no—
267C2#991211yescharacteristic
268C2#1186111yescharacteristic
269C2#1186611yescharacteristic
270C2#1189311yescharacteristic
271C2#1189411yescharacteristic
272C2#1327411yescharacteristic
273C2#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
#108222
#72592