Finite Group Databasebeta

← #152229

Subgroups — #152229

Order 512.

484 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 70 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#21690C211yesmaximal, characteristic
2C2×#1408C212yesmaximal
3#12797C212yesmaximal, complemented
4#7498C212yesmaximal, complemented
5C2×#493C2211yescharacteristic
6C2×#442—22no—
7C4×#127C411yescharacteristic, complemented
8C4×#127C2211yescharacteristic
9C2×#353C2214yes—
10#1972—22no—
11#1957C411yescharacteristic, complemented
12#1957C2211yescharacteristic
13#1844C411yescharacteristic, complemented
14#1799—22no—
15C4×#97C411yescharacteristic, complemented
16#1408—24no—
17#1393—22no—
18C22×D8—22no—
19C2×#127C2×C411yescharacteristic
20C2×#127C2311yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4—41no—
23C2×C4×D4—41no—
24C2×C4×D4D412yes—
25#493—22no—
26#493D412yes—
27#493Q812yes—
28#442—44no—
29#442—44no—
30C2×#106—22no—
31C2×#102—22no—
32C2×#102—22no—
33C2×#99—24no—
34C2×#97—21no—
35C2×#97—21no—
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38#390—21no—
39#390—41no—
40#390—41no—
41#390D412yes—
42#384C2×C411yescharacteristic
43#384C2×C411yescharacteristic
44#380—42no—
45C43C2×C411yescharacteristic
46#353—24no—
47#353—24no—
48#342—42no—
49C22×D4—21no—
50C22×D4—41no—
51C22×D4—41no—
52C22×D4C2×D412yes—
53C23×C4—22no—
54C23×C4—41no—
55C2×D8—24no—
56C2×D8—24no—
57C2×D8—44no—
58C2×D8—44no—
59C22×C8—21no—
60C22×C8—21no—
61#127—21no—
62#127—41no—
63#127—41no—
64#127C4⋊C412yescomplemented
65#127C2×D411yescharacteristic
66#127C2×Q811yescharacteristic
67#127#4112yes—
68C4×D4—42no—
69C4×D4—42no—
70C4×D4—42no—
71C4×D4—42no—
72C4×D4—84no—
73C4×D4—84no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—82no—
78C2×C4⋊C4C22⋊C412yes—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4—44no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42C22⋊C411yescharacteristic
87C2×C42C22⋊C411yescharacteristic
88C2×C42C22×C411yescharacteristic
89#106—44no—
90#102—44no—
91#102—44no—
92#99—44no—
93#99—44no—
94#97—24no—
95#97—42no—
96C4×C8—21no—
97C4×C8—41no—
98C4×C8C4⋊C412yescomplemented
99#95—82no—
100C24—22no—
101C24—41no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4—42no—
107C2×D4—42no—
108C2×D4—41no—
109C2×D4—41no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4—42no—
113C2×D4—41no—
114C2×D4—82no—
115C2×D4—82no—
116C2×D4—82no—
117C2×D4—82no—
118C2×D4#12112yes—
119C2×D4C22⋊Q812yes—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—41no—
126C22×C4—81no—
127C22×C4—81no—
128C22×C4—84no—
129C22×C4—84no—
130C22×C4—84no—
131C22×C4#9911yescharacteristic
132C22×C4#9911yescharacteristic
133C22×C4C2×C22⋊C411yescharacteristic
134D8—48no—
135D8—48no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—41no—
141C2×C8—41no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—82no—
145C4⋊C4—82no—
146C4⋊C4—164no—
147C22⋊C4—82no—
148C22⋊C4—84no—
149C22⋊C4—84no—
150C22⋊C4—82no—
151C22⋊C4—84no—
152C22⋊C4—84no—
153C42—22no—
154C42—21no—
155C42—41no—
156C42—42no—
157C42—42no—
158C42C2×C4⋊C411yescharacteristic
159C42#11711yescharacteristic
160C23—22no—
161C23—22no—
162C23—42no—
163C23—42no—
164C23—42no—
165C23—82no—
166C23—82no—
167C23—82no—
168C23—82no—
169C23—81no—
170C23—81no—
171C23C2×#9911yescharacteristic
172D4—42no—
173D4—42no—
174D4—44no—
175D4—82no—
176D4—82no—
177D4—82no—
178D4—82no—
179D4—84no—
180D4—84no—
181D4—84no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—84no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—81no—
199C2×C4—162no—
200C2×C4—162no—
201C2×C4—162no—
202C2×C4—164no—
203C2×C4—164no—
204C2×C4—164no—
205C2×C4#35312yes—
206C2×C4#38011yescharacteristic
207C2×C4#41011yescharacteristic
208C8—24no—
209C8—42no—
210C22—21no—
211C22—21no—
212C22—42no—
213C22—42no—
214C22—82no—
215C22—82no—
216C22—84no—
217C22—82no—
218C22—82no—
219C22—82no—
220C22—162no—
221C22—162no—
222C22—161no—
223C22—161no—
224C22#179911yescharacteristic
225C22C2×#35311yescharacteristic
226C22#212611yescharacteristic
227C4—22no—
228C4—22no—
229C4—22no—
230C4—21no—
231C4—21no—
232C4—41no—
233C4—81no—
234C4—82no—
235C4—161no—
236C4—324no—
237C2—21no—
238C2—21no—
239C2—82no—
240C2—82no—
241C2—162no—
242C2#1279711yescharacteristic
243C2#1358411yescharacteristic
244C2#1358511yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocleC2×#127C23
prime radicalC2×#493C22
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G
LECSG ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12797C2semidirect
4. #7498C2semidirect
7. C4×#127C4semidirect
11. #1957C4semidirect
13. #1844C4semidirect
15. C4×#97C4semidirect
64. #127C4⋊C4semidirect
98. C4×C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#216902
C2×#14082
#127972
#74982