Finite Group Databasebeta

← #152228

Subgroups — #152228

Order 512.

448 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C4×#492C211yesmaximal, characteristic
2C2×#1409C212yesmaximal
3#12797C212yesmaximal
4#7498C212yesmaximal
5C2×#492C2211yescharacteristic
6C2×C4×SD16—22no—
7C4×C4⋊Q8C411yescharacteristic, complemented
8C4×#127C2211yescharacteristic
9C2×#353C2214yes—
10#1968—22no—
11#1957C2211yescharacteristic
12#1956C411yescharacteristic, complemented
13#1844C411yescharacteristic, complemented
14#1799—22no—
15C42×C8C411yescharacteristic, complemented
16#1409—24no—
17#1393—22no—
18C22×SD16—22no—
19C2×C4⋊Q8C2×C411yescharacteristic
20C2×#127C2311yescharacteristic
21C2×C4×Q8—21no—
22C2×C4×D4—41no—
23C2×C4×D4D412yes—
24#492—22no—
25#492D412yes—
26#492Q812yes—
27C4×SD16—44no—
28C4×SD16—44no—
29C2×#106—22no—
30C2×#103—22no—
31C2×#102—22no—
32C2×#99—24no—
33C2×C4×C8—21no—
34C2×C4×C8—21no—
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37#391—21no—
38#390—41no—
39#390D412yes—
40#389—41no—
41#384C2×C411yescharacteristic
42#384C2×C411yescharacteristic
43#380—42no—
44C4×C4⋊C4—41no—
45C43C2×C411yescharacteristic
46#353—24no—
47#353—24no—
48#342—42no—
49C22×Q8—21no—
50C22×D4—41no—
51C22×D4C2×D412yes—
52C23×C4—22no—
53C2×SD16—24no—
54C2×SD16—24no—
55C2×SD16—44no—
56C2×SD16—44no—
57C22×C8—21no—
58C22×C8—21no—
59C4⋊Q8—21no—
60C4⋊Q8—41no—
61C4⋊Q8C4⋊C412yescomplemented
62#127—41no—
63#127C2×D411yescharacteristic
64#127C2×Q811yescharacteristic
65#127#4112yes—
66C4×Q8—42no—
67C4×Q8—42no—
68C4×D4—42no—
69C4×D4—42no—
70C4×D4—84no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—82no—
77C2×C4⋊C4C22⋊C412yes—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—44no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—41no—
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22⋊C411yescharacteristic
86C2×C42C22×C411yescharacteristic
87#106—44no—
88#103—44no—
89#102—44no—
90#99—44no—
91#99—44no—
92C4×C8—22no—
93C4×C8—22no—
94C4×C8—21no—
95C4×C8—41no—
96C4×C8—42no—
97C4×C8C4⋊C412yescomplemented
98#95—42no—
99#95—82no—
100C24—22no—
101C2×Q8—22no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8—41no—
105C2×D4—22no—
106C2×D4—42no—
107C2×D4—42no—
108C2×D4—41no—
109C2×D4—41no—
110C2×D4—42no—
111C2×D4—82no—
112C2×D4—82no—
113C2×D4#12112yes—
114C2×D4C22⋊Q812yes—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—41no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—41no—
122C22×C4—84no—
123C22×C4—84no—
124C22×C4#9911yescharacteristic
125C22×C4#9911yescharacteristic
126C22×C4C2×C22⋊C411yescharacteristic
127SD16—48no—
128SD16—48no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—41no—
134C2×C8—41no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—82no—
140C4⋊C4—82no—
141C4⋊C4—82no—
142C4⋊C4—164no—
143C22⋊C4—82no—
144C22⋊C4—84no—
145C22⋊C4—84no—
146C22⋊C4—84no—
147C42—24no—
148C42—21no—
149C42—21no—
150C42—21no—
151C42—41no—
152C42—42no—
153C42—81no—
154C42—81no—
155C42C2×C4⋊C411yescharacteristic
156C42#11711yescharacteristic
157C23—22no—
158C23—22no—
159C23—42no—
160C23—42no—
161C23—82no—
162C23—82no—
163C23C2×#9911yescharacteristic
164Q8—44no—
165Q8—82no—
166D4—42no—
167D4—42no—
168D4—82no—
169D4—82no—
170D4—82no—
171D4—84no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—84no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—81no—
192C2×C4—162no—
193C2×C4—164no—
194C2×C4—164no—
195C2×C4—164no—
196C2×C4#35311yescharacteristic
197C2×C4#35311yescharacteristic
198C2×C4#38011yescharacteristic
199C2×C4#41011yescharacteristic
200C8—24no—
201C8—42no—
202C22—21no—
203C22—21no—
204C22—42no—
205C22—42no—
206C22—82no—
207C22—82no—
208C22—82no—
209C22—82no—
210C22—162no—
211C22#179911yescharacteristic
212C22C2×#35311yescharacteristic
213C22#212611yescharacteristic
214C4—22no—
215C4—22no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—42no—
220C4—41no—
221C4—82no—
222C4—82no—
223C4—161no—
224C4—324no—
225C4#140912yescomplemented
226C2—21no—
227C2—21no—
228C2—82no—
229C2—82no—
230C2#1279711yescharacteristic
231C2C2×#140911yescharacteristic
232C2#1358311yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, prime radical, cosocleC2×#127C23
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
7. C4×C4⋊Q8C4semidirect
12. #1956C4semidirect
13. #1844C4semidirect
15. C42×C8C4semidirect
61. C4⋊Q8C4⋊C4semidirect
97. C4×C8C4⋊C4semidirect
225. C4#1409semidirect

Maximal subgroups

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

subgroupindex
C4×#4922
C2×#14092
#127972
#74982