Finite Group Databasebeta

← #152231

Subgroups — #152231

Order 512.

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

normal subgroups only

This group has 426 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×#1410C212yesmaximal
3#12802C212yesmaximal, complemented
4#7499C212yesmaximal, complemented
5C2×#492C2211yescharacteristic
6C2×C4×SD16—22no—
7C4×C4⋊Q8C2211yescharacteristic
8C4×#127C411yescharacteristic
9C2×#356C2214yes—
10#1968—22no—
11#1957C411yescharacteristic
12#1956C2211yescharacteristic
13#1844C411yescharacteristic
14#1801—22no—
15C42×C8C411yescharacteristic
16#1410—24no—
17#1394—22no—
18C22×SD16—22no—
19C2×C4⋊Q8C2311yescharacteristic
20C2×#127C2×C411yescharacteristic
21C2×C4×Q8D412yes—
22C2×C4×D4—21no—
23C2×C4×D4—41no—
24#492—22no—
25#492D412yes—
26#492Q812yes—
27C4×SD16—44no—
28C4×SD16—44no—
29C2×#106—22no—
30#423—42no—
31C2×#103—22no—
32C2×#102—22no—
33C2×#101—24no—
34C2×C4×C8—21no—
35C2×C4×C8—21no—
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38#391D412yes—
39#390—21no—
40#390—41no—
41#389—41no—
42#384C2×C411yescharacteristic
43#384C2×C411yescharacteristic
44C4×C4⋊C4—41no—
45C43C2×C411yescharacteristic
46#356—24no—
47#356—24no—
48#343—42no—
49C22×Q8C2×D412yes—
50C22×D4—21no—
51C22×D4—41no—
52C23×C4—41no—
53C2×SD16—24no—
54C2×SD16—24no—
55C2×SD16—44no—
56C2×SD16—44no—
57C2×M16—44no—
58C22×C8—21no—
59C22×C8—21no—
60C4⋊Q8—41no—
61C4⋊Q8C2×D411yescharacteristic
62C4⋊Q8C2×Q811yescharacteristic
63C4⋊Q8#4112yes—
64#127—21no—
65#127—41no—
66#127C4⋊C412yes—
67C4×Q8—42no—
68C4×Q8—42no—
69C4×D4—42no—
70C4×D4—42no—
71C4×D4—84no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4C22⋊C412yes—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—41no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—22no—
83C2×C42C22⋊C411yescharacteristic
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22×C411yescharacteristic
86#106—44no—
87C4⋊C8—82no—
88C4⋊C8—82no—
89#103—44no—
90#102—44no—
91#101—44no—
92#101—44no—
93C4×C8—22no—
94C4×C8—22no—
95C4×C8—21no—
96C4×C8—41no—
97C4×C8—42no—
98C4×C8C4⋊C412yes—
99#95—42no—
100C24—41no—
101C2×Q8—22no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8#12112yes—
105C2×Q8C22⋊Q812yes—
106C2×D4—22no—
107C2×D4—42no—
108C2×D4—42no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—41no—
112C2×D4—82no—
113C2×D4—82no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—41no—
119C22×C4—84no—
120C22×C4—81no—
121C22×C4—81no—
122C22×C4#9911yescharacteristic
123C22×C4#9911yescharacteristic
124C22×C4C2×C22⋊C411yescharacteristic
125SD16—48no—
126SD16—48no—
127M16—84no—
128M16—84no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—84no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—82no—
142C4⋊C4—82no—
143C4⋊C4—82no—
144C22⋊C4—82no—
145C22⋊C4—84no—
146C42—24no—
147C42—21no—
148C42—21no—
149C42—21no—
150C42—41no—
151C42—42no—
152C42—42no—
153C42—42no—
154C42C2×C4⋊C411yescharacteristic
155C42#11711yescharacteristic
156C23—42no—
157C23—82no—
158C23—82no—
159C23—81no—
160C23—81no—
161C23C2×#9911yescharacteristic
162Q8—42no—
163Q8—42no—
164Q8—82no—
165D4—44no—
166D4—82no—
167D4—84no—
168D4—84no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—81no—
189C2×C4—162no—
190C2×C4—162no—
191C2×C4#35311yescharacteristic
192C2×C4#35311yescharacteristic
193C2×C4#38011yescharacteristic
194C2×C4#41011yescharacteristic
195C8—24no—
196C8—42no—
197C8—164no—
198C22—21no—
199C22—21no—
200C22—84no—
201C22—82no—
202C22—162no—
203C22—161no—
204C22—161no—
205C22#179911yescharacteristic
206C22C2×#35311yescharacteristic
207C22#212611yescharacteristic
208C4—22no—
209C4—22no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—42no—
214C4—41no—
215C4—42no—
216C4—42no—
217C4—82no—
218C4—161no—
219C4#141012yescomplemented
220C2—21no—
221C2—21no—
222C2—162no—
223C2#1279711yescharacteristic
224C2C2×#141011yescharacteristic
225C2#1358911yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4⋊Q8C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12802C2semidirect
4. #7499C2semidirect
219. C4#1410semidirect

Maximal subgroups

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

subgroupindex
C4×#4922
C2×#14102
#128022
#74992