Finite Group Databasebeta

← #152232

Subgroups — #152232

Order 512.

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

normal subgroups only

This group has 384 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×#494C211yesmaximal, characteristic
2C2×#1411C212yesmaximal
3#12802C212yesmaximal
4#7499C212yesmaximal
5C2×#494C2211yescharacteristic
6C2×C4×Q16—22no—
7C4×C4⋊Q8C411yescharacteristic
8C4×C4⋊Q8C2211yescharacteristic
9C2×#356C2214yes—
10#1969—22no—
11#1956C411yescharacteristic
12#1956C2211yescharacteristic
13#1845C411yescharacteristic
14#1801—22no—
15C42×C8C411yescharacteristic
16#1411—24no—
17#1394—22no—
18C22×Q16—22no—
19C2×C4⋊Q8C2×C411yescharacteristic
20C2×C4⋊Q8C2311yescharacteristic
21C2×C4×Q8—21no—
22C2×C4×Q8D412yes—
23#494—22no—
24#494D412yes—
25#494Q812yes—
26C4×Q16—44no—
27C4×Q16—44no—
28C2×#107—22no—
29#423—42no—
30C2×#103—22no—
31C2×#103—22no—
32C2×#101—24no—
33C2×C4×C8—21no—
34C2×C4×C8—21no—
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37#391—21no—
38#391D412yes—
39#389—41no—
40#389—41no—
41#384C2×C411yescharacteristic
42#384C2×C411yescharacteristic
43C4×C4⋊C4—41no—
44C4×C4⋊C4—41no—
45C43C2×C411yescharacteristic
46#356—24no—
47#356—24no—
48#343—42no—
49C22×Q8—21no—
50C22×Q8C2×D412yes—
51C2×Q16—24no—
52C2×Q16—24no—
53C2×Q16—44no—
54C2×Q16—44no—
55C2×M16—44no—
56C22×C8—21no—
57C22×C8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—41no—
60C4⋊Q8—41no—
61C4⋊Q8C4⋊C412yes—
62C4⋊Q8C2×D411yescharacteristic
63C4⋊Q8C2×Q811yescharacteristic
64C4⋊Q8#4112yes—
65C4×Q8—42no—
66C4×Q8—42no—
67C4×Q8—42no—
68C4×Q8—42no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4C22⋊C412yes—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42—22no—
80C2×C42—41no—
81C2×C42C22⋊C411yescharacteristic
82C2×C42C22⋊C411yescharacteristic
83C2×C42C22×C411yescharacteristic
84#107—44no—
85C4⋊C8—82no—
86C4⋊C8—82no—
87#103—44no—
88#103—44no—
89#101—44no—
90#101—44no—
91C4×C8—22no—
92C4×C8—22no—
93C4×C8—21no—
94C4×C8—41no—
95C4×C8—42no—
96C4×C8C4⋊C412yes—
97#95—42no—
98#95—42no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8—42no—
105C2×Q8—41no—
106C2×Q8#12112yes—
107C2×Q8C22⋊Q812yes—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4#9911yescharacteristic
115C22×C4#9911yescharacteristic
116C22×C4C2×C22⋊C411yescharacteristic
117Q16—48no—
118Q16—48no—
119M16—84no—
120M16—84no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—84no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—82no—
136C4⋊C4—82no—
137C4⋊C4—82no—
138C4⋊C4—82no—
139C42—24no—
140C42—21no—
141C42—21no—
142C42—21no—
143C42—41no—
144C42—42no—
145C42—42no—
146C42—42no—
147C42—81no—
148C42—81no—
149C42C2×C4⋊C411yescharacteristic
150C42#11711yescharacteristic
151C23C2×#9911yescharacteristic
152Q8—42no—
153Q8—42no—
154Q8—44no—
155Q8—82no—
156Q8—82no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—81no—
176C2×C4—81no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—81no—
180C2×C4#35311yescharacteristic
181C2×C4#35311yescharacteristic
182C2×C4#38011yescharacteristic
183C2×C4#41011yescharacteristic
184C8—24no—
185C8—42no—
186C8—164no—
187C22—21no—
188C22—21no—
189C22#179911yescharacteristic
190C22C2×#35311yescharacteristic
191C22#212611yescharacteristic
192C4—22no—
193C4—22no—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—42no—
198C4—41no—
199C4—42no—
200C4—42no—
201C4—82no—
202C4—82no—
203C4—161no—
204C4#141112yescomplemented
205C2—21no—
206C2—21no—
207C2#1279711yescharacteristic
208C2C2×#141111yescharacteristic
209C2#1358911yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4⋊Q8C23
prime radicalC23C2×#99
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
204. C4#1411semidirect

Maximal subgroups

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

subgroupindex
C4×#4942
C2×#14112
#128022
#74992