Finite Group Databasebeta

← #122385

Subgroups — #122385

Order 512.

296 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C8×C4⋊Q8C211yesmaximal, characteristic
2#11494C212yesmaximal
3#7063C212yesmaximal
4#7057C212yesmaximal
5C2×C8×Q8C411yescharacteristic
6C2×C8×Q8C2211yescharacteristic
7C4×C4⋊Q8C411yescharacteristic
8#2163—22no—
9#1984C411yescharacteristic
10#1984C2211yescharacteristic
11#1939—22no—
12#1842C2211yescharacteristic
13C8×C4⋊C4—22no—
14C42×C8C411yescharacteristic
15#1322—22no—
16#1322C2214yes—
17C2×C4⋊Q8C811yescharacteristic
18C2×C4×Q8C2×C411yescharacteristic
19C2×C4×Q8C2×C411yescharacteristic
20C8×Q8—41no—
21C8×Q8—41no—
22C8×Q8—41no—
23C8×Q8—41no—
24C2×C4⋊C8—21no—
25C2×C4⋊C8—22no—
26C2×C4⋊C8C2×C411yescharacteristic
27C2×C4⋊C8D412yes—
28C2×C4⋊C8C2311yescharacteristic
29C2×C4×C8—21no—
30C2×C4×C8—21no—
31C2×C4×C8—21no—
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8D412yes—
35#391C811yescharacteristic
36#391C811yescharacteristic
37#389—22no—
38#384C811yescharacteristic
39C4×C4⋊C4—22no—
40C43C2×C411yescharacteristic
41C4⋊C16—42no—
42C22⋊C16—24no—
43#346—42no—
44#336—22no—
45#336—22no—
46C22×Q8C2×C811yescharacteristic
47C22×Q8C2×C811yescharacteristic
48C22×C8—21no—
49C22×C8—21no—
50C22×C8C22⋊C411yescharacteristic
51C22×C8C22⋊C411yescharacteristic
52C22×C8C2×D412yes—
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C4×Q8—41no—
56C4×Q8—41no—
57C4×Q8—41no—
58C4×Q8—41no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4C2×C811yescharacteristic
63C2×C4⋊C4C2×C811yescharacteristic
64C2×C4⋊C4M1612yes—
65C2×C42—21no—
66C2×C42—21no—
67C2×C42—21no—
68C2×C42C22⋊C412yes—
69C2×C42C2×C811yescharacteristic
70C2×C42C2×C811yescharacteristic
71C2×C42C22×C411yescharacteristic
72C2×C16—44no—
73C4⋊C8—42no—
74C4⋊C8—41no—
75C4⋊C8—41no—
76C4⋊C8—44no—
77C4⋊C8—42no—
78C4×C8—24no—
79C4×C8—22no—
80C4×C8—22no—
81C4×C8—21no—
82C4×C8—21no—
83C4×C8—41no—
84C4×C8—42no—
85C4×C8—41no—
86C4×C8—41no—
87#95—22no—
88#95—22no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—41no—
93C2×Q8—41no—
94C2×Q8—41no—
95C2×Q8—41no—
96C2×Q8M3212yes—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4C22⋊C812yes—
102C22×C4C22⋊C812yes—
103C22×C4C2×C22⋊C411yescharacteristic
104C22×C4C22×C811yescharacteristic
105C22×C4C2×M1611yescharacteristic
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—21no—
109C2×C8—22no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—42no—
114C2×C8—42no—
115C2×C8—41no—
116C2×C8#9911yescharacteristic
117C2×C8#9911yescharacteristic
118C4⋊C4—28no—
119C4⋊C4—41no—
120C4⋊C4—42no—
121C4⋊C4—41no—
122C4⋊C4—44no—
123C4⋊C4—42no—
124C42—22no—
125C42—22no—
126C42—21no—
127C42—22no—
128C42—21no—
129C42—22no—
130C42—21no—
131C42—41no—
132C42—42no—
133C42—41no—
134C42—41no—
135C42M3212yes—
136C16—84no—
137C23C2×C22⋊C811yescharacteristic
138Q8—42no—
139Q8—42no—
140Q8—42no—
141Q8—42no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4C2×#9911yescharacteristic
165C2×C4#41311yescharacteristic
166C2×C4C2×M3211yescharacteristic
167C2×C4C2×M3211yescharacteristic
168C8—21no—
169C8—42no—
170C8—42no—
171C8—41no—
172C8—81no—
173C8#35612yes—
174C22—21no—
175C22—21no—
176C22#145911yescharacteristic
177C22#211211yescharacteristic
178C22#211511yescharacteristic
179C4—24no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—42no—
184C4—42no—
185C4—42no—
186C4—41no—
187C4—42no—
188C4—81no—
189C4#212611yescharacteristic
190C4C2×#35611yescharacteristic
191C2—21no—
192C2—21no—
193C2#1146611yescharacteristic
194C2#1147811yescharacteristic
195C2#1148511yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4⋊C8C23
socleC22#1459
prime radicalC23C2×C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C8×C4⋊Q82
#114942
#70632
#70572