Finite Group Databasebeta

← #149979

Subgroups — #149979

Order 512.

351 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 58 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#28440C211yesmaximal, characteristic
2#11532C211yesmaximal, characteristic
3#11478C211yesmaximal, characteristic
4#7355C214yesmaximal
5C2×#494C412yes—
6C2×#462—21no—
7C2×#462—21no—
8#2868C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10#2016—21no—
11#2016—21no—
12#1956C2212yes—
13C4×C4⋊C8C411yescharacteristic
14#1467—21no—
15#1461—21no—
16#1325C2212yes—
17#1325C2212yes—
18#1319—21no—
19#1319—21no—
20C22×Q16—22no—
21C22×Q16—22no—
22C2×C4⋊Q8—21no—
23C2×C4⋊Q8—21no—
24C2×C4⋊Q8C2×C411yescharacteristic
25C2×C4⋊Q8C2×C412yes—
26C2×C4×Q8D411yescharacteristic
27C2×C4×Q8D411yescharacteristic
28#494—44no—
29#462—42no—
30#462—42no—
31#462—42no—
32#462—42no—
33#432—41no—
34#431—41no—
35C2×C4⋊C8—21no—
36C2×#103—22no—
37C2×#103—22no—
38C2×C4×C8C2×C412yes—
39#391D411yescharacteristic
40#391D411yescharacteristic
41#389—41no—
42#384—41no—
43#384—41no—
44#384C2311yescharacteristic
45C4×C4⋊C4—41no—
46C43C2×C411yescharacteristic
47#336—21no—
48#324—22no—
49#324—22no—
50#324—41no—
51#324—41no—
52C22×Q8—22no—
53C22×Q8C22⋊C411yescharacteristic
54C22×Q8C22⋊C411yescharacteristic
55C2×Q16—44no—
56C2×Q16—44no—
57C2×Q16—44no—
58C2×Q16—44no—
59C2×Q16—82no—
60C2×Q16—82no—
61C22×C8—22no—
62C4⋊Q8—22no—
63C4⋊Q8—22no—
64C4⋊Q8—24no—
65C4⋊Q8—42no—
66C4⋊Q8—41no—
67C4⋊Q8—41no—
68C4⋊Q8—41no—
69C4⋊Q8—41no—
70C4⋊Q8—82no—
71C4×Q8—41no—
72C4×Q8—41no—
73C4×Q8—41no—
74C4×Q8—41no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4C2×D411yescharacteristic
84C2×C4⋊C4C2×D411yescharacteristic
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42C22⋊C411yescharacteristic
90C2×C42C22⋊C411yescharacteristic
91C2×C42C22×C411yescharacteristic
92C4⋊C8—22no—
93C4⋊C8—22no—
94C4⋊C8—81no—
95C4⋊C8—81no—
96#103—44no—
97#103—44no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100#97—81no—
101C4×C8—42no—
102C4×C8—81no—
103#95—41no—
104#95—41no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—41no—
109C2×Q8—41no—
110C2×Q8—41no—
111C2×Q8—41no—
112C2×Q8—42no—
113C2×Q8—42no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8#10012yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—42no—
126C22×C4#9911yescharacteristic
127C22×C4#9911yescharacteristic
128C22×C4C2×C22⋊C411yescharacteristic
129Q16—88no—
130Q16—88no—
131C2×C8—24no—
132C2×C8—42no—
133C2×C8—42no—
134C2×C8—82no—
135C2×C8—82no—
136C4⋊C4—44no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—81no—
140C4⋊C4—81no—
141C4⋊C4—84no—
142C4⋊C4—82no—
143C4⋊C4—81no—
144C4⋊C4—81no—
145C4⋊C4—84no—
146C4⋊C4—81no—
147C4⋊C4—81no—
148C42—24no—
149C42—21no—
150C42—21no—
151C42—21no—
152C42—41no—
153C42—42no—
154C42—41no—
155C42—41no—
156C42—41no—
157C42—41no—
158C42#13112yes—
159C23C2×#9911yescharacteristic
160Q8—42no—
161Q8—42no—
162Q8—81no—
163Q8—81no—
164Q8—82no—
165Q8—82no—
166Q8—82no—
167Q8—82no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4#35311yescharacteristic
193C2×C4#35311yescharacteristic
194C2×C4#40811yescharacteristic
195C2×C4C2×#10011yescharacteristic
196C8—44no—
197C8—162no—
198C8—162no—
199C22—21no—
200C22—21no—
201C22#147111yescharacteristic
202C22C2×#35311yescharacteristic
203C22#213411yescharacteristic
204C4—24no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—42no—
209C4—41no—
210C4—42no—
211C4—81no—
212C4—81no—
213C4—81no—
214C4—162no—
215C4#141112yes—
216C2—21no—
217C2—21no—
218C2#1156311yescharacteristic
219C2#1358611yescharacteristic
220C2C2×#141111yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC23C2×#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#284402
#115322
#114782
#73552