Finite Group Databasebeta

← #150550

Subgroups — #150550

Order 512.

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

normal subgroups only

This group has 334 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#28519C211yesmaximal, characteristic
2C4×#355C211yesmaximal, characteristic
3#11531C211yesmaximal, characteristic
4#7437C214yesmaximal, complemented
5C2×#477—21no—
6C2×#476—21no—
7#2868C411yescharacteristic
8C4×#124C2211yescharacteristic
9C2×#355—21no—
10#2096C2212yes—
11#2093C412yes—
12#1959—21no—
13#1847—21no—
14#1767C411yescharacteristic
15C4×#101—21no—
16#1465—21no—
17#1394C2212yes—
18#1324—21no—
19#1319C2212yes—
20C2×C4⋊Q8—21no—
21C2×C4⋊Q8—41no—
22C2×#124D411yescharacteristic
23C2×C4×Q8D411yescharacteristic
24C2×C4×D4D411yescharacteristic
25#477—42no—
26#477—42no—
27#476—42no—
28#476—42no—
29C2×#107—22no—
30C2×#106—22no—
31C2×C4⋊C8—21no—
32C2×#103—22no—
33C2×#102—22no—
34C2×#101—22no—
35#407—41no—
36C4×M16—41no—
37C2×#97—21no—
38#391C2311yescharacteristic
39#390C2×C411yescharacteristic
40#388—41no—
41#384—21no—
42#384—41no—
43#384C2×C412yes—
44#382D411yescharacteristic
45C4×C22⋊C4—41no—
46C43C2×C411yescharacteristic
47#355—24no—
48#343—41no—
49#336C2×C412yes—
50#324—22no—
51#323—41no—
52C22×Q8—41no—
53C22×Q8C2×D411yescharacteristic
54C22×D4C22⋊C411yescharacteristic
55C23×C4—21no—
56C2×M16—42no—
57C2×M16—81no—
58C22×C8—22no—
59C4⋊Q8—41no—
60C4⋊Q8—41no—
61C4⋊Q8—41no—
62C4⋊Q8—41no—
63C4⋊Q8—84no—
64#124—22no—
65#124—22no—
66C4×Q8—41no—
67C4×Q8—41no—
68C4×D4—41no—
69C4×D4—41no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4C22⋊C411yescharacteristic
76C2×C4⋊C4C2×D411yescharacteristic
77C2×C22⋊C4—21no—
78C2×C22⋊C4—41no—
79C2×C22⋊C4—41no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42C22⋊C411yescharacteristic
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22×C411yescharacteristic
86#107—44no—
87#106—44no—
88C4⋊C8—22no—
89C4⋊C8—22no—
90#103—44no—
91#102—44no—
92#101—24no—
93#101—42no—
94C22⋊C8—42no—
95C22⋊C8—81no—
96#97—42no—
97#97—81no—
98C4×C8—81no—
99#95—41no—
100#95—41no—
101C24—21no—
102C2×Q8—21no—
103C2×Q8—41no—
104C2×Q8—41no—
105C2×Q8—81no—
106C2×Q8—81no—
107C2×Q8—84no—
108C2×Q8#13712yes—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—41no—
112C2×D4—41no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—42no—
121C22×C4—82no—
122C22×C4#9911yescharacteristic
123C22×C4#9911yescharacteristic
124C22×C4C2×C22⋊C411yescharacteristic
125M16—44no—
126M16—82no—
127M16—162no—
128C2×C8—24no—
129C2×C8—42no—
130C2×C8—42no—
131C2×C8—82no—
132C2×C8—82no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—81no—
138C4⋊C4—84no—
139C4⋊C4—84no—
140C4⋊C4—81no—
141C4⋊C4—81no—
142C22⋊C4—44no—
143C22⋊C4—81no—
144C22⋊C4—82no—
145C42—21no—
146C42—21no—
147C42—21no—
148C42—41no—
149C42—42no—
150C42—42no—
151C42—41no—
152C42—41no—
153C42#10412yes—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—81no—
158C23—81no—
159C23C2×#9911yescharacteristic
160Q8—42no—
161Q8—81no—
162Q8—162no—
163Q8—162no—
164D4—42no—
165D4—81no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—82no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—84no—
186C2×C4—82no—
187C2×C4—161no—
188C2×C4#35311yescharacteristic
189C2×C4#35311yescharacteristic
190C2×C4#41911yescharacteristic
191C2×C4C2×#10411yescharacteristic
192C8—44no—
193C8—84no—
194C8—162no—
195C22—21no—
196C22—21no—
197C22—82no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22—161no—
202C22#150811yescharacteristic
203C22#212311yescharacteristic
204C22C2×#35311yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—42no—
209C4—42no—
210C4—42no—
211C4—41no—
212C4—81no—
213C4—81no—
214C4—162no—
215C4#140712yes—
216C2—21no—
217C2—21no—
218C2—161no—
219C2#1178811yescharacteristic
220C2C2×#140711yescharacteristic
221C2#1359111yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7437C2semidirect

Maximal subgroups

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

subgroupindex
#285192
C4×#3552
#115312
#74372