Finite Group Databasebeta

← #139663

Subgroups — #139663

Order 512.

392 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 96 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#33486C211yesmaximal, characteristic
2#28142C211yesmaximal, characteristic
3#11658C212yesmaximal, complemented
4#8616C211yesmaximal, characteristic
5#7217C212yesmaximal, complemented
6C2×C4×Q16C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2538—21no—
9D4×C16—24no—
10#2162C2212yes—
11C2×C22⋊C16—22no—
12#2076C412yes—
13#2001C412yes—
14#1892—22no—
15#1834C2211yescharacteristic
16#1803—21no—
17#1800—21no—
18#1567C2212yes—
19#1332—24no—
20#1292—21no—
21C22×Q16C2×C411yescharacteristic
22C23×C8D412yes—
23C2×C22⋊Q8—22no—
24C2×C4×Q8—21no—
25C2×C4×D4D411yescharacteristic
26C22×C16—24no—
27C4×Q16—22no—
28C4×Q16—22no—
29C8×D4—22no—
30C8×D4—22no—
31C2×#107C2×C411yescharacteristic
32#423—22no—
33C2×C4⋊C8D411yescharacteristic
34C2×#103—21no—
35C2×#103—22no—
36C2×#100—41no—
37C2×C22⋊C8C2×C412yes—
38C2×C4×C8C2311yescharacteristic
39#396—42no—
40#390—41no—
41#389—41no—
42#384—21no—
43#380—22no—
44C4⋊C16—22no—
45C22⋊C16—24no—
46C22⋊C16—22no—
47C4×C16D414yes—
48#343—41no—
49#340—22no—
50#340—21no—
51#340C2×C412yes—
52#335—22no—
53#335—22no—
54C22×Q8—21no—
55C22×D4D811yescharacteristic
56C23×C4C22⋊C412yes—
57C2×Q16—22no—
58C2×Q16—42no—
59C2×M16—42no—
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—82no—
63C22×C8C22⋊C412yes—
64C22×C8C22×C411yescharacteristic
65C22×C8C2×D411yescharacteristic
66C22⋊Q8—44no—
67C22⋊Q8—82no—
68C4×Q8—42no—
69C4×Q8—42no—
70C4×D4—22no—
71C4×D4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4D811yescharacteristic
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4SD1612yes—
82C2×C42—41no—
83C2×C42C2×D411yescharacteristic
84C2×C16—24no—
85C2×C16—84no—
86#107—22no—
87C4⋊C8—22no—
88C4⋊C8—42no—
89C4⋊C8—42no—
90#103—41no—
91#103—41no—
92#103—82no—
93#100—82no—
94C22⋊C8—24no—
95C4×C8C2×D412yes—
96C4×C8#4112yes—
97#95—42no—
98C24#10212yes—
99C2×Q8—22no—
100C2×Q8—41no—
101C2×Q8—41no—
102C2×Q8—41no—
103C2×Q8—41no—
104C2×Q8—41no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4SD3212yes—
108C2×D4Q3212yes—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4—42no—
116C22×C4#10212yes—
117C22×C4C2×C22⋊C411yescharacteristic
118C22×C4C2×D811yescharacteristic
119C22×C4C2×SD1611yescharacteristic
120Q16—24no—
121Q16—42no—
122M16—82no—
123M16—82no—
124C2×C8—22no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—44no—
128C2×C8—42no—
129C2×C8—82no—
130C2×C8—82no—
131C2×C8C4×D412yes—
132C2×C8#12011yescharacteristic
133C2×C8#12311yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—84no—
140C4⋊C4—82no—
141C4⋊C4—82no—
142C4⋊C4—161no—
143C22⋊C4—24no—
144C22⋊C4—84no—
145C22⋊C4—162no—
146C42—82no—
147C42#12112yes—
148C42#13512yes—
149C16—44no—
150C23—22no—
151C23—22no—
152C23—42no—
153C23—42no—
154C23#35814yes—
155C23C2×#10211yescharacteristic
156Q8—42no—
157Q8—42no—
158Q8—81no—
159Q8—81no—
160D4—24no—
161D4—24no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—82no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—82no—
174C2×C4—81no—
175C2×C4—82no—
176C2×C4—81no—
177C2×C4—82no—
178C2×C4—81no—
179C2×C4#38611yescharacteristic
180C2×C4#41611yescharacteristic
181C2×C4#46512yes—
182C2×C4#46612yes—
183C2×C4C2×SD3211yescharacteristic
184C2×C4C2×Q3211yescharacteristic
185C2×C4#50812yes—
186C8—22no—
187C8—21no—
188C8—21no—
189C8—42no—
190C8—162no—
191C22—21no—
192C22—21no—
193C22—24no—
194C22—44no—
195C22—42no—
196C22—42no—
197C22—82no—
198C22#189311yescharacteristic
199C22C2×#35811yescharacteristic
200C22#213811yescharacteristic
201C4—21no—
202C4—22no—
203C4—42no—
204C4—82no—
205C4—162no—
206C4—161no—
207C4#189111yescharacteristic
208C4#191111yescharacteristic
209C4#221612yes—
210C4#221712yes—
211C2—21no—
212C2—21no—
213C2—44no—
214C2#948511yescharacteristic
215C2#1306011yescharacteristic
216C2#1307611yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11658C2semidirect
5. #7217C2semidirect

Maximal subgroups

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

subgroupindex
#334862
#281422
#116582
#86162
#72172