Finite Group Databasebeta

← #101798

Subgroups — #101798

Order 512.

422 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 80 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#29331C211yesmaximal, characteristic
2#21913C211yesmaximal, characteristic, complemented
3#7684C212yesmaximal
4#7632C211yesmaximal, characteristic, complemented
5#7026C212yesmaximal, complemented
6#2856C2211yescharacteristic
7#2296C2211yescharacteristic
8#2100C412yes—
9#2041C412yes—
10#1934—22no—
11#1916—24no—
12#1875—21no—
13#1875—21no—
14#1818—21no—
15#1798—21no—
16C4×C22⋊C8C2211yescharacteristic
17C4×C22⋊C8C2212yes—
18#1728C2212yes—
19#1271—22no—
20#1270—24no—
21C22×M16—22no—
22C2×#127C2×C411yescharacteristic
23C2×#121—21no—
24C2×#121—22no—
25C2×#117D411yescharacteristic
26C2×#117D411yescharacteristic
27C22×C42C2311yescharacteristic
28C2×C4⋊C8—22no—
29C2×#102—22no—
30C2×#102—22no—
31C2×C22⋊C8—22no—
32C2×C22⋊C8—21no—
33C2×C4×C8—22no—
34C2×C4×C8—21no—
35#390—21no—
36#390—22no—
37#389C2×C412yes—
38#389D412yes—
39#385—22no—
40#384C2×C411yescharacteristic
41C4×C4⋊C4—22no—
42#340—22no—
43#340—22no—
44#336—22no—
45#336—24no—
46#336—22no—
47#336—22no—
48#336C2×C412yes—
49#336D414yes—
50C22×D4—21no—
51C22×D4—22no—
52C22×D4—41no—
53C23×C4C2×D411yescharacteristic
54C23×C4#4112yes—
55C2×M16—24no—
56C2×M16—42no—
57C2×M16—82no—
58C2×M16—82no—
59C22×C8—22no—
60C22×C8—24no—
61C22×C8—42no—
62#127—81no—
63#121—44no—
64#121—82no—
65#121—82no—
66#117—42no—
67#117—81no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4C22⋊C412yes—
73C2×C4⋊C4C22⋊C412yes—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—42no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—81no—
81C2×C42C22×C411yescharacteristic
82C2×C42C2×D411yescharacteristic
83C2×C42C2×D412yes—
84C4⋊C8—82no—
85#102—44no—
86#102—82no—
87C22⋊C8—44no—
88C22⋊C8—42no—
89C4×C8—81no—
90C4×C8—82no—
91#95—22no—
92#95—22no—
93C24—41no—
94C24#12311yescharacteristic
95C2×D4—42no—
96C2×D4—42no—
97C2×D4—42no—
98C2×D4—42no—
99C2×D4—42no—
100C2×D4—82no—
101C2×D4—82no—
102C2×D4—82no—
103C2×D4—84no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—41no—
113C22×C4—42no—
114C22×C4—41no—
115C22×C4C2×C22⋊C411yescharacteristic
116C22×C4C4×D412yes—
117C22×C4#12011yescharacteristic
118C22×C4#12112yes—
119M16—84no—
120M16—84no—
121C2×C8—42no—
122C2×C8—44no—
123C2×C8—42no—
124C2×C8—44no—
125C2×C8—44no—
126C2×C8—42no—
127C2×C8—44no—
128C2×C8—82no—
129C2×C8—82no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—82no—
139C22⋊C4—42no—
140C22⋊C4—81no—
141C22⋊C4—82no—
142C22⋊C4—82no—
143C22⋊C4—82no—
144C42—42no—
145C42—41no—
146C42—41no—
147C42—42no—
148C42—42no—
149C23—22no—
150C23—21no—
151C23—21no—
152C23—22no—
153C23—21no—
154C23—82no—
155C23—81no—
156C23—81no—
157C23—82no—
158C23—81no—
159C23#38611yescharacteristic
160D4—84no—
161D4—162no—
162D4—162no—
163D4—164no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—81no—
184C2×C4—82no—
185C2×C4#42112yes—
186C2×C4#44312yes—
187C2×C4#44412yes—
188C2×C4#45412yes—
189C2×C4#47112yes—
190C2×C4#47212yes—
191C8—84no—
192C8—82no—
193C8—84no—
194C22—42no—
195C22—41no—
196C22—41no—
197C22—41no—
198C22—42no—
199C22—162no—
200C22—161no—
201C22—161no—
202C22—161no—
203C22—162no—
204C22#187612yes—
205C22#188811yescharacteristic
206C22#189711yescharacteristic
207C22#190712yes—
208C22#191311yescharacteristic
209C4—22no—
210C4—22no—
211C4—22no—
212C4—22no—
213C4—22no—
214C4—22no—
215C4—82no—
216C4—82no—
217C4—161no—
218C2—81no—
219C2—321no—
220C2#907011yescharacteristic
221C2#921312yes—
222C2#926412yes—
223C2#941411yescharacteristic
224C2#943811yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21913C2semidirect
4. #7632C2semidirect
5. #7026C2semidirect

Maximal subgroups

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

subgroupindex
#293312
#219132
#76842
#76322
#70262