Finite Group Databasebeta

← #100909

Subgroups — #100909

Order 512.

393 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 393 conjugacy classes of subgroups — too many to draw. Only its 76 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#27981C211yesmaximal, characteristic
2#21918C211yesmaximal, characteristic, complemented
3#8729C211yesmaximal, characteristic, complemented
4#7724C212yesmaximal
5#7038C212yesmaximal, complemented
6#2856C2211yescharacteristic
7#2285C2211yescharacteristic
8#2071—22no—
9#1961C412yes—
10#1917—22no—
11#1916—22no—
12#1875—21no—
13#1875—21no—
14#1836C412yes—
15#1817C2212yes—
16#1798—21no—
17#1797C2212yes—
18#1750—21no—
19#1749C2211yescharacteristic
20#1296—22no—
21#1270—22no—
22#1270—22no—
23C2×#127C2×C411yescharacteristic
24C2×#121—21no—
25C2×#121—22no—
26C2×#117D411yescharacteristic
27C2×#117D411yescharacteristic
28C22×C42C2311yescharacteristic
29C2×C4⋊C8—22no—
30C2×C4⋊C8—22no—
31C2×#102—22no—
32C2×#102—22no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—21no—
35C2×C22⋊C8—22no—
36C2×#97—21no—
37C2×#97C2×C412yes—
38C2×C4×C8—21no—
39#390—21no—
40#390—22no—
41#384C2×C411yescharacteristic
42#384D412yes—
43#382C2×C412yes—
44#380—22no—
45C4×C4⋊C4—22no—
46#340—22no—
47#340—22no—
48#336—21no—
49#336—21no—
50#336—22no—
51#336—21no—
52#336—22no—
53#336—22no—
54#336D412yes—
55#336D412yes—
56C22×D4—21no—
57C22×D4—22no—
58C22×D4—41no—
59C23×C4C2×D411yescharacteristic
60C23×C4#4112yes—
61C22×C8—22no—
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—42no—
65#127—42no—
66#121—44no—
67#121—82no—
68#121—82no—
69#117—81no—
70#117—81no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4C22⋊C412yes—
76C2×C4⋊C4C22⋊C412yes—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—42no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—42no—
84C2×C42C22×C411yescharacteristic
85C2×C42C2×D411yescharacteristic
86C2×C42C2×D412yes—
87C4⋊C8—44no—
88C4⋊C8—82no—
89#102—82no—
90#102—82no—
91C22⋊C8—44no—
92C22⋊C8—42no—
93C22⋊C8—82no—
94#97—42no—
95#97—81no—
96C4×C8—81no—
97#95—24no—
98C24—41no—
99C24#12311yescharacteristic
100C2×D4—42no—
101C2×D4—42no—
102C2×D4—44no—
103C2×D4—42no—
104C2×D4—82no—
105C2×D4—82no—
106C2×D4—82no—
107C2×D4—84no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—41no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—41no—
118C22×C4C2×C22⋊C411yescharacteristic
119C22×C4C4×D412yes—
120C22×C4#12011yescharacteristic
121C22×C4#12112yes—
122C2×C8—42no—
123C2×C8—42no—
124C2×C8—42no—
125C2×C8—42no—
126C2×C8—42no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—42no—
131C2×C8—82no—
132C2×C8—82no—
133C2×C8—82no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—82no—
140C4⋊C4—82no—
141C22⋊C4—81no—
142C22⋊C4—81no—
143C22⋊C4—82no—
144C22⋊C4—82no—
145C22⋊C4—82no—
146C42—22no—
147C42—22no—
148C42—42no—
149C42—41no—
150C42—41no—
151C42—42no—
152C23—22no—
153C23—21no—
154C23—21no—
155C23—22no—
156C23—21no—
157C23—82no—
158C23—81no—
159C23—81no—
160C23—82no—
161C23—81no—
162C23#38611yescharacteristic
163D4—84no—
164D4—164no—
165D4—164no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—81no—
185C2×C4—82no—
186C2×C4#41012yes—
187C2×C4#45412yes—
188C2×C4#47112yes—
189C2×C4#47212yes—
190C8—82no—
191C8—82no—
192C8—82no—
193C8—162no—
194C22—41no—
195C22—42no—
196C22—42no—
197C22—41no—
198C22—41no—
199C22—161no—
200C22—162no—
201C22—162no—
202C22—161no—
203C22—161no—
204C22#188112yes—
205C22#188811yescharacteristic
206C22#189911yescharacteristic
207C22#190712yes—
208C22#191111yescharacteristic
209C4—22no—
210C4—22no—
211C4—22no—
212C4—22no—
213C4—42no—
214C4—82no—
215C4—82no—
216C4—161no—
217C2—81no—
218C2—321no—
219C2#900811yescharacteristic
220C2#934812yes—
221C2#937112yes—
222C2#943811yescharacteristic
223C2#946511yescharacteristic
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. #21918C2semidirect
3. #8729C2semidirect
5. #7038C2semidirect

Maximal subgroups

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

subgroupindex
#279812
#219182
#87292
#77242
#70382