Finite Group Databasebeta

← #100813

Subgroups — #100813

Order 512.

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

normal subgroups only

This group has 436 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#28447C211yesmaximal, characteristic
2#23366C211yesmaximal, characteristic, complemented
3#8594C211yesmaximal, characteristic, complemented
4#7719C212yesmaximal
5#7041C212yesmaximal, complemented
6C2×#497—21no—
7C2×#464—22no—
8#2858C2211yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10#2016C412yes—
11#2014C412yes—
12#1985—22no—
13#1959—21no—
14#1917—24no—
15#1839C2212yes—
16#1831C2212yes—
17#1767C2211yescharacteristic
18C2×#333—21no—
19#1296—21no—
20#1270—24no—
21C22×Q16—22no—
22C22×SD16—22no—
23C2×C4⋊Q8C2×C411yescharacteristic
24C2×C4⋊Q8D411yescharacteristic
25C2×#127C2×C411yescharacteristic
26C2×#124—21no—
27C2×#124—21no—
28C2×C4×Q8D412yes—
29#497—44no—
30#464—44no—
31#464—44no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8—22no—
34C2×C4⋊C8—22no—
35C2×#103—22no—
36C2×#102—22no—
37C2×#97—21no—
38C2×#97—22no—
39#391C2×C412yes—
40#390—22no—
41#390—22no—
42#389—22no—
43#384D411yescharacteristic
44C4×C4⋊C4—22no—
45C43C2311yescharacteristic
46#336—24no—
47#336—22no—
48#336—22no—
49#336—22no—
50#336—22no—
51#336C2×C412yes—
52#336D414yes—
53#333—42no—
54C22×Q8—21no—
55C22×Q8C22⋊C412yes—
56C22×D4—21no—
57C22×D4—22no—
58C2×Q16—44no—
59C2×Q16—44no—
60C2×Q16—82no—
61C2×SD16—44no—
62C2×SD16—44no—
63C2×SD16—82no—
64C22×C8—22no—
65C22×C8—24no—
66C22×C8—42no—
67C4⋊Q8—21no—
68C4⋊Q8—22no—
69C4⋊Q8—21no—
70C4⋊Q8—81no—
71#127—81no—
72#124—42no—
73#124—42no—
74#124—41no—
75#124—42no—
76#124—41no—
77C4×Q8—42no—
78C4×Q8—42no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4C22⋊C412yes—
84C2×C22⋊C4—41no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—41no—
87C2×C42—22no—
88C2×C42C22×C411yescharacteristic
89C2×C42C2×D411yescharacteristic
90C2×C42C2×D412yes—
91C2×C42C2×D411yescharacteristic
92C2×C42#4112yes—
93C4⋊C8—24no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C4⋊C8—82no—
97#103—44no—
98#102—44no—
99#97—42no—
100#97—82no—
101#95—24no—
102C24—41no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—42no—
106C2×Q8—42no—
107C2×Q8—42no—
108C2×Q8—42no—
109C2×Q8—41no—
110C2×Q8—41no—
111C2×D4—42no—
112C2×D4—42no—
113C2×D4—44no—
114C2×D4—41no—
115C2×D4—41no—
116C2×D4—82no—
117C2×D4—82no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—41no—
121C22×C4C2×C22⋊C411yescharacteristic
122C22×C4C4×D412yes—
123C22×C4#12011yescharacteristic
124C22×C4#12112yes—
125C22×C4#12311yescharacteristic
126Q16—88no—
127SD16—88no—
128C2×C8—24no—
129C2×C8—42no—
130C2×C8—42no—
131C2×C8—48no—
132C2×C8—44no—
133C2×C8—82no—
134C2×C8—82no—
135C2×C8—82no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—82no—
140C4⋊C4—82no—
141C4⋊C4—82no—
142C22⋊C4—81no—
143C22⋊C4—81no—
144C22⋊C4—82no—
145C22⋊C4—82no—
146C22⋊C4—82no—
147C42—22no—
148C42—41no—
149C42—42no—
150C42—42no—
151C42—42no—
152C42—42no—
153C42AGL(1,ℤ/8)11yescharacteristic
154C42AGL(1,ℤ/8)12yes—
155C42#13711yescharacteristic
156C23—82no—
157C23—81no—
158C23—81no—
159C23—82no—
160C23—81no—
161C23#38611yescharacteristic
162Q8—44no—
163Q8—82no—
164Q8—82no—
165Q8—82no—
166D4—82no—
167D4—82no—
168D4—164no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—44no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4#41711yescharacteristic
188C2×C4#41811yescharacteristic
189C2×C4#45312yes—
190C2×C4#46812yes—
191C2×C4#46912yes—
192C8—44no—
193C8—84no—
194C8—162no—
195C22—161no—
196C22—162no—
197C22—162no—
198C22—161no—
199C22—161no—
200C22#188112yes—
201C22#188711yescharacteristic
202C22#189511yescharacteristic
203C22#190512yes—
204C22#191411yescharacteristic
205C4—22no—
206C4—22no—
207C4—21no—
208C4—21no—
209C4—22no—
210C4—41no—
211C4—42no—
212C4—82no—
213C4—82no—
214C4—161no—
215C2—321no—
216C2#900711yescharacteristic
217C2#931012yes—
218C2#933412yes—
219C2#940511yescharacteristic
220C2#942911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23366C2semidirect
3. #8594C2semidirect
5. #7041C2semidirect

Maximal subgroups

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

subgroupindex
#284472
#233662
#85942
#77192
#70412