Finite Group Databasebeta

← #99335

Subgroups — #99335

Order 512.

357 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 92 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#23353C211yesmaximal, characteristic
2#9640C211yesmaximal, characteristic
3#8786C212yesmaximal, complemented
4C4×#329C211yesmaximal, characteristic
5#7030C212yesmaximal, complemented
6C4×#124C2211yescharacteristic
7#1985—21no—
8#1924—22no—
9#1870—21no—
10#1851—22no—
11#1842C2211yescharacteristic
12#1831C2212yes—
13#1744C2211yescharacteristic
14C4×#97C2212yes—
15C2×#329C412yes—
16C2×Q8⋊C8—21no—
17C2×D4⋊C8—21no—
18#1295C411yescharacteristic
19#1295C411yescharacteristic
20#1277—22no—
21#1277—22no—
22C22×M16—21no—
23C2×#124C2×C411yescharacteristic
24C2×C4×Q8D411yescharacteristic
25C2×C4×D4D411yescharacteristic
26C2×C4⋊C8—21no—
27C2×C4⋊C8—22no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8D412yes—
30C2×C4⋊C8D412yes—
31C2×C22⋊C8—22no—
32C2×C22⋊C8—21no—
33C2×#97—22no—
34C2×#97—21no—
35C2×#97C2×C412yes—
36C2×#97C2×C412yes—
37C2×C4×C8—21no—
38C2×C4×C8—22no—
39#391Q811yescharacteristic
40#390Q811yescharacteristic
41#388—22no—
42#382C2×C411yescharacteristic
43C4×C22⋊C4—22no—
44C43C2311yescharacteristic
45#336—22no—
46#336—22no—
47#336—22no—
48#336—22no—
49#329—24no—
50Q8⋊C8—42no—
51D4⋊C8—42no—
52C22×Q8C4⋊C411yescharacteristic
53C22×D4C4⋊C411yescharacteristic
54C23×C4—21no—
55C2×M16—81no—
56C2×M16—82no—
57C22×C8—22no—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—22no—
61C22×C8—41no—
62C22×C8—41no—
63#124—22no—
64#124—21no—
65#124—21no—
66C4×Q8—41no—
67C4×Q8—41no—
68C4×D4—41no—
69C4×D4—41no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4C4⋊C411yescharacteristic
72C2×C4⋊C4C4⋊C411yescharacteristic
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—21no—
76C2×C42—21no—
77C2×C42C22×C411yescharacteristic
78C2×C42C2×D411yescharacteristic
79C2×C42C2×D412yes—
80C2×C42C2×Q811yescharacteristic
81C2×C42#4112yes—
82C4⋊C8—42no—
83C4⋊C8—42no—
84C4⋊C8—41no—
85C4⋊C8—41no—
86C4⋊C8—44no—
87C4⋊C8—81no—
88C22⋊C8—81no—
89C22⋊C8—82no—
90#97—22no—
91#97—22no—
92#97—22no—
93#97—44no—
94#97—81no—
95#97M1614yescomplemented
96C4×C8—22no—
97C4×C8—42no—
98C4×C8—42no—
99C4×C8—41no—
100#95—22no—
101#95—22no—
102C24—21no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—41no—
106C2×Q8—41no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—41no—
110C2×D4—41no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—41no—
115C22×C4—42no—
116C22×C4—41no—
117C22×C4—42no—
118C22×C4C2×C4⋊C411yescharacteristic
119C22×C4C4×D412yes—
120C22×C4#12011yescharacteristic
121C22×C4C22⋊Q812yes—
122C22×C4#12311yescharacteristic
123M16—84no—
124C2×C8—24no—
125C2×C8—24no—
126C2×C8—42no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—42no—
131C2×C8—42no—
132C2×C8—42no—
133C2×C8—42no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8—82no—
138C2×C8—81no—
139C2×C8—82no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—81no—
143C22⋊C4—42no—
144C22⋊C4—42no—
145C22⋊C4—82no—
146C22⋊C4—81no—
147C42—21no—
148C42—22no—
149C42—22no—
150C42—21no—
151C42—42no—
152C42—41no—
153C42—41no—
154C42—41no—
155C42C2×M1612yes—
156C42#13111yescharacteristic
157C42#13111yescharacteristic
158C23—41no—
159C23—41no—
160C23—41no—
161C23—42no—
162C23—42no—
163C23#38511yescharacteristic
164Q8—42no—
165Q8—81no—
166D4—42no—
167D4—81no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—82no—
187C2×C4#42311yescharacteristic
188C2×C4#42411yescharacteristic
189C2×C4#43512yes—
190C2×C4#43612yes—
191C2×C4#45311yescharacteristic
192C2×C4#45412yes—
193C2×C4#45512yes—
194C2×C4#45511yescharacteristic
195C8—44no—
196C8—44no—
197C8—82no—
198C8—82no—
199C8—82no—
200C8—161no—
201C22—81no—
202C22—82no—
203C22—81no—
204C22—82no—
205C22—81no—
206C22#185111yescharacteristic
207C22#185611yescharacteristic
208C22#185811yescharacteristic
209C22#186012yes—
210C22#186112yes—
211C4—21no—
212C4—21no—
213C4—22no—
214C4—22no—
215C4—42no—
216C4—41no—
217C4—81no—
218C4—81no—
219C4—81no—
220C4#165711yescharacteristic
221C4#165911yescharacteristic
222C4#166212yes—
223C2—161no—
224C2#883711yescharacteristic
225C2#884212yes—
226C2#891411yescharacteristic
227C2#891712yes—
228C2#891811yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
socleC23#385
prime radicalC22×D4C4⋊C4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8786C2semidirect
5. #7030C2semidirect
95. #97M16semidirect

Maximal subgroups

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

subgroupindex
#233532
#96402
#87862
C4×#3292
#70302