Finite Group Databasebeta

← #141002

Subgroups — #141002

Order 512.

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

normal subgroups only

This group has 348 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#33234C211yesmaximal, characteristic
2#12659C211yesmaximal, characteristic, complemented
3#12595C211yesmaximal, characteristic, complemented
4C2×#1356C212yesmaximal, complemented
5C2×#1355C212yesmaximal
6C2×#491C411yescharacteristic
7C2×#491C2211yescharacteristic
8C2×#490C411yescharacteristic
9C2×#490C2211yescharacteristic
10#2859C411yescharacteristic
11#2858C411yescharacteristic
12C2×#364—21no—
13C2×#347C2212yes—
14C2×#347C2212yes—
15#1959—21no—
16#1959—21no—
17#1958—21no—
18#1958—21no—
19C4×#97C2211yescharacteristic
20#1377—21no—
21#1377—21no—
22#1376—21no—
23#1356—24no—
24#1355—24no—
25C2×C4⋊Q8C2×C411yescharacteristic
26C2×#127C2×C411yescharacteristic
27C2×#125C2×C411yescharacteristic
28C2×#125C2×C411yescharacteristic
29C2×#124—21no—
30C2×#124—21no—
31C2×M32—22no—
32C2×M32—22no—
33#491—22no—
34#491—22no—
35#491—42no—
36#490—22no—
37#490—22no—
38#490—42no—
39C2×#103—21no—
40C2×#103—21no—
41C2×#103—21no—
42C2×#103—21no—
43C2×#102—21no—
44C2×#102—21no—
45C2×#102—21no—
46C2×#102—21no—
47C2×#97C2×C411yescharacteristic
48C2×#97D411yescharacteristic
49C2×#97D411yescharacteristic
50C2×#97C2311yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53#390—42no—
54#389—42no—
55#384—21no—
56#382—21no—
57C43C2×C411yescharacteristic
58#364—24no—
59#347—22no—
60#347—22no—
61#347—22no—
62#347D414yes—
63C22×Q8—21no—
64C22×D4—21no—
65C22×D4—41no—
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22⋊C411yescharacteristic
68C22×C8C2×D411yescharacteristic
69C22×C8C2×D411yescharacteristic
70C4⋊Q8—22no—
71C4⋊Q8—41no—
72#127—22no—
73#127—41no—
74#125—22no—
75#125—22no—
76#125—41no—
77#125—41no—
78#124—42no—
79#124—42no—
80#124—42no—
81#124—42no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C42—22no—
91C2×C42C22⋊C411yescharacteristic
92C2×C42C22⋊C411yescharacteristic
93C2×C42C22×C411yescharacteristic
94M32—24no—
95M32—24no—
96M32—42no—
97M32—42no—
98C2×C16—42no—
99C2×C16—42no—
100#103—42no—
101#103—42no—
102#103—42no—
103#103—41no—
104#103—41no—
105#102—42no—
106#102—42no—
107#102—42no—
108#102—41no—
109#102—41no—
110#97—22no—
111#97—41no—
112#97—41no—
113#97C2×D412yes—
114#97#4112yes—
115C4×C8—22no—
116C4×C8—41no—
117#95—42no—
118C24—41no—
119C2×Q8—41no—
120C2×Q8—41no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8—41no—
124C2×Q8—41no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—82no—
132C2×D4—84no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4#10012yes—
139C22×C4C2×C22⋊C411yescharacteristic
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8#10012yes—
150C2×C8#12112yes—
151C2×C8#12411yescharacteristic
152C2×C8#12711yescharacteristic
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—82no—
158C4⋊C4—82no—
159C4⋊C4—82no—
160C4⋊C4—84no—
161C22⋊C4—84no—
162C22⋊C4—84no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—82no—
168C42#10412yes—
169C42C4×D412yes—
170C16—44no—
171C16—44no—
172C23—84no—
173C23—81no—
174C23—81no—
175C23—81no—
176C23C2×#10011yescharacteristic
177Q8—81no—
178Q8—81no—
179Q8—81no—
180Q8—81no—
181D4—81no—
182D4—81no—
183D4—81no—
184D4—81no—
185D4—164no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4#39011yescharacteristic
200C2×C4C2×#10011yescharacteristic
201C2×C4C2×#10411yescharacteristic
202C2×C4#42111yescharacteristic
203C2×C4#42111yescharacteristic
204C2×C4#42111yescharacteristic
205C8—22no—
206C8—21no—
207C8—21no—
208C8—22no—
209C8—41no—
210C22—22no—
211C22—162no—
212C22—162no—
213C22—162no—
214C22—161no—
215C22#152711yescharacteristic
216C22#153311yescharacteristic
217C22#196511yescharacteristic
218C4—42no—
219C4—42no—
220C4—42no—
221C4—161no—
222C4—161no—
223C4—161no—
224C4#153011yescharacteristic
225C4#153611yescharacteristic
226C4#196311yescharacteristic
227C4#198011yescharacteristic
228C2—21no—
229C2—21no—
230C2—321no—
231C2#968911yescharacteristic
232C2#1191712yes—
subgroupG/H
center, socleC22#1533
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×#97 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12659C2semidirect
3. #12595C2semidirect
4. C2×#1356C2semidirect

Maximal subgroups

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

subgroupindex
#332342
#126592
#125952
C2×#13562
C2×#13552