Finite Group Databasebeta

← #140993

Subgroups — #140993

Order 512.

372 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 84 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#12658C211yesmaximal, characteristic, complemented
3C4×#347C211yesmaximal, characteristic, complemented
4C2×#1356C212yesmaximal, complemented
5C2×#1355C212yesmaximal
6C2×#491C411yescharacteristic
7C2×#491C2211yescharacteristic
8C2×#490C411yescharacteristic
9C2×#490C2211yescharacteristic
10#2859C411yescharacteristic
11#2858C411yescharacteristic
12#2150—21no—
13C2×#347—21no—
14C2×#347C2212yes—
15C2×#347C2212yes—
16C4×M32—21no—
17#1959—21no—
18#1959—21no—
19#1958—21no—
20#1958—21no—
21C4×#97C2211yescharacteristic
22#1375—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
58C4⋊C16—24no—
59#347—24no—
60#347—22no—
61#347—22no—
62#347—22no—
63#347D414yes—
64#346—22no—
65C4×C16—22no—
66C22×Q8—21no—
67C22×D4—21no—
68C22×D4—41no—
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C22×C8C2×D411yescharacteristic
73C4⋊Q8—22no—
74C4⋊Q8—41no—
75#127—22no—
76#127—41no—
77#125—22no—
78#125—22no—
79#125—41no—
80#125—41no—
81#124—42no—
82#124—42no—
83#124—41no—
84#124—41no—
85#124—42no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C42—22no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42C22⋊C411yescharacteristic
97C2×C42C22×C411yescharacteristic
98M32—24no—
99M32—24no—
100M32—42no—
101M32—42no—
102C2×C16—24no—
103C2×C16—24no—
104#103—42no—
105#103—42no—
106#103—42no—
107#103—41no—
108#103—41no—
109#102—42no—
110#102—42no—
111#102—42no—
112#102—41no—
113#102—41no—
114#97—22no—
115#97—22no—
116#97—41no—
117#97C2×D412yes—
118#97#4112yes—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8—22no—
122#95—42no—
123C24—41no—
124C2×Q8—41no—
125C2×Q8—41no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—82no—
137C2×D4—84no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4#10012yes—
144C22×C4C2×C22⋊C411yescharacteristic
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8#10012yes—
155C2×C8#12112yes—
156C2×C8#12411yescharacteristic
157C2×C8#12711yescharacteristic
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—82no—
163C4⋊C4—82no—
164C4⋊C4—82no—
165C4⋊C4—84no—
166C22⋊C4—84no—
167C22⋊C4—82no—
168C22⋊C4—82no—
169C42—21no—
170C42—21no—
171C42—82no—
172C42#10412yes—
173C42#10411yescharacteristic
174C42#10411yescharacteristic
175C42#10412yes—
176C42C4×D412yes—
177C16—28no—
178C16—44no—
179C23—84no—
180C23—81no—
181C23—81no—
182C23—81no—
183C23C2×#10011yescharacteristic
184Q8—81no—
185Q8—81no—
186Q8—81no—
187Q8—81no—
188D4—81no—
189D4—81no—
190D4—81no—
191D4—81no—
192D4—164no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4#32912yes—
206C2×C4#32912yes—
207C2×C4#39011yescharacteristic
208C2×C4C2×#10011yescharacteristic
209C2×C4C2×#10411yescharacteristic
210C2×C4C2×#10411yescharacteristic
211C2×C4C2×#10411yescharacteristic
212C2×C4#42111yescharacteristic
213C8—22no—
214C8—21no—
215C8—21no—
216C8—22no—
217C8—41no—
218C22—22no—
219C22—162no—
220C22—162no—
221C22—162no—
222C22—161no—
223C22C2×#32911yescharacteristic
224C22#152711yescharacteristic
225C22#196511yescharacteristic
226C4—22no—
227C4—22no—
228C4—42no—
229C4—42no—
230C4—161no—
231C4—161no—
232C4—161no—
233C4#152511yescharacteristic
234C4#153011yescharacteristic
235C4#196211yescharacteristic
236C4#198011yescharacteristic
237C2—21no—
238C2—21no—
239C2—321no—
240C2#968511yescharacteristic
241C2#1191112yes—
subgroupG/H
center, socleC22C2×#329
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ 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. #12658C2semidirect
3. C4×#347C2semidirect
4. C2×#1356C2semidirect

Maximal subgroups

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

subgroupindex
#332342
#126582
C4×#3472
C2×#13562
C2×#13552