Finite Group Databasebeta

← #140994

Subgroups — #140994

Order 512.

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

normal subgroups only

This group has 365 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#33246C211yesmaximal, characteristic
2#12660C211yesmaximal, characteristic, complemented
3#12541C211yesmaximal, characteristic, complemented
4C2×#1356C212yesmaximal, complemented
5C2×#1355C212yesmaximal
6C2×#499C411yescharacteristic
7C2×#492C411yescharacteristic
8C2×#491C2211yescharacteristic
9C2×#490C2211yescharacteristic
10#2642C411yescharacteristic
11#2641C411yescharacteristic
12C2×#364—21no—
13C2×#347—21no—
14C2×#347C2212yes—
15C2×#347C2212yes—
16C4×M32—21no—
17#1973—21no—
18#1973—21no—
19#1968—21no—
20#1968—21no—
21#1726C2211yescharacteristic
22#1377—21no—
23#1356—24no—
24#1355—24no—
25C22×SD16—21no—
26C22×SD16—21no—
27C2×C4⋊Q8C2×C411yescharacteristic
28C2×#127C2×C411yescharacteristic
29C2×#125C2×C411yescharacteristic
30C2×#125C2×C411yescharacteristic
31C2×M32—22no—
32C2×M32—22no—
33#499—42no—
34#492—42no—
35#491—22no—
36#491—22no—
37#490—22no—
38#490—22no—
39C2×#107—21no—
40C2×#106—21no—
41C2×#103—21no—
42C2×#103—21no—
43C2×#102—21no—
44C2×#102—21no—
45C2×#97D411yescharacteristic
46C2×#97D411yescharacteristic
47C2×#97C2311yescharacteristic
48C2×C4×C8C2×C411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51#400—42no—
52#394—42no—
53#391—21no—
54#391—21no—
55#390—21no—
56#390—21no—
57#376C2×C411yescharacteristic
58#364—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—
69C2×SD16—42no—
70C2×SD16—42no—
71C2×SD16—42no—
72C2×SD16—42no—
73C2×SD16—41no—
74C2×SD16—41no—
75C2×SD16—42no—
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C22×C8C2×D411yescharacteristic
79C22×C8C2×D411yescharacteristic
80C4⋊Q8—22no—
81C4⋊Q8—41no—
82#127—22no—
83#127—41no—
84#125—22no—
85#125—22no—
86#125—41no—
87#125—41no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42C22⋊C411yescharacteristic
97C2×C42C22×C411yescharacteristic
98M32—24no—
99M32—24no—
100M32—24no—
101M32—42no—
102C2×C16—24no—
103C2×C16—42no—
104#107—42no—
105#106—42no—
106#103—42no—
107#103—41no—
108#103—41no—
109#102—42no—
110#102—41no—
111#102—41no—
112#97—22no—
113#97—41no—
114#97C2×D412yes—
115#97#4112yes—
116C4×C8—22no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—22no—
120#95—22no—
121#95—42no—
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—162no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4#10012yes—
144C22×C4C2×C22⋊C411yescharacteristic
145SD16—84no—
146SD16—82no—
147SD16—82no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8#10012yes—
155C2×C8#10412yes—
156C2×C8#10411yescharacteristic
157C2×C8#10411yescharacteristic
158C2×C8#10412yes—
159C2×C8#12112yes—
160C2×C8#12411yescharacteristic
161C2×C8#12711yescharacteristic
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C22⋊C4—162no—
170C22⋊C4—162no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42C4×D412yes—
177C16—28no—
178C16—44no—
179C23—82no—
180C23—82no—
181C23—81no—
182C23—81no—
183C23—81no—
184C23C2×#10011yescharacteristic
185Q8—81no—
186Q8—81no—
187Q8—81no—
188Q8—81no—
189D4—81no—
190D4—81no—
191D4—81no—
192D4—81no—
193D4—164no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
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#32912yes—
218C8#32912yes—
219C22—22no—
220C22—161no—
221C22—161no—
222C22—162no—
223C22—162no—
224C22—161no—
225C22#152511yescharacteristic
226C22#153011yescharacteristic
227C22#196511yescharacteristic
228C4—42no—
229C4—42no—
230C4—81no—
231C4—161no—
232C4—161no—
233C4—161no—
234C4C2×#32911yescharacteristic
235C4#152711yescharacteristic
236C4#196211yescharacteristic
237C4#198011yescharacteristic
238C2—21no—
239C2—21no—
240C2—321no—
241C2#968511yescharacteristic
242C2#1191212yes—
subgroupG/H
center, socleC22#1530
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G
LECSG ▷ C2×#97 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12660C2semidirect
3. #12541C2semidirect
4. C2×#1356C2semidirect

Maximal subgroups

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

subgroupindex
#332462
#126602
#125412
C2×#13562
C2×#13552