Finite Group Databasebeta

← #140997

Subgroups — #140997

Order 512.

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

normal subgroups only

This group has 343 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#33246C211yesmaximal, characteristic
2#12657C211yesmaximal, characteristic, complemented
3#12596C211yesmaximal, characteristic, complemented
4C2×#1356C212yesmaximal, complemented
5C2×#1355C212yesmaximal
6C2×#499C411yescharacteristic
7C2×#492C411yescharacteristic
8C2×#491C2211yescharacteristic
9C2×#490C2211yescharacteristic
10#2642C411yescharacteristic
11#2641C411yescharacteristic
12#2150—21no—
13C2×#347C2212yes—
14C2×#347C2212yes—
15#1973—21no—
16#1973—21no—
17#1968—21no—
18#1968—21no—
19#1726C2211yescharacteristic
20#1377—21no—
21#1376—21no—
22#1375—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
58C4⋊C16—24no—
59#347—22no—
60#347—22no—
61#347—22no—
62#347D414yes—
63C22×Q8—21no—
64C22×D4—21no—
65C22×D4—41no—
66C2×SD16—42no—
67C2×SD16—42no—
68C2×SD16—42no—
69C2×SD16—42no—
70C2×SD16—42no—
71C2×SD16—42no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C4⋊Q8—22no—
77C4⋊Q8—41no—
78#127—22no—
79#127—41no—
80#125—22no—
81#125—22no—
82#125—41no—
83#125—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C42C22⋊C411yescharacteristic
92C2×C42C22⋊C411yescharacteristic
93C2×C42C22×C411yescharacteristic
94M32—24no—
95M32—42no—
96M32—42no—
97M32—42no—
98C2×C16—24no—
99C2×C16—42no—
100#107—42no—
101#106—42no—
102#103—42no—
103#103—41no—
104#103—41no—
105#102—42no—
106#102—41no—
107#102—41no—
108#97—41no—
109#97—41no—
110#97C2×D412yes—
111#97#4112yes—
112C4×C8—22no—
113C4×C8—22no—
114C4×C8—41no—
115#95—22no—
116#95—42no—
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—162no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4#10012yes—
139C22×C4C2×C22⋊C411yescharacteristic
140SD16—84no—
141SD16—84no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8#10012yes—
151C2×C8#10412yes—
152C2×C8#12112yes—
153C2×C8#12411yescharacteristic
154C2×C8#12711yescharacteristic
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—82no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C22⋊C4—162no—
163C22⋊C4—162no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—21no—
169C42C4×D412yes—
170C16—44no—
171C16—44no—
172C23—82no—
173C23—82no—
174C23—81no—
175C23—81no—
176C23—81no—
177C23C2×#10011yescharacteristic
178Q8—81no—
179Q8—81no—
180Q8—81no—
181Q8—81no—
182D4—81no—
183D4—81no—
184D4—81no—
185D4—81no—
186D4—164no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4#39011yescharacteristic
201C2×C4C2×#10011yescharacteristic
202C2×C4C2×#10411yescharacteristic
203C2×C4#42111yescharacteristic
204C2×C4#42111yescharacteristic
205C2×C4#42111yescharacteristic
206C8—22no—
207C8—21no—
208C8—21no—
209C8—22no—
210C8—22no—
211C22—22no—
212C22—161no—
213C22—161no—
214C22—162no—
215C22—162no—
216C22—161no—
217C22#153011yescharacteristic
218C22#153611yescharacteristic
219C22#196511yescharacteristic
220C4—42no—
221C4—42no—
222C4—81no—
223C4—161no—
224C4—161no—
225C4—161no—
226C4#152711yescharacteristic
227C4#153311yescharacteristic
228C4#196311yescharacteristic
229C4#198011yescharacteristic
230C2—21no—
231C2—21no—
232C2—321no—
233C2#968911yescharacteristic
234C2#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. #12657C2semidirect
3. #12596C2semidirect
4. C2×#1356C2semidirect

Maximal subgroups

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

subgroupindex
#332462
#126572
#125962
C2×#13562
C2×#13552