Finite Group Databasebeta

← #98695

Subgroups — #98695

Order 512.

656 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 112 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#59864C211yesmaximal, characteristic
2#9644C211yesmaximal, characteristic
3#8591C212yesmaximal, complemented
4#7651C211yesmaximal, characteristic
5#7591C212yesmaximal, complemented
6#2804—22no—
7#2580—24no—
8#2571C412yes—
9#2563—22no—
10#2487—22no—
11#2366C412yes—
12D4×C4⋊C4—22no—
13C4×C4⋊Q8C2212yes—
14C4×#127C2211yescharacteristic
15#1842C2211yescharacteristic
16#1831C2212yes—
17#1744C2211yescharacteristic
18C2×#332—22no—
19C2×D4⋊C8—22no—
20#1296—22no—
21#1271—24no—
22#1270—22no—
23C2×C4⋊Q8D412yes—
24C2×#127C2×C411yescharacteristic
25C2×C4×Q8—22no—
26C2×C4×D4—24no—
27C2×C4×D4—22no—
28C2×C4×D4D412yes—
29C22×C4⋊C4—22no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—21no—
33C2×C4×C8—21no—
34#400—24no—
35#399—24no—
36#393—28no—
37#391—22no—
38#390—24no—
39#390—22no—
40#390D412yes—
41#389—24no—
42#389—22no—
43#389D414yes—
44#386—24no—
45#385—24no—
46#385—28no—
47#384C2×C411yescharacteristic
48#384D412yes—
49#382—24no—
50#380—22no—
51#380—24no—
52C4×C4⋊C4—22no—
53C4×C4⋊C4C2×C414yes—
54C43C2311yescharacteristic
55#336—22no—
56#336—22no—
57#336—22no—
58#332—44no—
59D4⋊C8—44no—
60C22×Q8—22no—
61C22×D4—22no—
62C22×D4C22⋊C412yes—
63C23×C4—24no—
64C23×C4—22no—
65C22×C8—41no—
66C22×C8—42no—
67C22×C8—41no—
68C4⋊Q8—22no—
69C4⋊Q8—22no—
70C4⋊Q8—22no—
71C4⋊Q8—22no—
72#127—24no—
73C4×Q8—84no—
74C4×D4—42no—
75C4×D4—42no—
76C4×D4—42no—
77C4×D4—42no—
78C4×D4—42no—
79C4×D4—42no—
80C4×D4—88no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—82no—
88C2×C4⋊C4—82no—
89C2×C4⋊C4C22⋊C414yes—
90C2×C4⋊C4C22⋊C414yes—
91C2×C4⋊C4C22⋊C412yes—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—28no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—22no—
96C2×C42—24no—
97C2×C42C22×C411yescharacteristic
98C2×C42C2×D411yescharacteristic
99C2×C42C2×D412yes—
100C2×C42C2×D411yescharacteristic
101C2×C42C2×D412yes—
102C4⋊C8—42no—
103C4⋊C8—42no—
104C4⋊C8—42no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107C4×C8—22no—
108C4×C8—41no—
109#95—24no—
110#95—28no—
111#95—24no—
112#95—24no—
113C24—22no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—82no—
117C2×Q8—82no—
118C2×D4—28no—
119C2×D4—22no—
120C2×D4—44no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4—42no—
125C2×D4#10114yes—
126C22×C4—24no—
127C22×C4—22no—
128C22×C4—24no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—84no—
134C22×C4—88no—
135C22×C4C2×C22⋊C411yescharacteristic
136C22×C4C2×C22⋊C412yes—
137C22×C4#12011yescharacteristic
138C22×C4#12012yes—
139C22×C4#12011yescharacteristic
140C2×C8—42no—
141C2×C8—82no—
142C2×C8—82no—
143C2×C8—82no—
144C2×C8—82no—
145C2×C8—82no—
146C2×C8—81no—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—42no—
150C4⋊C4—44no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—82no—
158C4⋊C4—84no—
159C4⋊C4—84no—
160C22⋊C4—44no—
161C22⋊C4—44no—
162C22⋊C4—82no—
163C22⋊C4—84no—
164C22⋊C4—88no—
165C42—22no—
166C42—21no—
167C42—21no—
168C42—41no—
169C42—42no—
170C42—48no—
171C42#13512yes—
172C42#13512yes—
173C42AGL(1,ℤ/8)12yes—
174C42#13712yes—
175C23—24no—
176C23—42no—
177C23—42no—
178C23—44no—
179C23—44no—
180C23#37911yescharacteristic
181Q8—84no—
182Q8—84no—
183D4—44no—
184D4—48no—
185D4—48no—
186D4—82no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—84no—
206C2×C4—168no—
207C2×C4C2×#10112yes—
208C2×C4#41612yes—
209C2×C4#41712yes—
210C2×C4#44912yes—
211C2×C4#44912yes—
212C2×C4#45212yes—
213C2×C4#45212yes—
214C2×C4#45311yescharacteristic
215C2×C4#45511yescharacteristic
216C8—82no—
217C8—162no—
218C8—161no—
219C22—44no—
220C22—48no—
221C22—82no—
222C22—84no—
223C22—82no—
224C22#178411yescharacteristic
225C22#178812yes—
226C22#179211yescharacteristic
227C22#179411yescharacteristic
228C22#179512yes—
229C4—22no—
230C4—22no—
231C4—22no—
232C4—22no—
233C4—22no—
234C4—41no—
235C4—84no—
236C4—84no—
237C4—82no—
238C4#162011yescharacteristic
239C4#163011yescharacteristic
240C4#164012yes—
241C2—84no—
242C2#839812yes—
243C2#843412yes—
244C2#844511yescharacteristic
245C2#845711yescharacteristic
246C2#849111yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8591C2semidirect
5. #7591C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#96442
#85912
#76512
#75912