Finite Group Databasebeta

← #97505

Subgroups — #97505

Order 512.

712 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 136 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#9638C211yesmaximal, characteristic
3#9637C212yesmaximal, complemented
4#8587C212yesmaximal, complemented
5#8585C211yesmaximal, characteristic
6#2804C414yes—
7#2580—24no—
8#2571—21no—
9#2563—22no—
10#2487—22no—
11#2366—21no—
12D4×C4⋊C4—22no—
13C4×C4⋊Q8C2212yes—
14C4×#127C2211yescharacteristic
15#1842C2212yes—
16#1842C2211yescharacteristic
17C42×C8C2211yescharacteristic
18C2×#333—22no—
19C2×#331—21no—
20C2×Q8⋊C8—24no—
21C2×D4⋊C8—22no—
22#1296—21no—
23#1296—22no—
24C2×C4⋊Q8D412yes—
25C2×#127D411yescharacteristic
26C2×C4×Q8D414yes—
27C2×C4×D4—24no—
28C2×C4×D4—22no—
29C2×C4×D4D412yes—
30C22×C4⋊C4—22no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—22no—
33C2×C4⋊C8—22no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—22no—
36C2×C4×C8—21no—
37C2×C4×C8—22no—
38#400—24no—
39#399—24no—
40#393—28no—
41#391C2×C414yes—
42#390—24no—
43#390—22no—
44#390C2×C412yes—
45#389—24no—
46#389—22no—
47#389—22no—
48#386—24no—
49#385—24no—
50#385—28no—
51#384D411yescharacteristic
52#384D412yes—
53#382—24no—
54#380—22no—
55#380—24no—
56C4×C4⋊C4—22no—
57C4×C4⋊C4—22no—
58C43C2311yescharacteristic
59#333—44no—
60#331—42no—
61Q8⋊C8—48no—
62D4⋊C8—44no—
63C22×Q8C22⋊C414yes—
64C22×D4—22no—
65C22×D4C22⋊C412yes—
66C23×C4—24no—
67C23×C4—22no—
68C22×C8—41no—
69C22×C8—42no—
70C22×C8—41no—
71C4⋊Q8—22no—
72C4⋊Q8—22no—
73C4⋊Q8—22no—
74C4⋊Q8—22no—
75#127—22no—
76#127SD1612yes—
77#127Q1612yes—
78C4×Q8—44no—
79C4×Q8—44no—
80C4×D4—42no—
81C4×D4—42no—
82C4×D4—42no—
83C4×D4—42no—
84C4×D4—42no—
85C4×D4—42no—
86C4×D4—84no—
87C4×D4—84no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—82no—
96C2×C4⋊C4—82no—
97C2×C4⋊C4C22⋊C412yes—
98C2×C4⋊C4C22⋊C414yes—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—28no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—22no—
103C2×C42—24no—
104C2×C42C22×C411yescharacteristic
105C2×C42C2×D411yescharacteristic
106C2×C42C2×D412yes—
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D412yes—
109C4⋊C8—42no—
110C4⋊C8—44no—
111C4⋊C8—42no—
112C4⋊C8—42no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117C4×C8—22no—
118C4×C8—24no—
119C4×C8—41no—
120C4×C8—42no—
121#95—24no—
122#95—28no—
123#95—24no—
124#95—24no—
125C24—22no—
126C2×Q8—24no—
127C2×Q8—24no—
128C2×Q8—44no—
129C2×Q8—44no—
130C2×D4—22no—
131C2×D4—24no—
132C2×D4—24no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4#10314yes—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—84no—
148C22×C4—88no—
149C22×C4C2×C22⋊C411yescharacteristic
150C22×C4C2×C22⋊C412yes—
151C22×C4#12011yescharacteristic
152C22×C4#12012yes—
153C22×C4#12011yescharacteristic
154C2×C8—42no—
155C2×C8—44no—
156C2×C8—82no—
157C2×C8—82no—
158C2×C8—82no—
159C2×C8—81no—
160C2×C8—82no—
161C4⋊C4—42no—
162C4⋊C4—44no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—84no—
174C4⋊C4—84no—
175C22⋊C4—44no—
176C22⋊C4—44no—
177C22⋊C4—82no—
178C22⋊C4—84no—
179C22⋊C4—88no—
180C42—21no—
181C42—22no—
182C42—21no—
183C42—22no—
184C42—41no—
185C42—44no—
186C42—44no—
187C42C2×SD1611yescharacteristic
188C42C2×Q1611yescharacteristic
189C42#13512yes—
190C42#13512yes—
191C42AGL(1,ℤ/8)12yes—
192C42#13712yes—
193C42#13712yes—
194C23—24no—
195C23—44no—
196C23—42no—
197C23—44no—
198C23—42no—
199C23#37911yescharacteristic
200Q8—48no—
201Q8—84no—
202D4—44no—
203D4—44no—
204D4—44no—
205D4—44no—
206D4—44no—
207D4—82no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—84no—
225C2×C4—168no—
226C2×C4C2×#10311yescharacteristic
227C2×C4#41612yes—
228C2×C4#41712yes—
229C2×C4#41911yescharacteristic
230C2×C4#44812yes—
231C2×C4#44912yes—
232C2×C4#44912yes—
233C2×C4#45112yes—
234C2×C4#45212yes—
235C2×C4#45212yes—
236C2×C4#45311yescharacteristic
237C2×C4#45312yes—
238C2×C4#45511yescharacteristic
239C2×C4#45512yes—
240C8—44no—
241C8—161no—
242C8—162no—
243C22—48no—
244C22—44no—
245C22—82no—
246C22—84no—
247C22—82no—
248C22#178711yescharacteristic
249C22#178812yes—
250C22#179111yescharacteristic
251C22#179212yes—
252C22#179211yescharacteristic
253C4—21no—
254C4—21no—
255C4—22no—
256C4—22no—
257C4—41no—
258C4—84no—
259C4—84no—
260C4—82no—
261C4#162011yescharacteristic
262C4#162212yes—
263C4#163011yescharacteristic
264C4#163212yes—
265C4#164012yes—
266C4#164312yes—
267C4#164412yes—
268C2—84no—
269C2#835511yescharacteristic
270C2#835812yes—
271C2#844511yescharacteristic
272C2#845012yes—
273C2#853011yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127D4
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. #9637C2semidirect
4. #8587C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#96382
#96372
#85872
#85852