Finite Group Databasebeta

← #100763

Subgroups — #100763

Order 512.

436 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 88 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#28520C211yesmaximal, characteristic
2#23353C211yesmaximal, characteristic
3#9703C212yesmaximal, complemented
4#8579C211yesmaximal, characteristic
5#7049C212yesmaximal, complemented
6C2×#475—21no—
7C2×#474—21no—
8#2868C2211yescharacteristic
9C4×#124C2211yescharacteristic
10#2095C412yes—
11#2094C412yes—
12#1985—21no—
13#1959—21no—
14#1924—22no—
15#1870—21no—
16#1851—22no—
17#1847—21no—
18#1831C2212yes—
19#1767C2212yes—
20#1767C2211yescharacteristic
21C2×#332—22no—
22C2×#332—22no—
23C2×#329—21no—
24#1295—21no—
25C22×M16—21no—
26C2×C4⋊Q8—21no—
27C2×C4⋊Q8—22no—
28C2×#124D411yescharacteristic
29C2×C4×Q8D411yescharacteristic
30C2×C4×D4D411yescharacteristic
31#475—42no—
32#475—42no—
33#474—42no—
34#474—42no—
35C2×#107—22no—
36C2×#106—22no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8D412yes—
41C2×C4⋊C8D412yes—
42C2×#103—22no—
43C2×#102—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—21no—
46C2×#97—21no—
47C2×#97—22no—
48C2×#97—21no—
49#391C2×C411yescharacteristic
50#390C2×C411yescharacteristic
51#388—22no—
52#384—21no—
53#384—22no—
54#384C2×C412yes—
55#382D411yescharacteristic
56C4×C22⋊C4—22no—
57C43C2311yescharacteristic
58#336—22no—
59#336—22no—
60#336—22no—
61#336—22no—
62#336C2×C412yes—
63#332—44no—
64#332—44no—
65#329—42no—
66C22×Q8—41no—
67C22×Q8C22⋊C411yescharacteristic
68C22×D4C22⋊C411yescharacteristic
69C23×C4—21no—
70C2×M16—82no—
71C2×M16—81no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—42no—
76C4⋊Q8—42no—
77C4⋊Q8—42no—
78C4⋊Q8—42no—
79C4⋊Q8—42no—
80C4⋊Q8—41no—
81C4⋊Q8—41no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84#124—21no—
85#124—21no—
86#124—22no—
87C4×Q8—41no—
88C4×Q8—41no—
89C4×D4—41no—
90C4×D4—41no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4C22⋊C411yescharacteristic
97C2×C4⋊C4C22⋊C411yescharacteristic
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—21no—
101C2×C42—21no—
102C2×C42C22×C411yescharacteristic
103C2×C42C2×D411yescharacteristic
104C2×C42C2×D412yes—
105C2×C42C2×D411yescharacteristic
106C2×C42#4112yes—
107#107—44no—
108#106—44no—
109C4⋊C8—22no—
110C4⋊C8—22no—
111C4⋊C8—24no—
112C4⋊C8—42no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115C4⋊C8—81no—
116#103—44no—
117#102—44no—
118C22⋊C8—81no—
119C22⋊C8—82no—
120#97—41no—
121#97—42no—
122#97—41no—
123#97—42no—
124#97—81no—
125#95—22no—
126#95—22no—
127C24—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—82no—
133C2×Q8—81no—
134C2×Q8—81no—
135C2×Q8—82no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—41no—
139C2×D4—41no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—41no—
144C22×C4—42no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4C4×D412yes—
150C22×C4#12011yescharacteristic
151C22×C4#12112yes—
152C22×C4#12311yescharacteristic
153M16—84no—
154C2×C8—24no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—44no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—82no—
165C2×C8—82no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—81no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—82no—
180C22⋊C4—81no—
181C42—22no—
182C42—21no—
183C42—21no—
184C42—22no—
185C42—41no—
186C42—42no—
187C42—41no—
188C42—41no—
189C42AGL(1,ℤ/8)11yescharacteristic
190C42#13711yescharacteristic
191C42#13712yes—
192C23—42no—
193C23—41no—
194C23—41no—
195C23—42no—
196C23—41no—
197C23#38611yescharacteristic
198Q8—42no—
199Q8—81no—
200Q8—161no—
201Q8—162no—
202Q8—161no—
203D4—42no—
204D4—81no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—81no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4—84no—
225C2×C4—82no—
226C2×C4#41711yescharacteristic
227C2×C4#41911yescharacteristic
228C2×C4#45312yes—
229C2×C4#45311yescharacteristic
230C2×C4#45311yescharacteristic
231C2×C4#45412yes—
232C2×C4#46812yes—
233C2×C4#47012yes—
234C8—44no—
235C8—84no—
236C8—82no—
237C8—82no—
238C22—81no—
239C22—82no—
240C22—82no—
241C22—81no—
242C22—81no—
243C22#188312yes—
244C22#188711yescharacteristic
245C22#189411yescharacteristic
246C22#190212yes—
247C22#191511yescharacteristic
248C4—22no—
249C4—22no—
250C4—21no—
251C4—21no—
252C4—42no—
253C4—41no—
254C4—81no—
255C4—81no—
256C4—81no—
257C4—162no—
258C4#165511yescharacteristic
259C4#165612yes—
260C4#165711yescharacteristic
261C2—161no—
262C2#900111yescharacteristic
263C2#932112yes—
264C2#933912yes—
265C2#940311yescharacteristic
266C2#942111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊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. #9703C2semidirect
5. #7049C2semidirect

Maximal subgroups

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

subgroupindex
#285202
#233532
#97032
#85792
#70492