Finite Group Databasebeta

← #100767

Subgroups — #100767

Order 512.

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

normal subgroups only

This group has 420 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#28519C211yesmaximal, characteristic
2#23353C211yesmaximal, characteristic
3#9903C212yesmaximal, complemented
4#8579C211yesmaximal, characteristic
5#7049C212yesmaximal, complemented
6C2×#477—21no—
7C2×#476—21no—
8#2868C2211yescharacteristic
9C4×#124C2211yescharacteristic
10#2096C412yes—
11#2093C412yes—
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×#329—21no—
22#1296—22no—
23#1296—22no—
24#1295—21no—
25C22×M16—21no—
26C2×C4⋊Q8—21no—
27C2×C4⋊Q8—22no—
28C2×#124D411yescharacteristic
29C2×C4×Q8D411yescharacteristic
30C2×C4×D4D411yescharacteristic
31#477—42no—
32#477—42no—
33#476—42no—
34#476—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—22no—
53#384—21no—
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#329—42no—
64C22×Q8—41no—
65C22×Q8C22⋊C411yescharacteristic
66C22×D4C22⋊C411yescharacteristic
67C23×C4—21no—
68C2×M16—81no—
69C2×M16—82no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—42no—
74C4⋊Q8—41no—
75C4⋊Q8—41no—
76C4⋊Q8—41no—
77C4⋊Q8—41no—
78C4⋊Q8—84no—
79#124—21no—
80#124—21no—
81#124—22no—
82C4×Q8—41no—
83C4×Q8—41no—
84C4×D4—41no—
85C4×D4—41no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4C22⋊C411yescharacteristic
92C2×C4⋊C4C22⋊C411yescharacteristic
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—21no—
96C2×C42—21no—
97C2×C42C22×C411yescharacteristic
98C2×C42C2×D411yescharacteristic
99C2×C42C2×D412yes—
100C2×C42C2×D411yescharacteristic
101C2×C42#4112yes—
102#107—44no—
103#106—44no—
104C4⋊C8—22no—
105C4⋊C8—22no—
106C4⋊C8—24no—
107C4⋊C8—42no—
108C4⋊C8—42no—
109C4⋊C8—42no—
110C4⋊C8—81no—
111#103—44no—
112#102—44no—
113C22⋊C8—81no—
114C22⋊C8—82no—
115#97—41no—
116#97—42no—
117#97—41no—
118#97—42no—
119#97—81no—
120#95—22no—
121#95—22no—
122C24—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—41no—
126C2×Q8—41no—
127C2×Q8—81no—
128C2×Q8—81no—
129C2×Q8—84no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—41no—
133C2×D4—41no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4C2×C22⋊C411yescharacteristic
143C22×C4C4×D412yes—
144C22×C4#12011yescharacteristic
145C22×C4#12112yes—
146C22×C4#12311yescharacteristic
147M16—84no—
148C2×C8—24no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—44no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—82no—
159C2×C8—82no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—81no—
165C4⋊C4—84no—
166C4⋊C4—84no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C22⋊C4—82no—
172C22⋊C4—81no—
173C42—21no—
174C42—21no—
175C42—22no—
176C42—22no—
177C42—41no—
178C42—42no—
179C42—41no—
180C42—41no—
181C42AGL(1,ℤ/8)11yescharacteristic
182C42#13711yescharacteristic
183C42#13712yes—
184C23—42no—
185C23—41no—
186C23—41no—
187C23—42no—
188C23—41no—
189C23#38611yescharacteristic
190Q8—42no—
191Q8—81no—
192Q8—162no—
193Q8—162no—
194D4—42no—
195D4—81no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—84no—
217C2×C4#41711yescharacteristic
218C2×C4#41911yescharacteristic
219C2×C4#45311yescharacteristic
220C2×C4#45311yescharacteristic
221C2×C4#45512yes—
222C2×C4#45612yes—
223C2×C4#46812yes—
224C2×C4#47012yes—
225C8—44no—
226C8—84no—
227C8—82no—
228C8—82no—
229C22—81no—
230C22—82no—
231C22—82no—
232C22—81no—
233C22—81no—
234C22#188212yes—
235C22#188711yescharacteristic
236C22#189411yescharacteristic
237C22#190312yes—
238C22#191511yescharacteristic
239C4—22no—
240C4—22no—
241C4—21no—
242C4—21no—
243C4—42no—
244C4—41no—
245C4—81no—
246C4—81no—
247C4—81no—
248C4—162no—
249C4#165911yescharacteristic
250C4#166011yescharacteristic
251C4#166112yes—
252C2—161no—
253C2#900311yescharacteristic
254C2#931912yes—
255C2#933812yes—
256C2#940311yescharacteristic
257C2#942311yescharacteristic
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. #9903C2semidirect
5. #7049C2semidirect

Maximal subgroups

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

subgroupindex
#285192
#233532
#99032
#85792
#70492