Finite Group Databasebeta

← #138487

Subgroups — #138487

Order 512.

354 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 90 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#29878C211yesmaximal, characteristic
2#17815C211yesmaximal, characteristic
3#11972C212yesmaximal, complemented
4#11459C211yesmaximal, characteristic
5#7217C212yesmaximal, complemented
6C2×#500C2211yescharacteristic
7#2920—21no—
8C2×#431C2211yescharacteristic
9#2857—21no—
10#2162C2212yes—
11#2087—21no—
12#2087C412yes—
13#2076—21no—
14#2076C412yes—
15#1990—21no—
16#1834C2211yescharacteristic
17#1572—24no—
18#1567C2212yes—
19#1566—21no—
20#1365—24no—
21#1322—21no—
22#1314—21no—
23C2×C4⋊Q8—21no—
24C2×C22⋊Q8—41no—
25C22×C42D411yescharacteristic
26#500—22no—
27#500—22no—
28#432—22no—
29#432—41no—
30#431—22no—
31#431—22no—
32#431—22no—
33#431—41no—
34C2×#107—21no—
35C2×#107C2×C412yes—
36#423—22no—
37C2×C4⋊C8C2×C411yescharacteristic
38C2×C4⋊C8D411yescharacteristic
39#407—41no—
40C2×C22⋊C8C2×C411yescharacteristic
41C2×C22⋊C8D411yescharacteristic
42C4×M16—41no—
43C2×C4×C8D411yescharacteristic
44C2×C4×C8C2311yescharacteristic
45#391—41no—
46#389—22no—
47#389—41no—
48#384—21no—
49#380—22no—
50#380—41no—
51C4⋊C16—22no—
52C22⋊C16—22no—
53C4×C16—22no—
54#340—22no—
55#340—22no—
56#340—21no—
57#340C2×C412yes—
58#335—22no—
59#335D414yes—
60#334—24no—
61C22×Q8—41no—
62C23×C4C22⋊C411yescharacteristic
63C23×C4D811yescharacteristic
64C23×C4SD1611yescharacteristic
65C2×M16—42no—
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22×C411yescharacteristic
69C22×C8C2×D411yescharacteristic
70C4⋊Q8—41no—
71C4⋊Q8—41no—
72C4⋊Q8—41no—
73C4⋊Q8—41no—
74C22⋊Q8—84no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C22⋊C4—42no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42C22⋊C411yescharacteristic
87C2×C42D811yescharacteristic
88C2×C42SD1611yescharacteristic
89C2×C42C2×D411yescharacteristic
90C2×C16—42no—
91#107—24no—
92#107—42no—
93C4⋊C8—22no—
94C4⋊C8—22no—
95C4⋊C8—24no—
96C4⋊C8—42no—
97C22⋊C8—22no—
98C22⋊C8—22no—
99C22⋊C8—42no—
100C22⋊C8—42no—
101#97—42no—
102C4×C8—21no—
103C4×C8—21no—
104C4×C8—42no—
105C4×C8C2×D412yes—
106C4×C8#4112yes—
107#95—42no—
108#95—42no—
109C24#10211yescharacteristic
110C2×Q8—82no—
111C2×Q8—82no—
112C2×Q8—81no—
113C2×Q8—81no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—42no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4#10211yescharacteristic
128C22×C4#10211yescharacteristic
129C22×C4#10211yescharacteristic
130C22×C4C2×C22⋊C411yescharacteristic
131C22×C4C2×D811yescharacteristic
132C22×C4C2×SD1611yescharacteristic
133M16—82no—
134M16—82no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—44no—
142C2×C8C4×D412yes—
143C2×C8#12112yes—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—82no—
149C4⋊C4—84no—
150C4⋊C4—81no—
151C4⋊C4—81no—
152C4⋊C4—82no—
153C22⋊C4—84no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42#12011yescharacteristic
161C42#12311yescharacteristic
162C42AGL(1,ℤ/8)11yescharacteristic
163C42#13711yescharacteristic
164C16—82no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—41no—
169C23—41no—
170C23C2×#10211yescharacteristic
171Q8—162no—
172Q8—162no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4#38611yescharacteristic
197C2×C4#41711yescharacteristic
198C2×C4#44711yescharacteristic
199C2×C4#44811yescharacteristic
200C2×C4#48011yescharacteristic
201C2×C4#48111yescharacteristic
202C2×C4#50812yes—
203C2×C4#51012yes—
204C8—22no—
205C8—22no—
206C8—41no—
207C8—41no—
208C8—84no—
209C22—21no—
210C22—21no—
211C22—42no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—81no—
216C22#187411yescharacteristic
217C22#213811yescharacteristic
218C22#214111yescharacteristic
219C4—21no—
220C4—21no—
221C4—41no—
222C4—41no—
223C4—41no—
224C4—162no—
225C4—162no—
226C4#190911yescharacteristic
227C4#191511yescharacteristic
228C4#218912yes—
229C4#223712yes—
230C4#223812yes—
231C2—21no—
232C2—21no—
233C2—81no—
234C2#919511yescharacteristic
235C2#1299411yescharacteristic
236C2#1301011yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11972C2semidirect
5. #7217C2semidirect

Maximal subgroups

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

subgroupindex
#298782
#178152
#119722
#114592
#72172