Finite Group Databasebeta

← #138510

Subgroups — #138510

Order 512.

376 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 84 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#29808C211yesmaximal, characteristic
2#17818C211yesmaximal, characteristic
3#11965C212yesmaximal, complemented
4#11460C211yesmaximal, characteristic
5#7222C212yesmaximal, complemented
6C2×#494C2211yescharacteristic
7C2×#432—21no—
8C2×#431C2211yescharacteristic
9#2857—21no—
10#2163C2212yes—
11#2056—21no—
12#2056C412yes—
13#2001—21no—
14#2001C412yes—
15#1956—21no—
16#1834C2211yescharacteristic
17#1573—24no—
18#1568—21no—
19#1568C2212yes—
20#1362—24no—
21#1322—21no—
22#1314—21no—
23C22×Q16—21no—
24C22×Q16C2×C412yes—
25C2×C4⋊Q8—21no—
26C2×C22⋊Q8—22no—
27C22×C42D411yescharacteristic
28#494—22no—
29#494—22no—
30#432—22no—
31#432—22no—
32#432—41no—
33#432—41no—
34#431—22no—
35#431—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8C2×C411yescharacteristic
38C2×C4⋊C8D411yescharacteristic
39C2×#103—22no—
40C2×#103—22no—
41C2×C22⋊C8—41no—
42C2×C22⋊C8C2×C411yescharacteristic
43C2×C22⋊C8D411yescharacteristic
44C2×#97—41no—
45C2×C4×C8D411yescharacteristic
46C2×C4×C8C2311yescharacteristic
47#391—22no—
48#389—41no—
49#389—41no—
50#384—21no—
51#380—41no—
52#380—41no—
53C4⋊C16—22no—
54C22⋊C16—22no—
55#346—22no—
56#340—21no—
57#340C2×C412yes—
58#335—22no—
59#335D414yes—
60#334—22no—
61#334—22no—
62C22×Q8—22no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4D811yescharacteristic
65C23×C4SD1611yescharacteristic
66C2×Q16—24no—
67C2×Q16—42no—
68C2×Q16—82no—
69C2×Q16—82no—
70C22×C8—42no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22×C411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C4⋊Q8—41no—
76C4⋊Q8—41no—
77C4⋊Q8—41no—
78C4⋊Q8—41no—
79C22⋊Q8—82no—
80C22⋊Q8—82no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C22⋊C4—42no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42C22⋊C411yescharacteristic
93C2×C42D811yescharacteristic
94C2×C42SD1611yescharacteristic
95C2×C42C2×D411yescharacteristic
96C2×C16—42no—
97C4⋊C8—22no—
98C4⋊C8—22no—
99C4⋊C8—24no—
100C4⋊C8—42no—
101#103—82no—
102#103—82no—
103C22⋊C8—22no—
104C22⋊C8—22no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107#97—42no—
108#97—42no—
109C4×C8—21no—
110C4×C8—21no—
111C4×C8C2×D412yes—
112C4×C8#4112yes—
113#95—42no—
114#95—42no—
115C24#10211yescharacteristic
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—42no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4#10211yescharacteristic
136C22×C4#10211yescharacteristic
137C22×C4#10211yescharacteristic
138C22×C4C2×C22⋊C411yescharacteristic
139C22×C4C2×D811yescharacteristic
140C22×C4C2×SD1611yescharacteristic
141Q16—84no—
142Q16—84no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—44no—
150C2×C8—84no—
151C2×C8C4×D412yes—
152C2×C8#12112yes—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—82no—
156C4⋊C4—82no—
157C4⋊C4—84no—
158C4⋊C4—81no—
159C4⋊C4—81no—
160C4⋊C4—81no—
161C4⋊C4—81no—
162C22⋊C4—84no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—21no—
169C42#12011yescharacteristic
170C42#12311yescharacteristic
171C42AGL(1,ℤ/8)11yescharacteristic
172C42#13711yescharacteristic
173C16—82no—
174C23—21no—
175C23—21no—
176C23—41no—
177C23—41no—
178C23#36212yes—
179C23C2×#10211yescharacteristic
180Q8—82no—
181Q8—82no—
182Q8—82no—
183Q8—82no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4#38611yescharacteristic
210C2×C4#41711yescharacteristic
211C2×C4#44711yescharacteristic
212C2×C4#44811yescharacteristic
213C2×C4#48011yescharacteristic
214C2×C4#48111yescharacteristic
215C8—22no—
216C8—22no—
217C8—41no—
218C8—41no—
219C8—84no—
220C22—21no—
221C22—21no—
222C22—22no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22—81no—
227C22#187411yescharacteristic
228C22C2×#36211yescharacteristic
229C22#214811yescharacteristic
230C4—21no—
231C4—21no—
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—162no—
238C4—162no—
239C4#191011yescharacteristic
240C4#191411yescharacteristic
241C4#219112yes—
242C2—21no—
243C2—21no—
244C2—42no—
245C2#919711yescharacteristic
246C2#1300011yescharacteristic
247C2#1300311yescharacteristic
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. #11965C2semidirect
5. #7222C2semidirect

Maximal subgroups

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

subgroupindex
#298082
#178182
#119652
#114602
#72222