Finite Group Databasebeta

← #138501

Subgroups — #138501

Order 512.

401 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 98 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#17814C211yesmaximal, characteristic
3#11962C212yesmaximal, complemented
4#11459C211yesmaximal, characteristic
5#7217C212yesmaximal, complemented
6C2×#494C2211yescharacteristic
7C2×#431C2211yescharacteristic
8C2×#423—21no—
9#2857—21no—
10#2162C2212yes—
11#2056—21no—
12#2056C412yes—
13#2001—21no—
14#2001C412yes—
15#1956—21no—
16#1834C2211yescharacteristic
17#1571—24no—
18#1567—21no—
19#1567C2212yes—
20#1362—24no—
21#1322—21no—
22#1314—21no—
23C22×Q16—21no—
24C22×Q16C2×C412yes—
25C22×M16—41no—
26C2×C4⋊Q8—21no—
27C2×C22⋊Q8—22no—
28C22×C42D411yescharacteristic
29#494—22no—
30#494—22no—
31#431—22no—
32#431—22no—
33#423—22no—
34#423—22no—
35#423—41no—
36#423—41no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—41no—
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8D411yescharacteristic
41C2×#103—22no—
42C2×#103—22no—
43C2×C22⋊C8C2×C411yescharacteristic
44C2×C22⋊C8D411yescharacteristic
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—
55C4×C16—22no—
56#340—21no—
57#340C2×C412yes—
58#335—24no—
59#335—22no—
60#335D414yes—
61C22×Q8—22no—
62C23×C4C22⋊C411yescharacteristic
63C23×C4D811yescharacteristic
64C23×C4SD1611yescharacteristic
65C2×Q16—24no—
66C2×Q16—42no—
67C2×Q16—82no—
68C2×Q16—82no—
69C2×M16—42no—
70C2×M16—42no—
71C2×M16—82no—
72C2×M16—82no—
73C22×C8—42no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22×C411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C4⋊Q8—41no—
79C4⋊Q8—41no—
80C4⋊Q8—41no—
81C4⋊Q8—41no—
82C22⋊Q8—82no—
83C22⋊Q8—82no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C22⋊C4—42no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42D811yescharacteristic
97C2×C42SD1611yescharacteristic
98C2×C42C2×D411yescharacteristic
99C2×C16—42no—
100C4⋊C8—22no—
101C4⋊C8—22no—
102C4⋊C8—24no—
103C4⋊C8—42no—
104C4⋊C8—42no—
105C4⋊C8—42no—
106#103—82no—
107#103—82no—
108C22⋊C8—22no—
109C22⋊C8—22no—
110C4×C8—21no—
111C4×C8—21no—
112C4×C8C2×D412yes—
113C4×C8#4112yes—
114#95—42no—
115#95—42no—
116C24#10211yescharacteristic
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—42no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4#10211yescharacteristic
137C22×C4#10211yescharacteristic
138C22×C4#10211yescharacteristic
139C22×C4SD3212yes—
140C22×C4Q3212yes—
141C22×C4C2×C22⋊C411yescharacteristic
142C22×C4C2×D811yescharacteristic
143C22×C4C2×SD1611yescharacteristic
144Q16—84no—
145Q16—84no—
146M16—84no—
147M16—82no—
148M16—82no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—44no—
156C2×C8—84no—
157C2×C8C4×D412yes—
158C2×C8#12112yes—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—82no—
162C4⋊C4—84no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C22⋊C4—84no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42#12011yescharacteristic
177C42#12311yescharacteristic
178C42AGL(1,ℤ/8)11yescharacteristic
179C42#13711yescharacteristic
180C16—82no—
181C23—21no—
182C23—21no—
183C23—41no—
184C23—41no—
185C23#35812yes—
186C23C2×#10211yescharacteristic
187Q8—82no—
188Q8—82no—
189Q8—82no—
190Q8—82no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4#35812yes—
216C2×C4#38611yescharacteristic
217C2×C4#41711yescharacteristic
218C2×C4#44711yescharacteristic
219C2×C4#44811yescharacteristic
220C2×C4#48011yescharacteristic
221C2×C4#48111yescharacteristic
222C2×C4C2×SD3211yescharacteristic
223C2×C4C2×Q3211yescharacteristic
224C2×C4#50911yescharacteristic
225C2×C4#51011yescharacteristic
226C8—22no—
227C8—22no—
228C8—41no—
229C8—41no—
230C8—84no—
231C22—21no—
232C22—21no—
233C22—22no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22#187411yescharacteristic
239C22C2×#35811yescharacteristic
240C22#213911yescharacteristic
241C4—22no—
242C4—21no—
243C4—21no—
244C4—41no—
245C4—41no—
246C4—162no—
247C4—162no—
248C4#191011yescharacteristic
249C4#191411yescharacteristic
250C4#218711yescharacteristic
251C4#218811yescharacteristic
252C4#223511yescharacteristic
253C4#223611yescharacteristic
254C4#223912yes—
255C2—21no—
256C2—21no—
257C2—42no—
258C2#919711yescharacteristic
259C2#1299311yescharacteristic
260C2#1300911yescharacteristic
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. #11962C2semidirect
5. #7217C2semidirect

Maximal subgroups

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

subgroupindex
#298082
#178142
#119622
#114592
#72172