Finite Group Databasebeta

← #125498

Subgroups — #125498

Order 512.

299 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 72 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#33133C211yesmaximal, characteristic
2#12786C211yesmaximal, characteristic
3#8601C211yesmaximal, characteristic
4#7126C212yesmaximal, complemented
5#7111C212yesmaximal, complemented
6C2×#467C411yescharacteristic
7C2×#405C2211yescharacteristic
8#2625—21no—
9#2150—21no—
10C2×#349—21no—
11#1971C411yescharacteristic
12#1971C2211yescharacteristic
13#1944C411yescharacteristic
14#1944C2211yescharacteristic
15#1933—21no—
16#1933—21no—
17#1894—21no—
18#1807C411yescharacteristic
19#1564—21no—
20#1458—21no—
21#1377C2212yes—
22#1275C2212yes—
23C22×Q16C2×C411yescharacteristic
24C22×M16D411yescharacteristic
25C23×C8C2×C411yescharacteristic
26C2×C22⋊Q8—21no—
27C2×#117D411yescharacteristic
28C2×M32—22no—
29#467—41no—
30#467—41no—
31#431—22no—
32C2×#107C2×C411yescharacteristic
33C2×#106C2×C411yescharacteristic
34C2×#106C2×C411yescharacteristic
35C2×#103—21no—
36C2×#103—21no—
37C2×#103—21no—
38#405—41no—
39#405—41no—
40C2×#97C2×C411yescharacteristic
41C2×#97C2311yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#400—41no—
45#398—41no—
46#397—41no—
47#397—41no—
48#391—21no—
49#391—21no—
50#389—21no—
51#389—21no—
52#380—21no—
53C4⋊C16—41no—
54C4⋊C16—41no—
55#349—42no—
56#340—21no—
57#340—21no—
58#340—21no—
59#335—41no—
60#334—41no—
61#324—41no—
62#323—41no—
63C22×Q8—21no—
64C23×C4C22⋊C411yescharacteristic
65C2×Q16—22no—
66C2×Q16—41no—
67C2×Q16—41no—
68C2×M16—21no—
69C2×M16—21no—
70C2×M16—41no—
71C2×M16—41no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—22no—
75C22×C8—41no—
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22×C411yescharacteristic
79C22×C8C2×D411yescharacteristic
80C22⋊Q8—42no—
81C22⋊Q8—42no—
82#117—41no—
83#117—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4D811yescharacteristic
91C2×C4⋊C4SD1611yescharacteristic
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4D811yescharacteristic
95C2×C22⋊C4SD1611yescharacteristic
96C2×C42C22⋊C411yescharacteristic
97C2×C42C2×D411yescharacteristic
98M32—42no—
99M32—42no—
100C2×C16—42no—
101#107—41no—
102#106—22no—
103#106—41no—
104C4⋊C8—42no—
105#103—42no—
106#103—41no—
107#103—41no—
108#103—41no—
109#103—41no—
110C22⋊C8—42no—
111#97—41no—
112#97—41no—
113C4×C8—21no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8—42no—
118#95—41no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—41no—
123#95—41no—
124#95—41no—
125C24#10211yescharacteristic
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4#10211yescharacteristic
143C22×C4#10211yescharacteristic
144C22×C4#10211yescharacteristic
145C22×C4C2×C22⋊C411yescharacteristic
146C22×C4C2×D811yescharacteristic
147C22×C4C2×SD1611yescharacteristic
148Q16—82no—
149M16—42no—
150M16—42no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—82no—
163C2×C8#9911yescharacteristic
164C2×C8#9911yescharacteristic
165C2×C8#13112yes—
166C2×C8AGL(1,ℤ/8)11yescharacteristic
167C2×C8#13711yescharacteristic
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—82no—
177C4⋊C4—161no—
178C4⋊C4—161no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—82no—
182C22⋊C4—161no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C16—82no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—41no—
192C23—41no—
193C23C2×#10211yescharacteristic
194Q8—81no—
195Q8—81no—
196Q8—81no—
197Q8—81no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4#40811yescharacteristic
226C2×C4C2×#9911yescharacteristic
227C2×C4#41311yescharacteristic
228C2×C4#41711yescharacteristic
229C2×C4#42111yescharacteristic
230C2×C4#42111yescharacteristic
231C8—21no—
232C8—21no—
233C8—22no—
234C8—21no—
235C8—21no—
236C8—41no—
237C8—162no—
238C22—21no—
239C22—21no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—81no—
244C22—81no—
245C22#148011yescharacteristic
246C22#151911yescharacteristic
247C22#154711yescharacteristic
248C4—41no—
249C4—41no—
250C4—41no—
251C4—41no—
252C4—41no—
253C4—161no—
254C4—161no—
255C4—161no—
256C4—161no—
257C4#146811yescharacteristic
258C4#149511yescharacteristic
259C4#151311yescharacteristic
260C4#153511yescharacteristic
261C2—21no—
262C2—21no—
263C2—81no—
264C2#771411yescharacteristic
265C2#1186911yescharacteristic
266C2#1187211yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC2×#97C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7126C2semidirect
5. #7111C2semidirect

Maximal subgroups

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

subgroupindex
#331332
#127862
#86012
#71262
#71112