Finite Group Databasebeta

← #499198

Subgroups — #499198

Order 512.

800 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60660C211yesmaximal, characteristic
2C2×#2159C212yesmaximal
3#21156C211yesmaximal, characteristic
4#13454C218yesmaximal, complemented
5#12567C212yesmaximal
6#12551C211yesmaximal, characteristic
7#3458—21no—
8C22×M32C2212yes—
9#3164—22no—
10#3164—22no—
11#3163—24no—
12#3162C2214yes—
13#3160—22no—
14#2911—21no—
15C2×#427C2212yes—
16C2×#427C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C2×#405C2211yescharacteristic
19#2317—21no—
20#2159—24no—
21#2159C2218yes—
22C2×#367—22no—
23C2×#366—22no—
24#1944C2214yes—
25#1942—22no—
26#1736—21no—
27#1386C2218yes—
28#1377C2214yes—
29C22×M16D411yescharacteristic
30C23×C8C2311yescharacteristic
31#544—24no—
32#544—24no—
33#544—41no—
34#544—41no—
35#543—42no—
36#533—24no—
37#533—44no—
38C2×#117—21no—
39C2×#117—22no—
40C2×#117D411yescharacteristic
41C22×C4⋊C4—21no—
42C2×M32—22no—
43C2×M32—24no—
44C2×M32Q818yescomplemented
45C2×M32C2314yes—
46C22×C16—22no—
47#501—28no—
48#501—28no—
49#500—22no—
50#500—22no—
51#499—24no—
52#499—24no—
53#498—24no—
54#428—24no—
55#428—24no—
56#427—22no—
57#427—24no—
58#427—22no—
59#427—22no—
60#427D414yes—
61#427Q814yescomplemented
62#427C2314yes—
63C2×#107—21no—
64C2×#107—22no—
65C2×#107—22no—
66C2×#107—21no—
67C2×#107D412yes—
68C2×#107D412yes—
69C2×#107C2312yes—
70C2×#106—22no—
71C2×#106C2312yes—
72#405—22no—
73#405—22no—
74C2×#97C2312yes—
75C2×C4×C8D412yes—
76#389—24no—
77#385—42no—
78#384—41no—
79#382—41no—
80#367—28no—
81#366—28no—
82#340—24no—
83C23×C4—41no—
84C23×C4C2×D411yescharacteristic
85C2×M16—22no—
86C2×M16—22no—
87C2×M16Q1614yes—
88C22×C8C2×D412yes—
89C22×C8C2×D411yescharacteristic
90C22×C8C2×D412yes—
91C22×C8C2×D412yes—
92C22×C8C2×Q814yes—
93C22×C8C2×Q812yes—
94C22×C8#4112yes—
95C22×C8C2411yescharacteristic
96C4⋊Q8—24no—
97C4⋊Q8—24no—
98C4⋊Q8—44no—
99C4⋊Q8—44no—
100#125—28no—
101#125—44no—
102#125—44no—
103C22⋊Q8—48no—
104C22⋊Q8—48no—
105C22⋊Q8—44no—
106C22⋊Q8—44no—
107C4×Q8—44no—
108C4×Q8—44no—
109#117—22no—
110#117—22no—
111#117—24no—
112#117—24no—
113#117—24no—
114#117—42no—
115#117—42no—
116#117—42no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4D811yescharacteristic
126C2×C4⋊C4D811yescharacteristic
127C2×C22⋊C4—42no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4D812yes—
130C2×C42—41no—
131C2×C42—42no—
132C2×C42C2×D412yes—
133M32—28no—
134M32—24no—
135M32C2×Q818yes—
136C2×C16—28no—
137C2×C16—24no—
138#107—24no—
139#107—24no—
140#107—22no—
141#107—22no—
142#107—24no—
143#107—24no—
144#107—41no—
145#107—41no—
146#107C2×D414yes—
147#107C2×D414yes—
148#106—28no—
149#106—24no—
150#106—24no—
151#106—22no—
152#106C2×Q814yes—
153#97—24no—
154C4×C8—22no—
155C4×C8—22no—
156#95—44no—
157C24C2×D811yescharacteristic
158C2×Q8—44no—
159C2×Q8—44no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—82no—
165C22×C4—162no—
166C22×C4#12012yes—
167C22×C4C22⋊Q812yes—
168C22×C4C4⋊Q812yes—
169C22×C4C2×D812yes—
170C22×C4C2×D812yes—
171C22×C4C2×D811yescharacteristic
172C22×C4C2×Q1612yes—
173C22×C4AGL(1,ℤ/8)12yes—
174C22×C4#13712yes—
175C22×C4C22×D411yescharacteristic
176M16—28no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8#12011yescharacteristic
180C2×C8#12011yescharacteristic
181C2×C8C22⋊Q814yes—
182C2×C8C22⋊Q814yes—
183C2×C8C22⋊Q812yes—
184C2×C8C4⋊Q814yes—
185C2×C8C4⋊Q814yes—
186C2×C8C4⋊Q812yes—
187C2×C8C2×Q1614yes—
188C2×C8C22×D412yes—
189C2×C8C22×Q812yes—
190C2×C8C22×Q811yescharacteristic
191C2×C8C2×#4111yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—48no—
198C4⋊C4—48no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—82no—
207C4⋊C4—164no—
208C22⋊C4—24no—
209C22⋊C4—44no—
210C22⋊C4—44no—
211C22⋊C4—44no—
212C42—22no—
213C42—22no—
214C42—44no—
215C42—44no—
216C42—44no—
217C16—28no—
218C23—41no—
219C23—41no—
220C23#44712yes—
221C23#47412yes—
222C23#50012yes—
223C23C22×D811yescharacteristic
224Q8—164no—
225Q8—164no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—84no—
236C2×C4—82no—
237C2×C4—162no—
238C2×C4—162no—
239C2×C4—164no—
240C2×C4#39514yes—
241C2×C4#44712yes—
242C2×C4#45114yes—
243C2×C4#47412yes—
244C2×C4#47714yes—
245C2×C4#50012yes—
246C2×C4#50114yes—
247C2×C4C2×#12011yescharacteristic
248C2×C4C2×C22⋊Q811yescharacteristic
249C2×C4C2×C4⋊Q811yescharacteristic
250C2×C4C22×Q1611yescharacteristic
251C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
252C2×C4C2×#13711yescharacteristic
253C8—24no—
254C8#39512yes—
255C8#39512yes—
256C8C2×C22⋊Q812yes—
257C8C2×C4⋊Q812yes—
258C22—21no—
259C22—22no—
260C22—21no—
261C22—22no—
262C22—42no—
263C22—42no—
264C22#205714yes—
265C22C2×#44711yescharacteristic
266C22C2×#47411yescharacteristic
267C22C2×#50011yescharacteristic
268C4—42no—
269C4—42no—
270C4—84no—
271C4—84no—
272C4—84no—
273C4—162no—
274C4#205714yes—
275C4C2×#39511yescharacteristic
276C4C2×#45111yescharacteristic
277C4C2×#47711yescharacteristic
278C4C2×#50111yescharacteristic
279C2—21no—
280C2—24no—
281C2—21no—
282C2C2×#205711yescharacteristic
283C2#3353111yescharacteristic
284C2#3353211yescharacteristic
subgroupG/H
center, socleC22C2×#447
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13454C2semidirect
44. C2×M32Q8semidirect
61. #427Q8semidirect

Maximal subgroups

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

subgroupindex
#606602
C2×#21592
#211562
#134542
#125672
#125512