Finite Group Databasebeta

← #138477

Subgroups — #138477

Order 512.

356 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 86 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#22738C211yesmaximal, characteristic
2#19265C211yesmaximal, characteristic
3#11857C211yesmaximal, characteristic
4#11745C211yesmaximal, characteristic
5#11639C211yesmaximal, characteristic
6#7299C211yesmaximal, characteristic
7#7222C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9#3113—21no—
10C2×C8×Q8C2211yescharacteristic
11Q8×C4⋊C4—21no—
12#2163C2211yescharacteristic
13#2162C2211yescharacteristic
14#1948—21no—
15#1948—22no—
16#1834—21no—
17#1834C2211yescharacteristic
18C4×#106—21no—
19C4×#106C2211yescharacteristic
20#1680—22no—
21#1674C412yescomplemented
22#1670C412yescomplemented
23#1568C2211yescharacteristic
24#1552—21no—
25#1348—22no—
26#1348—21no—
27#1348—21no—
28Q8⋊C16—22no—
29#533—21no—
30C2×C4×Q8—41no—
31C2×C4×Q8D411yescharacteristic
32#500C2×C411yescharacteristic
33#494C2×C411yescharacteristic
34#487—21no—
35#483—41no—
36#475—22no—
37#467—21no—
38#467—21no—
39#462—22no—
40#451—41no—
41C8×Q8—21no—
42C8×Q8—21no—
43C8×Q8C2×C412yes—
44C8×Q8D412yes—
45#441—42no—
46C4×Q16—21no—
47C4×Q16—21no—
48#432—21no—
49C2×#106—22no—
50C2×#106—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8D411yescharacteristic
53C2×C4×C8—21no—
54C2×C4×C8C2311yescharacteristic
55#389—42no—
56#389—41no—
57#384—22no—
58#384—21no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—21no—
61C4⋊C16—21no—
62C4⋊C16—21no—
63C22⋊C16—21no—
64C22⋊C16—21no—
65#346—21no—
66C4×C16—21no—
67#340—22no—
68#340—21no—
69#340D411yescharacteristic
70#340D412yes—
71#340D411yescharacteristic
72#335C2×C412yes—
73#334—21no—
74#332—22no—
75#332—22no—
76C22×Q8D811yescharacteristic
77C2×Q16—21no—
78C2×Q16—21no—
79C2×Q16—42no—
80C22×C8—21no—
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C4⋊Q8—21no—
84C4⋊Q8—21no—
85#125—41no—
86C22⋊Q8—41no—
87C22⋊Q8—41no—
88C4×Q8—21no—
89C4×Q8—21no—
90C4×Q8—41no—
91C4×Q8—41no—
92C4×Q8—84no—
93C4×Q8C22⋊C412yes—
94C4×Q8SD1612yes—
95#117—41no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4D811yescharacteristic
105C2×C42—21no—
106C2×C42—42no—
107C2×C42—41no—
108C2×C42C2×D411yescharacteristic
109C2×C16—41no—
110C2×C16—41no—
111#107—21no—
112#107—21no—
113#106—42no—
114#106—44no—
115#106—42no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—21no—
120C4⋊C8C22⋊C412yes—
121#103—42no—
122#103—41no—
123#103—41no—
124#103—42no—
125C22⋊C8—41no—
126#97—41no—
127C4×C8—22no—
128C4×C8—22no—
129C4×C8C22×C411yescharacteristic
130C4×C8C2×D411yescharacteristic
131C4×C8#4111yescharacteristic
132C4×C8#4111yescharacteristic
133#95—42no—
134#95—41no—
135C2×Q8—24no—
136C2×Q8—21no—
137C2×Q8—41no—
138C2×Q8—41no—
139C2×Q8#10212yes—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—41no—
145C22×C4#12011yescharacteristic
146C22×C4C2×D811yescharacteristic
147C22×C4AGL(1,ℤ/8)11yescharacteristic
148Q16—81no—
149Q16—81no—
150Q16—84no—
151C2×C8—22no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8C4×D411yescharacteristic
157C2×C8C4×D411yescharacteristic
158C2×C8#12111yescharacteristic
159C2×C8#12111yescharacteristic
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—42no—
167C4⋊C4—41no—
168C4⋊C4—42no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—84no—
174C4⋊C4—82no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4—161no—
178C4⋊C4—162no—
179C4⋊C4#10212yes—
180C22⋊C4—81no—
181C42—22no—
182C42—22no—
183C42—84no—
184C42—82no—
185C42—81no—
186C42C2×C22⋊C411yescharacteristic
187C42#12311yescharacteristic
188C42C2×SD1611yescharacteristic
189C42#13711yescharacteristic
190C16—81no—
191C16—81no—
192C23#44711yescharacteristic
193Q8—24no—
194Q8—22no—
195Q8—82no—
196Q8—81no—
197Q8—81no—
198Q8#36014yes—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—82no—
213C2×C4—81no—
214C2×C4—82no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—82no—
219C2×C4—81no—
220C2×C4#38611yescharacteristic
221C2×C4C2×#10211yescharacteristic
222C2×C4#41711yescharacteristic
223C2×C4#44811yescharacteristic
224C2×C4#48011yescharacteristic
225C2×C4#48111yescharacteristic
226C8—21no—
227C8—21no—
228C8—21no—
229C8—21no—
230C8—42no—
231C8—82no—
232C22—21no—
233C22—21no—
234C22#187411yescharacteristic
235C22#219411yescharacteristic
236C22#219511yescharacteristic
237C4—21no—
238C4—21no—
239C4—24no—
240C4—21no—
241C4—21no—
242C4—42no—
243C4—82no—
244C4—162no—
245C4—161no—
246C4—161no—
247C4—161no—
248C4#190511yescharacteristic
249C4#190611yescharacteristic
250C4C2×#36012yes—
251C2—21no—
252C2—21no—
253C2#918011yescharacteristic
254C2#1299711yescharacteristic
255C2#1300111yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#447
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1674C4semidirect
22. #1670C4semidirect

Maximal subgroups

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

subgroupindex
#227382
#192652
#118572
#117452
#116392
#72992
#72222