Finite Group Databasebeta

← #139129

Subgroups — #139129

Order 512.

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

normal subgroups only

This group has 338 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#33796C211yesmaximal, characteristic
2#22248C211yesmaximal, characteristic
3#11972C211yesmaximal, characteristic
4#11857C211yesmaximal, characteristic
5#9848C211yesmaximal, characteristic
6#7217C211yesmaximal, characteristic
7#7142C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9C2×#500C2211yescharacteristic
10C4×C4⋊Q8—21no—
11#2252C412yescomplemented
12#2241C412yescomplemented
13#2236—22no—
14#2227—22no—
15#2162C2211yescharacteristic
16#2162C2211yescharacteristic
17#1994—21no—
18#1990—21no—
19#1921—22no—
20#1921—22no—
21#1834C2211yescharacteristic
22#1744C2211yescharacteristic
23#1567C2211yescharacteristic
24#1493—21no—
25#1365—22no—
26#1348—22no—
27#533—21no—
28C2×C4⋊Q8—21no—
29C2×C4×Q8—41no—
30#500—21no—
31#500—21no—
32#500C2×C411yescharacteristic
33#500C2×C412yes—
34#500D412yes—
35#494C2×C411yescharacteristic
36#487—21no—
37#467—21no—
38#467—21no—
39C4×Q16—21no—
40C4×Q16—21no—
41C2×#107—21no—
42C2×#107D412yes—
43#423—21no—
44C2×C4×C8D411yescharacteristic
45C2×C4×C8C2311yescharacteristic
46#391—41no—
47#389—22no—
48#389—41no—
49#384—21no—
50C4×C4⋊C4—22no—
51C4×C4⋊C4—41no—
52C43D411yescharacteristic
53#366—24no—
54C4⋊C16—21no—
55C4⋊C16—21no—
56#358—24no—
57C22⋊C16—21no—
58C22⋊C16—21no—
59C4×C16—21no—
60C4×C16C2×C412yes—
61#340—24no—
62#340—21no—
63#340D412yes—
64#339—41no—
65#339—41no—
66#336—21no—
67#336—21no—
68#335—21no—
69#335—21no—
70Q8⋊C8—42no—
71C22×Q8—41no—
72C2×Q16—21no—
73C2×Q16—21no—
74C2×M16—41no—
75C22×C8—21no—
76C22×C8C2×D411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C4⋊Q8—22no—
79C4⋊Q8—21no—
80C4⋊Q8—21no—
81C4⋊Q8—22no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84#125—41no—
85C22⋊Q8—41no—
86C22⋊Q8—41no—
87C4×Q8—41no—
88C4×Q8—41no—
89C4×Q8—84no—
90#117—41no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—42no—
100C2×C42D811yescharacteristic
101C2×C42D811yescharacteristic
102C2×C42C2×D411yescharacteristic
103C2×C16—22no—
104C2×C16—41no—
105#107—22no—
106#107—22no—
107#107—22no—
108#107—21no—
109#107—21no—
110#107C22⋊C414yes—
111C4⋊C8—41no—
112C4⋊C8—41no—
113#103—41no—
114#103—41no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8C22×C411yescharacteristic
118C4×C8C2×D411yescharacteristic
119C4×C8#4111yescharacteristic
120C4×C8#4111yescharacteristic
121#95—42no—
122#95—42no—
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—41no—
126C2×Q8—41no—
127C2×Q8—82no—
128C2×Q8—82no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4#12011yescharacteristic
134C22×C4C2×D811yescharacteristic
135C22×C4AGL(1,ℤ/8)11yescharacteristic
136Q16—42no—
137Q16—81no—
138M16—81no—
139M16—81no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—81no—
146C2×C8C2×C22⋊C411yescharacteristic
147C2×C8C4×D411yescharacteristic
148C2×C8#12111yescharacteristic
149C2×C8#12311yescharacteristic
150C4⋊C4—24no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—82no—
161C4⋊C4—84no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C22⋊C4—81no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—41no—
171C42—41no—
172C42—42no—
173C42—84no—
174C42—81no—
175C42Q3212yes—
176C42C4×D411yescharacteristic
177C42#12111yescharacteristic
178C42#13511yescharacteristic
179C42AGL(1,ℤ/8)11yescharacteristic
180C16—42no—
181C16—81no—
182C23#44711yescharacteristic
183Q8—84no—
184Q8—84no—
185Q8—81no—
186Q8—81no—
187Q8—82no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—44no—
197C2×C4—41no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—82no—
205C2×C4#38611yescharacteristic
206C2×C4C4×D811yescharacteristic
207C2×C4#44211yescharacteristic
208C2×C4#44911yescharacteristic
209C2×C4#45911yescharacteristic
210C2×C4#46311yescharacteristic
211C2×C4C2×Q3211yescharacteristic
212C2×C4#50811yescharacteristic
213C2×C4#51011yescharacteristic
214C2×C4#51011yescharacteristic
215C8—21no—
216C8—21no—
217C8—21no—
218C8—21no—
219C8—81no—
220C8—161no—
221C22—21no—
222C22—21no—
223C22#187911yescharacteristic
224C22#218811yescharacteristic
225C22#218911yescharacteristic
226C4—21no—
227C4—21no—
228C4—42no—
229C4—81no—
230C4—84no—
231C4—162no—
232C4—161no—
233C4—161no—
234C4#190511yescharacteristic
235C4#191211yescharacteristic
236C4C4×Q3211yescharacteristic
237C4#217611yescharacteristic
238C4AGL(1,ℤ/16)12yes—
239C4#220911yescharacteristic
240C4#221111yescharacteristic
241C2—21no—
242C2—21no—
243C2#930011yescharacteristic
244C2#1301511yescharacteristic
245C2#1303711yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#447
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1744 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2252C4semidirect
12. #2241C4semidirect

Maximal subgroups

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

subgroupindex
#337962
#222482
#119722
#118572
#98482
#72172
#71422