Finite Group Databasebeta

← #136402

Subgroups — #136402

Order 512.

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

normal subgroups only

This group has 341 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#22674C211yesmaximal, characteristic
2#12886C211yesmaximal, characteristic
3#12819C211yesmaximal, characteristic
4#11594C211yesmaximal, characteristic
5#9630C211yesmaximal, characteristic
6#7278C211yesmaximal, characteristic
7#7168C211yesmaximal, characteristic
8C2×#441—21no—
9C2×C4×Q16C2211yescharacteristic
10Q8×C4⋊C4—21no—
11#2150C2211yescharacteristic
12C2×#362—21no—
13C4×M32C2211yescharacteristic
14#1943—21no—
15#1865—22no—
16#1848C2211yescharacteristic
17#1844C2211yescharacteristic
18#1803—21no—
19#1770C2211yescharacteristic
20#1571—21no—
21#1564C2211yescharacteristic
22#1387—21no—
23#1383—22no—
24#1332C414yescomplemented
25#1303—21no—
26C22×Q16D411yescharacteristic
27C2×C4×Q8—22no—
28C2×C4×Q8—21no—
29C2×M32—21no—
30C2×M32—21no—
31#441—42no—
32#441—42no—
33C4×Q16—21no—
34C4×Q16—21no—
35C4×Q16C2×C412yes—
36C4×Q16Q812yes—
37#431—21no—
38C2×#108—21no—
39C2×#108—21no—
40C2×#107D411yescharacteristic
41C2×#106—21no—
42C2×#106D412yes—
43#423—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×#103—21no—
46C2×#103—22no—
47C2×#97—21no—
48C2×C4×C8C2311yescharacteristic
49#389—22no—
50#389—42no—
51#384—21no—
52#384—41no—
53C4×C4⋊C4—41no—
54C4×C4⋊C4D411yescharacteristic
55C4⋊C16—21no—
56C4⋊C16—21no—
57#362—42no—
58#346—21no—
59C4×C16C2×C412yes—
60#340—22no—
61#336—21no—
62#335—42no—
63#335C2×C412yes—
64#334—21no—
65C22×Q8—21no—
66C2×Q16—21no—
67C2×Q16—21no—
68C2×Q16—21no—
69C2×Q16C4⋊C412yes—
70C2×Q16D812yes—
71C2×Q16Q1612yes—
72C2×M16—41no—
73C22×C8—21no—
74C22×C8C2×D411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C4×Q8—22no—
77C4×Q8—22no—
78C4×Q8—41no—
79C4×Q8—41no—
80C4×Q8—44no—
81C4×Q8—82no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C42—21no—
91C2×C42—41no—
92C2×C42—42no—
93C2×C42C2×D411yescharacteristic
94M32—41no—
95M32—41no—
96M32—41no—
97M32—41no—
98C2×C16—22no—
99C2×C16—41no—
100#108—42no—
101#108—41no—
102#108—41no—
103#107—21no—
104#107C4⋊C412yes—
105#106—41no—
106#106—41no—
107#106—42no—
108C4⋊C8—22no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111C4⋊C8—41no—
112C4⋊C8—41no—
113#103—22no—
114#103—22no—
115#103—44no—
116C22⋊C8—41no—
117#97—41no—
118#97—41no—
119C4×C8—41no—
120C4×C8C22×C411yescharacteristic
121C4×C8C2×Q811yescharacteristic
122C4×C8#4111yescharacteristic
123C4×C8#4111yescharacteristic
124#95—42no—
125#95SD1612yes—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
132C22×C4—22no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—42no—
136C22×C4#12011yescharacteristic
137C22×C4C2×SD1611yescharacteristic
138C22×C4AGL(1,ℤ/8)11yescharacteristic
139Q16—22no—
140Q16—22no—
141Q16—22no—
142Q16#10714yes—
143M16—81no—
144M16—81no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—42no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—81no—
152C2×C8C2×C4⋊C411yescharacteristic
153C2×C8C4×D411yescharacteristic
154C2×C8C22⋊Q811yescharacteristic
155C2×C8#12311yescharacteristic
156C2×C8C2×D811yescharacteristic
157C2×C8C2×Q1611yescharacteristic
158C2×C8AGL(1,ℤ/8)11yescharacteristic
159C2×C8#13711yescharacteristic
160C4⋊C4—22no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—82no—
168C4⋊C4—84no—
169C4⋊C4—161no—
170C4⋊C4—162no—
171C42—42no—
172C42—41no—
173C42—41no—
174C42—42no—
175C42—84no—
176C42C4×D411yescharacteristic
177C42C22⋊Q811yescharacteristic
178C42#13511yescharacteristic
179C42AGL(1,ℤ/8)11yescharacteristic
180C16—42no—
181C16—81no—
182C23#45011yescharacteristic
183Q8—24no—
184Q8—42no—
185Q8—42no—
186Q8—42no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4#38511yescharacteristic
202C2×C4C4×SD1611yescharacteristic
203C2×C4#44211yescharacteristic
204C2×C4#44711yescharacteristic
205C2×C4#44911yescharacteristic
206C2×C4#45111yescharacteristic
207C2×C4#45411yescharacteristic
208C2×C4#45511yescharacteristic
209C2×C4#47611yescharacteristic
210C2×C4#47811yescharacteristic
211C8—21no—
212C8—21no—
213C8—81no—
214C8—82no—
215C8—161no—
216C8C2×#10711yescharacteristic
217C8#42811yescharacteristic
218C8#48011yescharacteristic
219C8#48411yescharacteristic
220C22—21no—
221C22—21no—
222C22#163711yescharacteristic
223C22#163911yescharacteristic
224C22#186411yescharacteristic
225C4—21no—
226C4—21no—
227C4—21no—
228C4—44no—
229C4—82no—
230C4—82no—
231C4—162no—
232C4#162511yescharacteristic
233C4#164411yescharacteristic
234C4#185211yescharacteristic
235C4#186011yescharacteristic
236C4#218012yes—
237C2—21no—
238C2—21no—
239C2#889011yescharacteristic
240C2#1294911yescharacteristic
241C2#1295711yescharacteristic
subgroupG/H
center, socleC22#1864
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#450
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
24. #1332C4semidirect

Maximal subgroups

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

subgroupindex
#226742
#128862
#128192
#115942
#96302
#72782
#71682