Finite Group Databasebeta

← #136552

Subgroups — #136552

Order 512.

310 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 92 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#23161C211yesmaximal, characteristic
2#13290C211yesmaximal, characteristic
3#13185C211yesmaximal, characteristic
4#11940C211yesmaximal, characteristic
5#9630C211yesmaximal, characteristic
6#7175C211yesmaximal, characteristic
7#7168C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9#2365—21no—
10C2×#365—21no—
11#2150C2211yescharacteristic
12C4×M32C2211yescharacteristic
13#1988—21no—
14#1943—21no—
15#1921—22no—
16#1848C2211yescharacteristic
17#1776—21no—
18C4×#106C2211yescharacteristic
19#1770C2211yescharacteristic
20#1571—21no—
21#1564C2211yescharacteristic
22#1387—22no—
23#1383—21no—
24#1364C414yescomplemented
25#1303—21no—
26C2×C4⋊Q8—21no—
27C2×C4×Q8—41no—
28C2×M32—21no—
29C2×M32—21no—
30#500—21no—
31#500—21no—
32#500C2×C412yes—
33#500Q812yes—
34#431—21no—
35C2×#108—21no—
36C2×#108—21no—
37C2×#107—21no—
38C2×#107D412yes—
39C2×#106D411yescharacteristic
40C2×#106D411yescharacteristic
41#423—21no—
42C2×C4⋊C8D411yescharacteristic
43C2×#97—21no—
44C2×C4×C8C2311yescharacteristic
45#391—41no—
46#389—22no—
47#387—42no—
48C4×C4⋊C4—22no—
49C4×C4⋊C4—21no—
50C4×C4⋊C4D411yescharacteristic
51#365—42no—
52C4⋊C16—21no—
53C4⋊C16C2×C412yes—
54#346—21no—
55C4×C16—21no—
56#340—22no—
57#340—22no—
58#340—21no—
59#336—21no—
60#335—42no—
61#335C2×C412yes—
62#334—21no—
63C22×Q8—41no—
64C2×M16—41no—
65C22×C8—21no—
66C22×C8C2×D411yescharacteristic
67C22×C8C2×D411yescharacteristic
68C4⋊Q8—22no—
69C4⋊Q8—22no—
70C4⋊Q8—41no—
71C4⋊Q8—41no—
72C4×Q8—84no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—42no—
79C2×C42—21no—
80C2×C42—42no—
81C2×C42—41no—
82C2×C42C2×D411yescharacteristic
83M32—22no—
84M32—41no—
85M32—41no—
86M32—41no—
87C2×C16—22no—
88C2×C16—41no—
89#108—42no—
90#108—41no—
91#108—41no—
92#107—22no—
93#107—22no—
94#107—22no—
95#107C4⋊C414yes—
96#106—21no—
97#106—21no—
98#106—21no—
99#106Q1612yes—
100C4⋊C8—22no—
101C4⋊C8—21no—
102C4⋊C8—21no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C22⋊C8—41no—
106#97—41no—
107#97—41no—
108C4×C8—41no—
109C4×C8C22×C411yescharacteristic
110C4×C8C2×Q811yescharacteristic
111C4×C8#4111yescharacteristic
112C4×C8#4111yescharacteristic
113#95—42no—
114#95—42no—
115#95—41no—
116#95SD1612yes—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—82no—
120C2×Q8—82no—
121C22×C4—22no—
122C22×C4—41no—
123C22×C4—42no—
124C22×C4—41no—
125C22×C4#12011yescharacteristic
126C22×C4C2×SD1611yescharacteristic
127C22×C4AGL(1,ℤ/8)11yescharacteristic
128M16—81no—
129M16—81no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—42no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—81no—
137C2×C8C2×C4⋊C411yescharacteristic
138C2×C8C4×D411yescharacteristic
139C2×C8C22⋊Q811yescharacteristic
140C2×C8#12311yescharacteristic
141C2×C8C2×Q1611yescharacteristic
142C2×C8#13511yescharacteristic
143C2×C8AGL(1,ℤ/8)11yescharacteristic
144C2×C8#13711yescharacteristic
145C4⋊C4—24no—
146C4⋊C4—22no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—84no—
155C42—41no—
156C42—41no—
157C42—82no—
158C42—84no—
159C42C4×D411yescharacteristic
160C42C22⋊Q811yescharacteristic
161C42#13511yescharacteristic
162C42AGL(1,ℤ/8)11yescharacteristic
163C16—42no—
164C16—42no—
165C23#45011yescharacteristic
166Q8—84no—
167Q8—84no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—44no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—81no—
176C2×C4—82no—
177C2×C4—81no—
178C2×C4—82no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4#38511yescharacteristic
183C2×C4C4×SD1611yescharacteristic
184C2×C4#44211yescharacteristic
185C2×C4#44911yescharacteristic
186C2×C4#44911yescharacteristic
187C2×C4#45111yescharacteristic
188C2×C4#45411yescharacteristic
189C2×C4#45611yescharacteristic
190C2×C4#47611yescharacteristic
191C2×C4#47811yescharacteristic
192C8—21no—
193C8—21no—
194C8—81no—
195C8—82no—
196C8—161no—
197C8C4×Q1611yescharacteristic
198C8#44011yescharacteristic
199C8#47711yescharacteristic
200C8#47911yescharacteristic
201C22—21no—
202C22—21no—
203C22#163711yescharacteristic
204C22#165411yescharacteristic
205C22#186411yescharacteristic
206C4—21no—
207C4—21no—
208C4—21no—
209C4—82no—
210C4—82no—
211C4—84no—
212C4—161no—
213C4#162811yescharacteristic
214C4#164811yescharacteristic
215C4#185711yescharacteristic
216C4#186311yescharacteristic
217C4AGL(1,ℤ/16)12yes—
218C2—21no—
219C2—21no—
220C2#892611yescharacteristic
221C2#1295411yescharacteristic
222C2#1295611yescharacteristic
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. #1364C4semidirect

Maximal subgroups

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

subgroupindex
#231612
#132902
#131852
#119402
#96302
#71752
#71682