Finite Group Databasebeta

← #136513

Subgroups — #136513

Order 512.

338 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 88 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#23200C211yesmaximal, characteristic
2#13188C211yesmaximal, characteristic
3#13182C211yesmaximal, characteristic
4C2×#1358C211yesmaximal, characteristic
5#9772C211yesmaximal, characteristic
6#7181C211yesmaximal, characteristic
7#7180C211yesmaximal, characteristic
8C2×#494C2211yescharacteristic
9#2950—21no—
10#2950—21no—
11#2161—21no—
12#2155—21no—
13C2×#364C2211yescharacteristic
14C2×#364C2211yescharacteristic
15#1979—21no—
16#1979—21no—
17#1886—22no—
18#1848C2211yescharacteristic
19#1772C2211yescharacteristic
20#1772C2211yescharacteristic
21#1546—22no—
22#1450C2211yescharacteristic
23#1389—21no—
24#1388—21no—
25#1358C414yescomplemented
26C22×Q16—21no—
27C22×Q16—21no—
28C2×C4⋊Q8D412yes—
29#519—22no—
30C2×M32—21no—
31C2×M32—21no—
32#494—21no—
33#494—21no—
34#494C2×C412yes—
35#494Q812yes—
36#441—42no—
37#441—42no—
38C4×Q16—42no—
39C4×Q16—42no—
40#431—21no—
41C2×#108—21no—
42C2×#108—21no—
43C2×#108D411yescharacteristic
44C2×#108D411yescharacteristic
45#427D411yescharacteristic
46#427D411yescharacteristic
47#423—21no—
48#419—22no—
49#419—22no—
50C2×#101—22no—
51#405—21no—
52#405—21no—
53C2×C4×C8C2311yescharacteristic
54#368—42no—
55#365—42no—
56#364—21no—
57#364—21no—
58#364C2×C412yes—
59#364C2×C412yes—
60#339—21no—
61#339—21no—
62#332—44no—
63#322—21no—
64C82—21no—
65C22×Q8—22no—
66C2×Q16—22no—
67C2×Q16—22no—
68C2×Q16—22no—
69C2×Q16—22no—
70C2×Q16—41no—
71C2×Q16—41no—
72C2×Q16—41no—
73C2×Q16—41no—
74C2×Q16—41no—
75C2×Q16—41no—
76C2×Q16—41no—
77C2×Q16—41no—
78C2×M16—21no—
79C2×M16—21no—
80C2×M16—41no—
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C4⋊Q8—22no—
84C4⋊Q8—42no—
85C4⋊Q8C4⋊C414yes—
86C4×Q8—44no—
87C4×Q8—44no—
88#117—21no—
89#117—21no—
90#117—82no—
91C2×C4⋊C4—42no—
92C2×C42C2×D411yescharacteristic
93M32—22no—
94M32—22no—
95M32—41no—
96M32—41no—
97C2×C16—41no—
98C2×C16—41no—
99#108—21no—
100#108—21no—
101#108—21no—
102#108—21no—
103#108—41no—
104#108—41no—
105#108—41no—
106#108—41no—
107#107—21no—
108#107—21no—
109#106D812yes—
110#106SD1612yes—
111C4⋊C8—22no—
112C4⋊C8—42no—
113#103—44no—
114#103—44no—
115#101—82no—
116C22⋊C8—41no—
117#97—41no—
118#97—41no—
119C4×C8—41no—
120C4×C8—41no—
121C4×C8—41no—
122C4×C8C22×C411yescharacteristic
123C4×C8C2×Q811yescharacteristic
124C4×C8#4111yescharacteristic
125C4×C8#4111yescharacteristic
126C2×Q8—24no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C22×C4—21no—
133C22×C4—42no—
134C22×C4#12011yescharacteristic
135Q16—42no—
136Q16—42no—
137Q16—42no—
138Q16—42no—
139Q16—42no—
140Q16—42no—
141Q16—42no—
142Q16—42no—
143M16—41no—
144M16—41no—
145M16—41no—
146M16—41no—
147M16—81no—
148M16—81no—
149C2×C8—42no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—81no—
153C2×C8C4×D411yescharacteristic
154C2×C8C4×D411yescharacteristic
155C2×C8C22⋊Q811yescharacteristic
156C2×C8C22⋊Q811yescharacteristic
157C2×C8C2×D811yescharacteristic
158C2×C8C2×SD1611yescharacteristic
159C2×C8#13511yescharacteristic
160C2×C8#13511yescharacteristic
161C2×C8AGL(1,ℤ/8)11yescharacteristic
162C2×C8AGL(1,ℤ/8)11yescharacteristic
163C2×C8AGL(1,ℤ/8)11yescharacteristic
164C2×C8#13711yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—44no—
168C4⋊C4—82no—
169C4⋊C4—84no—
170C22⋊C4—41no—
171C22⋊C4—41no—
172C42—21no—
173C42—41no—
174C42—41no—
175C42—84no—
176C42C2×C4⋊C411yescharacteristic
177C42#12311yescharacteristic
178C16—42no—
179C16—42no—
180C23#45511yescharacteristic
181Q8—44no—
182Q8—44no—
183Q8—82no—
184Q8—82no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4#38511yescharacteristic
194C2×C4#44711yescharacteristic
195C2×C4#44911yescharacteristic
196C2×C4#45011yescharacteristic
197C2×C4#45211yescharacteristic
198C2×C4#45311yescharacteristic
199C8—82no—
200C8—81no—
201C8—81no—
202C8—161no—
203C8C4×D811yescharacteristic
204C8C4×SD1611yescharacteristic
205C8#44011yescharacteristic
206C8#44211yescharacteristic
207C8#47411yescharacteristic
208C8#47611yescharacteristic
209C8#47811yescharacteristic
210C8#47811yescharacteristic
211C22—41no—
212C22#163911yescharacteristic
213C22#164011yescharacteristic
214C22#185611yescharacteristic
215C4—22no—
216C4—41no—
217C4—82no—
218C4—82no—
219C4—84no—
220C4#162111yescharacteristic
221C4#163111yescharacteristic
222C4#186211yescharacteristic
223C4#186411yescharacteristic
224C2—21no—
225C2—21no—
226C2—21no—
227C2#892111yescharacteristic
subgroupG/H
center, socleC2#8921
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#455
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1358C4semidirect

Maximal subgroups

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

subgroupindex
#232002
#131882
#131822
C2×#13582
#97722
#71812
#71802