Finite Group Databasebeta

← #145874

Subgroups — #145874

Order 512.

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

normal subgroups only

This group has 311 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#28809C211yesmaximal, characteristic
2#13185C211yesmaximal, characteristic
3#12886C211yesmaximal, characteristic
4#9627C211yesmaximal, characteristic
5#7278C211yesmaximal, characteristic
6#7175C211yesmaximal, characteristic
7#7169C211yesmaximal, characteristic
8C2×#498—21no—
9#2551—21no—
10C2×#365—21no—
11#2150C2211yescharacteristic
12C4×M32C2211yescharacteristic
13#2058—22no—
14#1948—21no—
15#1941—21no—
16#1848C2211yescharacteristic
17#1844C2211yescharacteristic
18C4×#106C2211yescharacteristic
19C8×C4⋊C4C2211yescharacteristic
20#1571—21no—
21#1564C2211yescharacteristic
22#1383—22no—
23#1383—21no—
24#1303—21no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—41no—
27C2×M32—21no—
28C2×M32—21no—
29#498—42no—
30#498—42no—
31#431—21no—
32C2×#108—21no—
33C2×#108—21no—
34C2×#106—22no—
35C2×#106—21no—
36C2×#106D411yescharacteristic
37C2×#106D411yescharacteristic
38C2×#106D412yes—
39#423—21no—
40C2×C4⋊C8D411yescharacteristic
41C2×C4×C8—21no—
42C2×C4×C8C2311yescharacteristic
43#395—42no—
44#389—22no—
45#389—41no—
46#384—21no—
47C4×C4⋊C4—21no—
48C4×C4⋊C4D411yescharacteristic
49#365—42no—
50C4⋊C16—21no—
51C4⋊C16Q812yescomplemented
52#346Q812yescomplemented
53C4×C16—21no—
54#340—22no—
55#340—21no—
56#336—21no—
57#335—42no—
58#335Q812yescomplemented
59#334—21no—
60C22×Q8—42no—
61C2×M16—41no—
62C22×C8—21no—
63C22×C8C2×D411yescharacteristic
64C22×C8C2×D411yescharacteristic
65C4⋊Q8—44no—
66C4⋊Q8—82no—
67C4⋊Q8—84no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C42—21no—
77C2×C42—41no—
78C2×C42C2×D411yescharacteristic
79M32—41no—
80M32—41no—
81M32—41no—
82M32—41no—
83C2×C16—22no—
84C2×C16—22no—
85#108—42no—
86#108—41no—
87#108—41no—
88#106—21no—
89#106—21no—
90#106—44no—
91#106—41no—
92#106—41no—
93#106—42no—
94#106D812yes—
95#106Q1612yes—
96C4⋊C8—22no—
97C4⋊C8—21no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100C4⋊C8SD1612yes—
101C22⋊C8—41no—
102C4×C8—41no—
103C4×C8—41no—
104C4×C8—41no—
105C4×C8C2×Q811yescharacteristic
106C4×C8C2×Q811yescharacteristic
107C4×C8C2×Q811yescharacteristic
108C4×C8#4111yescharacteristic
109#95—42no—
110#95—41no—
111#95SD1612yes—
112C2×Q8—84no—
113C2×Q8—84no—
114C2×Q8—162no—
115C22×C4—22no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—42no—
119C22×C4#12011yescharacteristic
120C22×C4C2×SD1611yescharacteristic
121C22×C4AGL(1,ℤ/8)11yescharacteristic
122M16—81no—
123M16—81no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—22no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—42no—
130C2×C8—81no—
131C2×C8C22⋊Q811yescharacteristic
132C2×C8C22⋊Q811yescharacteristic
133C2×C8C4⋊Q811yescharacteristic
134C2×C8C4⋊Q811yescharacteristic
135C2×C8C2×D811yescharacteristic
136C2×C8C2×Q1611yescharacteristic
137C2×C8AGL(1,ℤ/8)11yescharacteristic
138C2×C8#13711yescharacteristic
139C4⋊C4—24no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—82no—
147C4⋊C4—84no—
148C4⋊C4—84no—
149C4⋊C4—161no—
150C42—41no—
151C42—41no—
152C42—82no—
153C42C22⋊Q811yescharacteristic
154C42C4⋊Q811yescharacteristic
155C42C2×SD1611yescharacteristic
156C42#13711yescharacteristic
157C16—42no—
158C16—42no—
159C23#45011yescharacteristic
160Q8—168no—
161C2×C4—22no—
162C2×C4—24no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4#39511yescharacteristic
176C2×C4#44711yescharacteristic
177C2×C4#44811yescharacteristic
178C2×C4#45111yescharacteristic
179C2×C4#45411yescharacteristic
180C2×C4#45511yescharacteristic
181C2×C4#47511yescharacteristic
182C2×C4#47611yescharacteristic
183C2×C4#49811yescharacteristic
184C2×C4#50111yescharacteristic
185C8—21no—
186C8—21no—
187C8—42no—
188C8—82no—
189C8—161no—
190C8#47411yescharacteristic
191C8#47711yescharacteristic
192C8#50011yescharacteristic
193C8#50111yescharacteristic
194C22—21no—
195C22—21no—
196C22#163711yescharacteristic
197C22#163911yescharacteristic
198C22#205811yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—44no—
203C4—82no—
204C4—82no—
205C4—162no—
206C4#162711yescharacteristic
207C4#164511yescharacteristic
208C4#205711yescharacteristic
209C4#206111yescharacteristic
210C4#223012yes—
211C2—21no—
212C2—21no—
213C2#1057511yescharacteristic
214C2#1345711yescharacteristic
215C2#1346011yescharacteristic
subgroupG/H
center, socleC22#2058
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#450
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
51. C4⋊C16Q8semidirect
52. #346Q8semidirect
58. #335Q8semidirect

Maximal subgroups

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

subgroupindex
#288092
#131852
#128862
#96272
#72782
#71752
#71692