Finite Group Databasebeta

← #128607

Subgroups — #128607

Order 512.

363 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 108 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#21277C211yesmaximal, characteristic
2#12594C211yesmaximal, characteristic, complemented
3#11599C211yesmaximal, characteristic
4#11593C211yesmaximal, characteristic
5#8269C211yesmaximal, characteristic, complemented
6#7175C211yesmaximal, characteristic, complemented
7#7174C211yesmaximal, characteristic, complemented
8C2×C4×D8C2211yescharacteristic
9#2293—21no—
10C4×M32C2211yescharacteristic
11C4×M32C2211yescharacteristic
12#1972C412yes—
13#1943C412yes—
14#1936—22no—
15C4×#106C2211yescharacteristic
16C4×#102—21no—
17C4×#97C2211yescharacteristic
18#1564C2211yescharacteristic
19#1564C2211yescharacteristic
20#1452—21no—
21#1375—22no—
22#1333C414yescomplemented
23#1331C414yescomplemented
24#1276—21no—
25C22×D8C2×C411yescharacteristic
26C2×C4×D4—21no—
27C2×#117—41no—
28C2×M32—21no—
29C2×M32—21no—
30C4×D8—21no—
31C4×D8—21no—
32C4×D8C2×C412yes—
33C4×D8C2×C412yes—
34#431—21no—
35#431—21no—
36C2×#107C2×C411yescharacteristic
37C2×#106C2×C412yes—
38C2×#102—21no—
39C2×#102—22no—
40C2×#97C2×C412yes—
41C2×#97D412yes—
42C2×C4×C8D411yescharacteristic
43C2×C4×C8C2311yescharacteristic
44#390—22no—
45#389—22no—
46#388—42no—
47C4×C4⋊C4—21no—
48C4×C22⋊C4—41no—
49C43D411yescharacteristic
50#346—21no—
51#346C2×C412yes—
52C4×C16—21no—
53C4×C16C2×C412yes—
54#340—22no—
55#340—21no—
56#335—21no—
57#335C2×C412yes—
58#334—21no—
59#334C2×C412yes—
60#322—42no—
61C22×D4—21no—
62C23×C4—41no—
63C2×D8—21no—
64C2×D8—24no—
65C2×D8C4212yes—
66C22×C8C22⋊C412yes—
67C22×C8C22×C411yescharacteristic
68C22×C8C2×D411yescharacteristic
69C4×D4—22no—
70C4×D4—22no—
71C4×D4—41no—
72C4×D4—41no—
73#117—84no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—42no—
77C2×C22⋊C4—41no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—41no—
83C2×C42C22⋊C412yes—
84C2×C42C2×D411yescharacteristic
85M32—42no—
86M32—42no—
87C2×C16—22no—
88C2×C16—22no—
89#107—21no—
90#107C4212yes—
91#106—24no—
92C4⋊C8—22no—
93C4⋊C8—22no—
94#102—22no—
95#102—22no—
96#102—44no—
97C22⋊C8—41no—
98C22⋊C8—41no—
99#97—24no—
100#97—42no—
101C4×C8—21no—
102C4×C8—21no—
103C4×C8—41no—
104C4×C8—41no—
105C4×C8C22×C411yescharacteristic
106C4×C8C22×C411yescharacteristic
107C4×C8#4111yescharacteristic
108C4×C8#4111yescharacteristic
109#95—42no—
110#95—41no—
111C24—41no—
112C2×D4—22no—
113C2×D4—42no—
114C2×D4—41no—
115C2×D4—42no—
116C22×C4—22no—
117C22×C4—41no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—82no—
122C22×C4—82no—
123C22×C4C2×C22⋊C411yescharacteristic
124C22×C4AGL(1,ℤ/8)11yescharacteristic
125C22×C4AGL(1,ℤ/8)11yescharacteristic
126D8—24no—
127D8—42no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—42no—
133C2×C8—42no—
134C2×C8C2×C4211yescharacteristic
135C2×C8#11711yescharacteristic
136C2×C8C4×D411yescharacteristic
137C2×C8C4×D411yescharacteristic
138C2×C8#13114yes—
139C4⋊C4—22no—
140C4⋊C4—41no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—84no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C22⋊C4—84no—
147C22⋊C4—84no—
148C42—22no—
149C42—22no—
150C42—42no—
151C42—82no—
152C42—81no—
153C42C4×D411yescharacteristic
154C42C4×D411yescharacteristic
155C42AGL(1,ℤ/8)11yescharacteristic
156C42#13711yescharacteristic
157C16—42no—
158C16—42no—
159C23—42no—
160C23—81no—
161C23—82no—
162C23—81no—
163C23—82no—
164C23#41811yescharacteristic
165D4—42no—
166D4—42no—
167D4—82no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—84no—
175C2×C4—81no—
176C2×C4—81no—
177C2×C4—84no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4C4×C22⋊C411yescharacteristic
182C2×C4#40812yes—
183C2×C4#41711yescharacteristic
184C2×C4#42112yes—
185C2×C4#44011yescharacteristic
186C2×C4#44011yescharacteristic
187C2×C4#44211yescharacteristic
188C2×C4#44211yescharacteristic
189C8—21no—
190C8—21no—
191C8—42no—
192C8—82no—
193C8—82no—
194C8#40512yes—
195C8#43312yes—
196C22—21no—
197C22—21no—
198C22—82no—
199C22—82no—
200C22—82no—
201C22—162no—
202C22—162no—
203C22#149612yes—
204C22#176211yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—82no—
209C4—82no—
210C4—162no—
211C4#149512yes—
212C4#175011yescharacteristic
213C4#175911yescharacteristic
214C4AGL(1,ℤ/16)12yes—
215C2—22no—
216C2—162no—
217C2#817911yescharacteristic
218C2#1264711yescharacteristic
219C2#1264911yescharacteristic
subgroupG/H
centerC2×C4#442
commutatorC2×C8C2×C42
Frattini, cosocleC2×C4×C8C23
socleC22#1762
prime radicalC22×D8C2×C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12594C2semidirect
5. #8269C2semidirect
6. #7175C2semidirect
7. #7174C2semidirect
22. #1333C4semidirect
23. #1331C4semidirect

Maximal subgroups

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

subgroupindex
#212772
#125942
#115992
#115932
#82692
#71752
#71742