Finite Group Databasebeta

← #135370

Subgroups — #135370

Order 512.

319 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 82 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#20829C211yesmaximal, characteristic
2#12850C211yesmaximal, characteristic
3#10800C211yesmaximal, characteristic
4#7257C212yesmaximal
5#7256C212yesmaximal
6#2921—21no—
7C2×#405C2211yescharacteristic
8C2×#349C2212yes—
9#1849C411yescharacteristic, complemented
10#1849C411yescharacteristic, complemented
11#1837—21no—
12#1811C2212yes—
13#1810—21no—
14#1778—21no—
15C4×#108C412yescomplemented
16C2×#338C2211yescharacteristic
17C2×H(3,ℤ/4)—21no—
18C2×H(3,ℤ/4)C2211yescharacteristic
19#1729—21no—
20#1376—21no—
21#1375—21no—
22C22×M16—21no—
23C22×M16C2311yescharacteristic
24C23×C8D411yescharacteristic
25C2×#117—21no—
26C2×#117D411yescharacteristic
27C2×M32—22no—
28#433—42no—
29#433—42no—
30C2×#108—21no—
31C2×#108—21no—
32C2×#108C2×C412yes—
33C2×#108C2×C412yes—
34#423—21no—
35#423—21no—
36#407—22no—
37#405—22no—
38#405—22no—
39C4×M16—22no—
40C2×#97C2×C411yescharacteristic
41C2×#97D411yescharacteristic
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#349—22no—
45#349—22no—
46#341—22no—
47#341—22no—
48#338—22no—
49H(3,ℤ/4)—22no—
50H(3,ℤ/4)—42no—
51#336—24no—
52C23×C4C2×D411yescharacteristic
53C2×M16—22no—
54C2×M16—22no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16—41no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16#4112yes—
64C2×M16#4112yes—
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—41no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74#117—21no—
75#117—21no—
76#117—41no—
77#117—41no—
78#117—41no—
79#117—41no—
80#117D812yes—
81#117SD1612yes—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—41no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—41no—
86C2×C42—22no—
87C2×C42C22⋊C411yescharacteristic
88C2×C42C22⋊C411yescharacteristic
89M32—42no—
90M32—42no—
91C2×C16—42no—
92#108—42no—
93#108—42no—
94#108—42no—
95#108—42no—
96C4⋊C8—42no—
97C4⋊C8—42no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100#97—22no—
101#97—22no—
102#97—42no—
103C4×C8—22no—
104C4×C8—22no—
105C4×C8—42no—
106C24#13711yescharacteristic
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—42no—
114C22×C4C2×C22⋊C411yescharacteristic
115C22×C4#12311yescharacteristic
116C22×C4#12311yescharacteristic
117C22×C4C2×D811yescharacteristic
118C22×C4C2×SD1611yescharacteristic
119C22×C4#13711yescharacteristic
120M16—42no—
121M16—42no—
122M16—42no—
123M16—42no—
124M16—42no—
125M16—42no—
126M16—81no—
127M16—81no—
128M16—81no—
129M16—81no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8#10412yes—
147C2×C8#11712yes—
148C2×C8#12312yes—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—81no—
152C4⋊C4—81no—
153C22⋊C4—22no—
154C22⋊C4—22no—
155C22⋊C4—81no—
156C22⋊C4—81no—
157C42—21no—
158C42—21no—
159C42—42no—
160C42—42no—
161C42#10212yes—
162C42#10212yes—
163C16—82no—
164C23—21no—
165C23—21no—
166C23—81no—
167C23#41911yescharacteristic
168C23#48011yescharacteristic
169C23#48311yescharacteristic
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—82no—
185C2×C4—84no—
186C2×C4#38111yescharacteristic
187C2×C4#41011yescharacteristic
188C2×C4#41011yescharacteristic
189C2×C4C2×#10211yescharacteristic
190C2×C4C2×#10411yescharacteristic
191C2×C4#42111yescharacteristic
192C2×C4#48011yescharacteristic
193C2×C4#48311yescharacteristic
194C8—22no—
195C8—22no—
196C8—22no—
197C8—41no—
198C8—82no—
199C8—82no—
200C8—82no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—42no—
207C22—81no—
208C22#150611yescharacteristic
209C22#151511yescharacteristic
210C22#182311yescharacteristic
211C4—21no—
212C4—21no—
213C4—42no—
214C4—42no—
215C4—162no—
216C4#150211yescharacteristic
217C4#151711yescharacteristic
218C4#182011yescharacteristic
219C4#182611yescharacteristic
220C2—21no—
221C2—21no—
222C2—21no—
223C2—41no—
224C2—41no—
225C2#871611yescharacteristic
subgroupG/H
centerC4#1502
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
socleC2#8716
prime radicalC24#137
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C42 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #1849C4semidirect
10. #1849C4semidirect
15. C4×#108C4semidirect

Maximal subgroups

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

subgroupindex
#208292
#128502
#108002
#72572
#72562