Finite Group Databasebeta

← #139130

Subgroups — #139130

Order 512.

278 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 72 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#33797C211yesmaximal, characteristic
2#22258C211yesmaximal, characteristic
3#11970C211yesmaximal, characteristic
4#11865C211yesmaximal, characteristic
5#9765C211yesmaximal, characteristic
6#7220C211yesmaximal, characteristic
7#7144C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9#3160C2211yescharacteristic
10#2961—21no—
11#2254C411yescharacteristic, complemented
12#2252C411yescharacteristic, complemented
13#2241C411yescharacteristic, complemented
14#2241C411yescharacteristic, complemented
15#2237—22no—
16#2229—22no—
17#2163C2211yescharacteristic
18#2162C2211yescharacteristic
19#1995—21no—
20#1922—22no—
21#1922—22no—
22#1837C2211yescharacteristic
23#1746C2211yescharacteristic
24#1568C2211yescharacteristic
25#1531—21no—
26#1506—21no—
27#1363—22no—
28#1350—22no—
29#533—21no—
30#533—21no—
31#500C2×C411yescharacteristic
32#500C2×C411yescharacteristic
33#500C2×C411yescharacteristic
34#499—21no—
35#499D412yes—
36#498—21no—
37#494C2×C411yescharacteristic
38#487—21no—
39#467—21no—
40#467—21no—
41#446—42no—
42#446—42no—
43C4×Q16—21no—
44C4×Q16—21no—
45#432—21no—
46#427—21no—
47#427D412yes—
48#424—22no—
49#423—21no—
50#405—22no—
51C2×#97D411yescharacteristic
52C2×C4×C8C2311yescharacteristic
53#376D411yescharacteristic
54#367—22no—
55#366—22no—
56C4⋊C16—21no—
57C4⋊C16—21no—
58#358—22no—
59#358—22no—
60C22⋊C16—21no—
61C22⋊C16—21no—
62#346—21no—
63C4×C16C2×C411yescharacteristic
64C4×C16C2×C411yescharacteristic
65#341—21no—
66#341D412yes—
67#339—24no—
68#336—21no—
69#336—21no—
70#335—21no—
71#334—21no—
72#330—42no—
73#328—42no—
74C2×Q16—21no—
75C2×Q16—21no—
76C2×M16—22no—
77C22×C8—21no—
78C22×C8C2×D411yescharacteristic
79C22×C8C2×D411yescharacteristic
80C4⋊Q8—21no—
81C4⋊Q8—21no—
82C4⋊Q8—21no—
83C4⋊Q8—21no—
84#125—22no—
85#125—41no—
86C22⋊Q8—41no—
87C22⋊Q8—41no—
88C22⋊Q8—41no—
89C22⋊Q8—41no—
90C4×Q8—41no—
91C4×Q8—41no—
92C4×Q8—41no—
93C4×Q8—41no—
94#117—22no—
95#117—41no—
96C2×C42—21no—
97C2×C42C2×D411yescharacteristic
98C2×C16—21no—
99C2×C16—21no—
100C2×C16—41no—
101#107—21no—
102#107—21no—
103#107—21no—
104#107—21no—
105#107C22⋊C412yes—
106#107C22⋊C412yes—
107#106—22no—
108#106—22no—
109C4⋊C8—44no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112C4⋊C8—41no—
113#103—41no—
114#103—41no—
115C22⋊C8—41no—
116#97—21no—
117#97—42no—
118#97—41no—
119#97D812yes—
120C4×C8—42no—
121C4×C8C22×C411yescharacteristic
122C4×C8C2×D411yescharacteristic
123C4×C8#4111yescharacteristic
124C4×C8#4111yescharacteristic
125#95—21no—
126#95—21no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C22×C4—41no—
132C22×C4#12011yescharacteristic
133C22×C4#13511yescharacteristic
134C22×C4AGL(1,ℤ/8)11yescharacteristic
135Q16—42no—
136Q16—81no—
137M16—42no—
138M16—42no—
139C2×C8—22no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—42no—
145C2×C8—81no—
146C2×C8C2×C22⋊C411yescharacteristic
147C2×C8C4×D411yescharacteristic
148C2×C8#12111yescharacteristic
149C2×C8#12311yescharacteristic
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—81no—
161C4⋊C4—81no—
162C22⋊C4—42no—
163C22⋊C4—81no—
164C42—41no—
165C42—41no—
166C42—42no—
167C42—81no—
168C42C4×D411yescharacteristic
169C42#12111yescharacteristic
170C42C2×D811yescharacteristic
171C42AGL(1,ℤ/8)11yescharacteristic
172C16—41no—
173C16—41no—
174C16—81no—
175C23#44911yescharacteristic
176Q8—81no—
177Q8—81no—
178Q8—161no—
179Q8—161no—
180Q8—161no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4#38611yescharacteristic
193C2×C4C4×D811yescharacteristic
194C2×C4#44211yescharacteristic
195C2×C4#44711yescharacteristic
196C2×C4#45911yescharacteristic
197C2×C4#46311yescharacteristic
198C8—21no—
199C8—21no—
200C8—21no—
201C8—21no—
202C8—42no—
203C8—82no—
204C8—161no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22#187911yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—82no—
215C4—82no—
216C4—161no—
217C4—161no—
218C4—161no—
219C4#190511yescharacteristic
220C4#191211yescharacteristic
221C2—41no—
222C2#930011yescharacteristic
223C2#1301611yescharacteristic
224C2#1303511yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#449
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1746 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2254C4semidirect
12. #2252C4semidirect
13. #2241C4semidirect
14. #2241C4semidirect

Maximal subgroups

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

subgroupindex
#337972
#222582
#119702
#118652
#97652
#72202
#71442