Finite Group Databasebeta

← #139075

Subgroups — #139075

Order 512.

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

normal subgroups only

This group has 274 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#17991C211yesmaximal, characteristic
3#11970C211yesmaximal, characteristic
4#11859C211yesmaximal, characteristic
5#9759C211yesmaximal, characteristic
6#7218C211yesmaximal, characteristic
7#7147C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9#3160C2211yescharacteristic
10#2985—21no—
11#2254C411yescharacteristic, complemented
12#2252C411yescharacteristic, complemented
13#2241C411yescharacteristic, complemented
14#2241C411yescharacteristic, complemented
15#2237—22no—
16#2229—22no—
17#2163C2211yescharacteristic
18#2162C2211yescharacteristic
19#1835C2211yescharacteristic
20#1587—22no—
21#1584—22no—
22#1568C2211yescharacteristic
23#1518—21no—
24#1518—21no—
25#1497—21no—
26#1452C2211yescharacteristic
27#1363—22no—
28#1348—22no—
29#533—21no—
30#533—21no—
31#500C2×C411yescharacteristic
32#500C2×C411yescharacteristic
33#500C2×C411yescharacteristic
34#499—21no—
35#499—21no—
36#498—21no—
37#494C2×C411yescharacteristic
38#487—21no—
39#467—21no—
40#467D412yes—
41#446—22no—
42#446—42no—
43C8×Q8—22no—
44C4×Q16—21no—
45C4×Q16D412yes—
46#433—42no—
47#432—21no—
48#431D411yescharacteristic
49#431D411yescharacteristic
50#427—21no—
51#427—21no—
52C2×C4×C8C2311yescharacteristic
53#382—21no—
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#340—21no—
66#340—21no—
67#335—21no—
68#334D412yes—
69#328—42no—
70#328—42no—
71Q8⋊C8—24no—
72#322—21no—
73#322—21no—
74C2×Q16—21no—
75C2×Q16C22⋊C412yes—
76C22×C8C2×D411yescharacteristic
77C22×C8#4111yescharacteristic
78C4⋊Q8—21no—
79C4⋊Q8—21no—
80C4⋊Q8—21no—
81C4⋊Q8—21no—
82#125—41no—
83#125—41no—
84C22⋊Q8—22no—
85C22⋊Q8—41no—
86C22⋊Q8—41no—
87C22⋊Q8—41no—
88C4×Q8—22no—
89C4×Q8—41no—
90C4×Q8—41no—
91C4×Q8—41no—
92#117—41no—
93#117—41no—
94C2×C4⋊C4—41no—
95C2×C42C2×D411yescharacteristic
96C2×C16—21no—
97C2×C16—21no—
98C2×C16—41no—
99#107—21no—
100#107—21no—
101#107—21no—
102#107—21no—
103#107—21no—
104#107C22⋊C412yes—
105#106—41no—
106#106—41no—
107C4⋊C8—22no—
108C4⋊C8—21no—
109C4⋊C8—21no—
110C4⋊C8—42no—
111C4⋊C8—42no—
112#103—22no—
113#103—22no—
114C22⋊C8—21no—
115C22⋊C8—41no—
116C22⋊C8D812yes—
117#97—42no—
118#97—41no—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8—42no—
122C4×C8C22×C411yescharacteristic
123C4×C8C2×D411yescharacteristic
124C4×C8C2×D411yescharacteristic
125C4×C8#4111yescharacteristic
126#95—41no—
127#95—41no—
128C2×Q8—22no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
132C22×C4—81no—
133C22×C4#12111yescharacteristic
134C22×C4C2×D811yescharacteristic
135C22×C4AGL(1,ℤ/8)11yescharacteristic
136Q16—42no—
137Q16—81no—
138C2×C8—22no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—42no—
142C2×C8—41no—
143C2×C8C2×C22⋊C411yescharacteristic
144C2×C8C4×D411yescharacteristic
145C2×C8#12111yescharacteristic
146C2×C8#12311yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—81no—
158C4⋊C4—81no—
159C4⋊C4—81no—
160C4⋊C4—81no—
161C22⋊C4—42no—
162C22⋊C4—81no—
163C42—42no—
164C42—81no—
165C42C4×D411yescharacteristic
166C42#12011yescharacteristic
167C42#13511yescharacteristic
168C42AGL(1,ℤ/8)11yescharacteristic
169C16—41no—
170C16—41no—
171C16—81no—
172C23#45911yescharacteristic
173Q8—42no—
174Q8—42no—
175Q8—161no—
176Q8—161no—
177Q8—161no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—161no—
187C2×C4—161no—
188C2×C4—161no—
189C2×C4#38611yescharacteristic
190C2×C4C4×D811yescharacteristic
191C2×C4#44211yescharacteristic
192C2×C4#44711yescharacteristic
193C2×C4#44911yescharacteristic
194C2×C4#46311yescharacteristic
195C8—21no—
196C8—21no—
197C8—21no—
198C8—21no—
199C8—42no—
200C8—81no—
201C8—82no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22#187911yescharacteristic
206C4—21no—
207C4—21no—
208C4—21no—
209C4—21no—
210C4—21no—
211C4—82no—
212C4—82no—
213C4—161no—
214C4—161no—
215C4—321no—
216C4#190211yescharacteristic
217C4#190911yescharacteristic
218C2—41no—
219C2#929111yescharacteristic
220C2#1301611yescharacteristic
221C2#1303411yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#459
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1452 ◁ 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
#179912
#119702
#118592
#97592
#72182
#71472