Finite Group Databasebeta

← #144579

Subgroups — #144579

Order 512.

330 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 74 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#23835C211yesmaximal, characteristic, complemented
2#11742C212yesmaximal
3#9847C211yesmaximal, characteristic
4#7296C212yesmaximal, complemented
5#7142C211yesmaximal, characteristic, complemented
6#3169C2211yescharacteristic
7C4×C4⋊Q8—21no—
8#2162C2212yes—
9#2083—22no—
10#2057—22no—
11#1962—21no—
12C4×#107—21no—
13C4×#107C2212yes—
14#1744C2211yescharacteristic
15#1567C2211yescharacteristic
16#1492—21no—
17#1347—22no—
18#1347—22no—
19#532—21no—
20C2×C4⋊Q8—21no—
21C2×C4×Q8—41no—
22#500Q811yescharacteristic, complemented
23#493Q811yescharacteristic, complemented
24#486—21no—
25#466—22no—
26C4×D8—22no—
27C2×#107—22no—
28C2×#107—22no—
29#423—21no—
30C2×#104—41no—
31C2×#104—41no—
32C2×C4×C8D411yescharacteristic
33C2×C4×C8C2311yescharacteristic
34#391—41no—
35#389—22no—
36#389—41no—
37#384—21no—
38C4×C4⋊C4—22no—
39C4×C4⋊C4—41no—
40C43D411yescharacteristic
41C4⋊C16—22no—
42C22⋊C16—22no—
43C4×C16—22no—
44#340—21no—
45#340—21no—
46#340D412yes—
47#340D412yes—
48#336—22no—
49#335—22no—
50D4⋊C8—42no—
51C2×#41—41no—
52C22×Q8—41no—
53C2×D8—22no—
54C2×M16—41no—
55C22×C8—21no—
56C22×C8C2×D412yes—
57C4⋊Q8—21no—
58C4⋊Q8—44no—
59#127—21no—
60#124—41no—
61#121—41no—
62#121—41no—
63C4×Q8—82no—
64C4×Q8—82no—
65C4×D4—41no—
66C4×D4—41no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—41no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—42no—
76C2×C42D811yescharacteristic
77C2×C42D811yescharacteristic
78C2×C42C2×D411yescharacteristic
79C2×C16—42no—
80#107—22no—
81#107—44no—
82#107—44no—
83C4⋊C8—41no—
84C4⋊C8—41no—
85#104—82no—
86#104—82no—
87#102—42no—
88C4×C8—21no—
89C4×C8SD1612yes—
90C4×C8C2×Q811yescharacteristic
91C4×C8#4111yescharacteristic
92C4×C8#4112yes—
93#95—42no—
94#95—42no—
95#41—81no—
96#41—81no—
97#41—81no—
98#41—81no—
99C2×Q8—44no—
100C2×Q8—41no—
101C2×Q8—82no—
102C2×Q8—82no—
103C2×D4—41no—
104C2×D4—41no—
105C2×D4—41no—
106C2×D4—81no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—81no—
112C22×C4#12011yescharacteristic
113C22×C4C2×D811yescharacteristic
114C22×C4AGL(1,ℤ/8)11yescharacteristic
115D8—82no—
116M16—81no—
117M16—81no—
118C2×C8—22no—
119C2×C8—42no—
120C2×C8—42no—
121C2×C8—81no—
122C2×C8C22⋊Q812yes—
123C2×C8#12412yes—
124C4⋊C4—24no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—44no—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—84no—
132C4⋊C4—82no—
133C4⋊C4—82no—
134C22⋊C4—81no—
135C22⋊C4—81no—
136C42—21no—
137C42—21no—
138C42—42no—
139C42—81no—
140C42—82no—
141C42—82no—
142C42D1612yes—
143C42SD3212yes—
144C42C22⋊Q811yescharacteristic
145C42#12411yescharacteristic
146C42C2×SD1611yescharacteristic
147C42#13711yescharacteristic
148C16—82no—
149C23—81no—
150C23#44711yescharacteristic
151Q8—88no—
152Q8—81no—
153Q8—81no—
154D4—82no—
155D4—81no—
156D4—81no—
157D4—82no—
158D4—161no—
159D4—161no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—44no—
169C2×C4—82no—
170C2×C4—82no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—81no—
175C2×C4—161no—
176C2×C4—161no—
177C2×C4#39311yescharacteristic
178C2×C4#44811yescharacteristic
179C2×C4#47411yescharacteristic
180C2×C4#47611yescharacteristic
181C2×C4#48611yescharacteristic
182C2×C4#48911yescharacteristic
183C2×C4C2×D1611yescharacteristic
184C2×C4C2×SD3211yescharacteristic
185C2×C4#50911yescharacteristic
186C2×C4#51011yescharacteristic
187C8—22no—
188C8—22no—
189C8—42no—
190C8—161no—
191C22—21no—
192C22—21no—
193C22—161no—
194C22—161no—
195C22—161no—
196C22#202311yescharacteristic
197C22#218411yescharacteristic
198C22#218511yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—42no—
203C4—42no—
204C4—84no—
205C4—162no—
206C4—161no—
207C4#203112yes—
208C4#222411yescharacteristic
209C4#222611yescharacteristic
210C4#224011yescharacteristic
211C4#224311yescharacteristic
212C2—21no—
213C2—21no—
214C2—321no—
215C2#1037011yescharacteristic
216C2#1336311yescharacteristic
217C2#1337311yescharacteristic
subgroupG/H
centerC2×C4#486
commutator, Frattini, cosocleC2×C4×C8C23
socleC22#2023
prime radical#3169C22
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1744 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23835C2semidirect
4. #7296C2semidirect
5. #7142C2semidirect
22. #500Q8semidirect
23. #493Q8semidirect

Maximal subgroups

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

subgroupindex
#238352
#117422
#98472
#72962
#71422