Finite Group Databasebeta

← #145229

Subgroups — #145229

Order 512.

283 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 60 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#19203C211yesmaximal, characteristic
2#12031C211yesmaximal, characteristic, complemented
3#11964C211yesmaximal, characteristic
4#11964C211yesmaximal, characteristic
5#11643C211yesmaximal, characteristic, complemented
6#8635C211yesmaximal, characteristic, complemented
7#7293C211yesmaximal, characteristic, complemented
8#3208—21no—
9C2×#493C2211yescharacteristic
10#2960C2211yescharacteristic
11#2163C2211yescharacteristic
12#2163C2211yescharacteristic
13#1688—22no—
14#1668—22no—
15#1568C2211yescharacteristic
16#1568C2211yescharacteristic
17#1564C2211yescharacteristic
18#1545—21no—
19#1537—21no—
20#1368—22no—
21#1361—22no—
22#1361—22no—
23#1338—22no—
24#1307—21no—
25C22×D8—21no—
26C22×D8—21no—
27#533D411yescharacteristic
28C2×#127—21no—
29#493—21no—
30#493—21no—
31#493—21no—
32#493D412yes—
33#492—41no—
34#486—41no—
35#460—24no—
36#450—42no—
37#446—21no—
38#446—21no—
39C8×Q8—21no—
40C8×Q8—21no—
41#432—21no—
42#432—21no—
43#431—21no—
44#424—21no—
45#423D411yescharacteristic
46#405—21no—
47C2×C4×C8C2311yescharacteristic
48C4⋊C16—21no—
49C4⋊C16D412yes—
50C22⋊C16—21no—
51C22⋊C16—21no—
52#346—21no—
53#346—21no—
54#335—21no—
55#335—21no—
56#335—21no—
57#334—21no—
58#334—21no—
59#334D412yes—
60#331—24no—
61#330—42no—
62C22×D4—41no—
63C22×D4—41no—
64C22×D4—41no—
65C2×SD16—44no—
66C2×D8—22no—
67C2×D8—22no—
68C2×D8—41no—
69C2×D8—41no—
70C2×D8—82no—
71C2×D8—82no—
72C2×M16—21no—
73C22×C8#4111yescharacteristic
74C22×C8#4111yescharacteristic
75C4⋊Q8SD1611yescharacteristic
76C4⋊Q8SD1611yescharacteristic
77#127—22no—
78#127—41no—
79#127—41no—
80#127—41no—
81#125—21no—
82C22⋊Q8—21no—
83C22⋊Q8—21no—
84C4×Q8—21no—
85C4×Q8—21no—
86#117—21no—
87C2×C42C2×D411yescharacteristic
88C2×C16—22no—
89C2×C16—41no—
90C4⋊C8—21no—
91C4⋊C8—22no—
92C4⋊C8—21no—
93C4⋊C8—41no—
94C4⋊C8—41no—
95C4⋊C8—41no—
96C4⋊C8—41no—
97#102—44no—
98C22⋊C8—41no—
99C22⋊C8—41no—
100C22⋊C8—41no—
101#97—41no—
102#97—41no—
103#97—41no—
104C4×C8—41no—
105C4×C8—41no—
106C4×C8C2×D411yescharacteristic
107C4×C8C2×D411yescharacteristic
108C4×C8C2×D411yescharacteristic
109C4×C8#4111yescharacteristic
110C24—81no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×D4—42no—
114C2×D4—42no—
115C2×D4—42no—
116C2×D4—82no—
117C2×D4—82no—
118C2×D4—81no—
119C2×D4—81no—
120C2×D4—81no—
121C2×D4—162no—
122C2×D4—162no—
123C22×C4#12311yescharacteristic
124C22×C4#13511yescharacteristic
125C22×C4AGL(1,ℤ/8)11yescharacteristic
126SD16—164no—
127D8—82no—
128D8—82no—
129D8—82no—
130D8—82no—
131M16—81no—
132C2×C8—22no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—42no—
136C2×C8—81no—
137C2×C8—81no—
138C2×C8#12111yescharacteristic
139C2×C8#12111yescharacteristic
140C2×C8#12311yescharacteristic
141C2×C8#12411yescharacteristic
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C22⋊C4—41no—
149C42—41no—
150C42#12011yescharacteristic
151C42#12111yescharacteristic
152C42C2×SD1611yescharacteristic
153C42AGL(1,ℤ/8)11yescharacteristic
154C16—42no—
155C16—81no—
156C23—82no—
157C23—162no—
158C23—162no—
159C23—162no—
160C23#48211yescharacteristic
161Q8—42no—
162Q8—81no—
163D4—84no—
164D4—82no—
165D4—82no—
166D4—162no—
167D4—162no—
168D4—162no—
169D4—162no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4#39411yescharacteristic
177C2×C4#44911yescharacteristic
178C2×C4#45011yescharacteristic
179C2×C4#46511yescharacteristic
180C2×C4#46911yescharacteristic
181C2×C4#48111yescharacteristic
182C8—21no—
183C8—21no—
184C8—21no—
185C8—82no—
186C8—81no—
187C8—161no—
188C8—161no—
189C8#47112yes—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—162no—
194C22—162no—
195C22—162no—
196C22—324no—
197C22#203411yescharacteristic
198C4—21no—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—81no—
204C4—81no—
205C4#204611yescharacteristic
206C4#204811yescharacteristic
207C2—41no—
208C2—322no—
209C2#1044311yescharacteristic
210C2#1339611yescharacteristic
211C2#1340211yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC2×#493C22
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12031C2semidirect
5. #11643C2semidirect
6. #8635C2semidirect
7. #7293C2semidirect

Maximal subgroups

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

subgroupindex
#192032
#120312
#119642
#116432
#86352
#72932