Finite Group Databasebeta

← #139376

Subgroups — #139376

Order 512.

311 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 80 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#33494C211yesmaximal, characteristic
2#28731C211yesmaximal, characteristic
3#11975C211yesmaximal, characteristic, complemented
4#11974C211yesmaximal, characteristic, complemented
5#8632C211yesmaximal, characteristic
6#7303C211yesmaximal, characteristic, complemented
7#7303C211yesmaximal, characteristic, complemented
8C2×#500C2211yescharacteristic
9#3141—21no—
10#2931C2211yescharacteristic
11#2168—22no—
12#2168—22no—
13#2163C2211yescharacteristic
14#2163C2211yescharacteristic
15#2113—22no—
16#2078C414yes—
17#1997—21no—
18#1816—22no—
19C4×#108C2211yescharacteristic
20#1568C2211yescharacteristic
21#1568C2211yescharacteristic
22#1536—21no—
23#1365—22no—
24#1364—22no—
25#1307—21no—
26C2×#131D412yes—
27#532D411yescharacteristic
28C2×C4⋊Q8—21no—
29C2×M32—22no—
30C22×C16—22no—
31#500—21no—
32#500—21no—
33#500—21no—
34#500—21no—
35#489—42no—
36#487—41no—
37#486—41no—
38#436—21no—
39#436—21no—
40C8×D4—21no—
41C8×D4—21no—
42#432—21no—
43#432—21no—
44C2×#108—21no—
45C2×#107—21no—
46C2×#107C2×C412yes—
47#423D411yescharacteristic
48#417—24no—
49#408C2×C412yes—
50C4×M16—21no—
51C2×C4×C8C2311yescharacteristic
52C4⋊C16D412yes—
53C4⋊C16D412yes—
54C22⋊C16—24no—
55C22⋊C16—21no—
56C22⋊C16—21no—
57#346—21no—
58#346—21no—
59#343—24no—
60#341C2×C412yes—
61#335—21no—
62#335—21no—
63#334—21no—
64#334—21no—
65#329—42no—
66C2×#41C22⋊C412yes—
67C22×Q8—41no—
68#131—42no—
69#131—42no—
70C2×M16—22no—
71C2×M16—22no—
72C2×M16C22⋊C412yes—
73C22×C8—22no—
74C22×C8C22×C411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C4⋊Q8—41no—
77C4⋊Q8—41no—
78C4⋊Q8—41no—
79C4⋊Q8—41no—
80C4⋊Q8SD1611yescharacteristic
81#127SD1611yescharacteristic
82#124D812yes—
83#121—21no—
84#121—21no—
85C4×D4—21no—
86C4×D4—21no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—41no—
89C2×C42C2×D411yescharacteristic
90M32—82no—
91C2×C16—22no—
92C2×C16—22no—
93C2×C16—82no—
94#108—42no—
95#107—22no—
96#107—22no—
97#107—41no—
98#107—41no—
99C4⋊C8—21no—
100C4⋊C8—21no—
101C4⋊C8—41no—
102C4⋊C8—41no—
103#103—44no—
104#102—44no—
105C22⋊C8—22no—
106C22⋊C8—22no—
107C22⋊C8—41no—
108C22⋊C8—41no—
109#97—41no—
110#97—41no—
111#97—41no—
112C4×C8—41no—
113C4×C8C2×D411yescharacteristic
114C4×C8C2×D411yescharacteristic
115C4×C8#4111yescharacteristic
116C4×C8#4111yescharacteristic
117#41—42no—
118#41—42no—
119C2×Q8—81no—
120C2×Q8—81no—
121C2×Q8—162no—
122C2×Q8#10212yes—
123C2×D4—22no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4#10212yes—
127C22×C4—22no—
128C22×C4—42no—
129C22×C4C2×C22⋊C411yescharacteristic
130C22×C4#13511yescharacteristic
131C22×C4#13511yescharacteristic
132M16—42no—
133M16—42no—
134M16—42no—
135M16—82no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—41no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—81no—
143C2×C8—81no—
144C2×C8C4×D411yescharacteristic
145C2×C8C4×D411yescharacteristic
146C2×C8#12011yescharacteristic
147C2×C8#12311yescharacteristic
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—81no—
153C4⋊C4—81no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C42#12111yescharacteristic
157C42#12111yescharacteristic
158C42C2×D811yescharacteristic
159C42C2×SD1611yescharacteristic
160C16—42no—
161C16—42no—
162C23—22no—
163C23#41611yescharacteristic
164Q8—42no—
165Q8—162no—
166Q8—162no—
167D4—42no—
168D4—81no—
169D4—81no—
170D4—82no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—82no—
179C2×C4—162no—
180C2×C4#38611yescharacteristic
181C2×C4C2×#10211yescharacteristic
182C2×C4#46511yescharacteristic
183C2×C4#46511yescharacteristic
184C2×C4#46611yescharacteristic
185C2×C4#46611yescharacteristic
186C8—21no—
187C8—21no—
188C8—21no—
189C8—42no—
190C8—82no—
191C8—161no—
192C8—161no—
193C8#45512yes—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22#189311yescharacteristic
201C4—21no—
202C4—21no—
203C4—21no—
204C4—21no—
205C4—21no—
206C4—42no—
207C4—162no—
208C4#188511yescharacteristic
209C4#191511yescharacteristic
210C2—41no—
211C2—82no—
212C2#939311yescharacteristic
213C2#1306811yescharacteristic
214C2#1306911yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#532D4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11975C2semidirect
4. #11974C2semidirect
6. #7303C2semidirect
7. #7303C2semidirect

Maximal subgroups

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

subgroupindex
#334942
#287312
#119752
#119742
#86322
#73032