Finite Group Databasebeta

← #140332

Subgroups — #140332

Order 512.

227 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 66 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#11951C211yesmaximal, characteristic
2#11680C211yesmaximal, characteristic
3#11414C211yesmaximal, characteristic, complemented
4#9721C211yesmaximal, characteristic
5#9635C211yesmaximal, characteristic, complemented
6#7302C211yesmaximal, characteristic, complemented
7#7175C211yesmaximal, characteristic, complemented
8#3169C2211yescharacteristic
9#2163C2211yescharacteristic
10C4×M32C2211yescharacteristic
11#1980—21no—
12C4×#106C2211yescharacteristic
13#1574—21no—
14#1566C2211yescharacteristic
15#1564C2211yescharacteristic
16#1502—21no—
17#1451C2211yescharacteristic
18#1364C411yescharacteristic, complemented
19#1364C411yescharacteristic, complemented
20#1361C411yescharacteristic, complemented
21#1361C411yescharacteristic, complemented
22#1333—21no—
23#1333—21no—
24#1313—21no—
25#1313—21no—
26#1299—21no—
27#532—21no—
28C2×M32—21no—
29#500C2×C411yescharacteristic
30#493C2×C411yescharacteristic
31#486—21no—
32#466—21no—
33#466—21no—
34C4×D8—21no—
35C4×D8—21no—
36#431—21no—
37#431D411yescharacteristic
38C2×#106—21no—
39#423—21no—
40C2×#104—41no—
41C2×#104—41no—
42C4×M16D411yescharacteristic
43C2×C4×C8C2311yescharacteristic
44C4×C4⋊C4—21no—
45C4⋊C16C2×C411yescharacteristic
46C4⋊C16C2×C411yescharacteristic
47C22⋊C16—21no—
48#346—21no—
49#346D411yescharacteristic
50#346D411yescharacteristic
51C4×C16—21no—
52#340—21no—
53#335—41no—
54#335C2×C411yescharacteristic
55#335C2×C411yescharacteristic
56#334—21no—
57#334—21no—
58#334—21no—
59#334—41no—
60#327—42no—
61#322—21no—
62#322—21no—
63C2×#41—41no—
64C2×D8—21no—
65C2×D8—21no—
66C2×M16—21no—
67C2×M16—41no—
68C22×C8#4111yescharacteristic
69C22×C8#4111yescharacteristic
70C4⋊Q8—21no—
71#127—21no—
72#124—41no—
73#121—41no—
74#121—41no—
75C4×D4—41no—
76C4×D4—41no—
77C2×C4⋊C4—41no—
78C2×C42—41no—
79C2×C42C2×D411yescharacteristic
80M32—42no—
81C2×C16—21no—
82C2×C16—21no—
83C2×C16—21no—
84C2×C16—21no—
85#107—21no—
86#107—21no—
87#106—22no—
88#106—41no—
89C4⋊C8—21no—
90C4⋊C8—21no—
91C4⋊C8—21no—
92C4⋊C8—41no—
93C4⋊C8—41no—
94#104—82no—
95#104—82no—
96#102—41no—
97#102—41no—
98C22⋊C8—21no—
99C22⋊C8—41no—
100#97—21no—
101C4×C8—21no—
102C4×C8—21no—
103C4×C8—41no—
104C4×C8C22×C411yescharacteristic
105C4×C8C2×D411yescharacteristic
106C4×C8#4111yescharacteristic
107C4×C8#4111yescharacteristic
108#95—41no—
109#41—81no—
110#41—81no—
111#41—81no—
112#41—81no—
113C2×Q8—41no—
114C2×D4—41no—
115C2×D4—41no—
116C2×D4—41no—
117C2×D4—81no—
118C22×C4—81no—
119C22×C4—81no—
120C22×C4#12411yescharacteristic
121C22×C4AGL(1,ℤ/8)11yescharacteristic
122C22×C4AGL(1,ℤ/8)11yescharacteristic
123D8—41no—
124D8—41no—
125D8—81no—
126M16—41no—
127M16—41no—
128M16—161no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—81no—
137C2×C8#11711yescharacteristic
138C2×C8C4×D411yescharacteristic
139C2×C8#12311yescharacteristic
140C2×C8#12611yescharacteristic
141C4⋊C4—42no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—81no—
145C22⋊C4—81no—
146C22⋊C4—81no—
147C42—81no—
148C42—81no—
149C42C4×D411yescharacteristic
150C42#12111yescharacteristic
151C42AGL(1,ℤ/8)11yescharacteristic
152C42#13711yescharacteristic
153C16—22no—
154C16—41no—
155C16—41no—
156C16—41no—
157C23—81no—
158C23#49011yescharacteristic
159Q8—81no—
160Q8—81no—
161D4—81no—
162D4—81no—
163D4—81no—
164D4—81no—
165D4—161no—
166D4—161no—
167D4—161no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—82no—
174C2×C4—81no—
175C2×C4—161no—
176C2×C4—161no—
177C2×C4—161no—
178C2×C4—161no—
179C2×C4#38811yescharacteristic
180C2×C4#44011yescharacteristic
181C2×C4#44211yescharacteristic
182C2×C4#46811yescharacteristic
183C2×C4#46911yescharacteristic
184C2×C4#48911yescharacteristic
185C8—21no—
186C8—21no—
187C8—21no—
188C8—81no—
189C8—81no—
190C8—81no—
191C8—81no—
192C8—161no—
193C8#44311yescharacteristic
194C8#44411yescharacteristic
195C22—21no—
196C22—21no—
197C22—21no—
198C22—161no—
199C22—161no—
200C22—161no—
201C22#193611yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—21no—
206C4—162no—
207C4—161no—
208C4#192911yescharacteristic
209C4#193211yescharacteristic
210C4AGL(1,ℤ/16)11yescharacteristic
211C4AGL(1,ℤ/16)11yescharacteristic
212C2—41no—
213C2—321no—
214C2#959211yescharacteristic
215C2#1315511yescharacteristic
216C2#1315611yescharacteristic
subgroupG/H
center, socleC22#1936
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#3169C22
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11414C2semidirect
5. #9635C2semidirect
6. #7302C2semidirect
7. #7175C2semidirect
18. #1364C4semidirect
19. #1364C4semidirect
20. #1361C4semidirect
21. #1361C4semidirect

Maximal subgroups

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

subgroupindex
#119512
#116802
#114142
#97212
#96352
#73022
#71752