Finite Group Databasebeta

← #145484

Subgroups — #145484

Order 512.

272 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 64 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#33793C211yesmaximal, characteristic
2#13279C211yesmaximal, characteristic, complemented
3#12018C211yesmaximal, characteristic, complemented
4#11986C211yesmaximal, characteristic
5#9799C211yesmaximal, characteristic, complemented
6#9724C211yesmaximal, characteristic
7#7298C211yesmaximal, characteristic, complemented
8#3169C2211yescharacteristic
9C2×#500C2211yescharacteristic
10#2254—21no—
11#2240—21no—
12#2233—22no—
13#2226—21no—
14#2226—21no—
15#2162C2211yescharacteristic
16#2150C2211yescharacteristic
17#1966—21no—
18C4×#108C2211yescharacteristic
19#1567C2211yescharacteristic
20#1546—21no—
21#1525—21no—
22#1484—21no—
23C8×M16C2211yescharacteristic
24#1387—22no—
25#1370—21no—
26#1367—21no—
27#1365—21no—
28#1361—21no—
29#532—21no—
30C2×C4⋊Q8—21no—
31C2×M32—21no—
32#500—21no—
33#500—21no—
34#500—21no—
35#500—21no—
36#500D411yescharacteristic
37#493D411yescharacteristic
38#486—21no—
39#466—21no—
40#466—21no—
41C4×D8—21no—
42C4×D8—21no—
43#431D411yescharacteristic
44C2×#108—21no—
45C2×#107—21no—
46C2×#107D412yes—
47#423—21no—
48C2×#101—42no—
49C4×M16—21no—
50C4×M16D411yescharacteristic
51C2×C4×C8C2311yescharacteristic
52#367—22no—
53#367—22no—
54C4⋊C16—21no—
55C4⋊C16—21no—
56C4⋊C16D411yescharacteristic
57C4⋊C16D411yescharacteristic
58#357—22no—
59#357—22no—
60C22⋊C16—21no—
61C4×C16—21no—
62#341—21no—
63#335—21no—
64#335—21no—
65#332—42no—
66#329—42no—
67D4⋊C8—42no—
68#322—21no—
69C82—21no—
70C2×#41—41no—
71C22×Q8—41no—
72C2×D8—21no—
73C2×D8—21no—
74C2×M16—21no—
75C2×M16—41no—
76C2×M16—41no—
77C22×C8C2×D411yescharacteristic
78C22×C8#4111yescharacteristic
79C4⋊Q8—21no—
80C4⋊Q8—41no—
81C4⋊Q8—41no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84#127—21no—
85#124—41no—
86#121—41no—
87#121—41no—
88C4×D4—41no—
89C4×D4—41no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—41no—
92C2×C42C2×D411yescharacteristic
93M32—42no—
94C2×C16—21no—
95C2×C16—21no—
96C2×C16—41no—
97#108—22no—
98#108—41no—
99#107—22no—
100#107—21no—
101#107—22no—
102#107—21no—
103#107—41no—
104#107—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8D812yes—
108#102—41no—
109#102—41no—
110#101—84no—
111C22⋊C8—21no—
112#97—21no—
113#97—41no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8—41no—
117C4×C8C2×D411yescharacteristic
118C4×C8C2×D411yescharacteristic
119C4×C8#4111yescharacteristic
120C4×C8#4111yescharacteristic
121#41—162no—
122C2×Q8—41no—
123C2×Q8—82no—
124C2×Q8—82no—
125C2×Q8—81no—
126C2×Q8—81no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—81no—
131C22×C4—42no—
132C22×C4—81no—
133C22×C4#12111yescharacteristic
134C22×C4#13511yescharacteristic
135C22×C4AGL(1,ℤ/8)11yescharacteristic
136D8—41no—
137D8—41no—
138D8—81no—
139M16—22no—
140M16—41no—
141M16—42no—
142M16—81no—
143M16—82no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—22no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—81no—
150C2×C8—81no—
151C2×C8#12011yescharacteristic
152C2×C8#12311yescharacteristic
153C2×C8#12311yescharacteristic
154C2×C8#12411yescharacteristic
155C4⋊C4—41no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—41no—
159C4⋊C4—81no—
160C4⋊C4—81no—
161C22⋊C4—81no—
162C22⋊C4—81no—
163C42#12111yescharacteristic
164C42#12111yescharacteristic
165C42C2×D811yescharacteristic
166C42AGL(1,ℤ/8)11yescharacteristic
167C16—41no—
168C16—41no—
169C16—81no—
170C23—81no—
171C23#46311yescharacteristic
172Q8—161no—
173Q8—162no—
174Q8—161no—
175Q8—161no—
176D4—81no—
177D4—81no—
178D4—81no—
179D4—81no—
180D4—161no—
181D4—162no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—84no—
188C2×C4—81no—
189C2×C4—82no—
190C2×C4—162no—
191C2×C4#39411yescharacteristic
192C2×C4#45911yescharacteristic
193C2×C4#45911yescharacteristic
194C2×C4#46311yescharacteristic
195C2×C4#46611yescharacteristic
196C2×C4#46911yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—42no—
201C8—42no—
202C8—82no—
203C8—161no—
204C8#45411yescharacteristic
205C8#45611yescharacteristic
206C22—21no—
207C22—21no—
208C22—21no—
209C22—161no—
210C22—161no—
211C22—161no—
212C22#205411yescharacteristic
213C4—21no—
214C4—21no—
215C4—21no—
216C4—161no—
217C4—162no—
218C4#203811yescharacteristic
219C4#204911yescharacteristic
220C4#221211yescharacteristic
221C4#221311yescharacteristic
222C4#222312yes—
223C2—41no—
224C2—321no—
225C2#1050311yescharacteristic
226C2#1344011yescharacteristic
227C2#1344511yescharacteristic
subgroupG/H
center, socleC22#2054
commutator, Frattini, cosocleC2×C4×C8C23
prime radical#3169C22
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13279C2semidirect
3. #12018C2semidirect
5. #9799C2semidirect
7. #7298C2semidirect

Maximal subgroups

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

subgroupindex
#337932
#132792
#120182
#119862
#97992
#97242
#72982