Finite Group Databasebeta

← #145485

Subgroups — #145485

Order 512.

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

normal subgroups only

This group has 264 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#33792C211yesmaximal, characteristic
2#13279C211yesmaximal, characteristic, complemented
3#12018C211yesmaximal, characteristic, complemented
4#11986C211yesmaximal, characteristic
5#9801C211yesmaximal, characteristic, complemented
6#9724C211yesmaximal, characteristic
7#7298C211yesmaximal, characteristic, complemented
8#3169C2211yescharacteristic
9C2×#500C2211yescharacteristic
10#2252—21no—
11#2240—21no—
12#2232—22no—
13#2224—21no—
14#2224—21no—
15#2162C2211yescharacteristic
16#2150C2211yescharacteristic
17#1997—21no—
18C4×#108C2211yescharacteristic
19#1567C2211yescharacteristic
20#1549—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—
48C4×M16—21no—
49C4×M16D411yescharacteristic
50C2×C4×C8C2311yescharacteristic
51#366—22no—
52#366—22no—
53C4⋊C16—21no—
54C4⋊C16—21no—
55C4⋊C16D411yescharacteristic
56C4⋊C16D411yescharacteristic
57#357—22no—
58#357—22no—
59C22⋊C16—21no—
60C4×C16—21no—
61#343—42no—
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—
110C22⋊C8—21no—
111#97—21no—
112#97—41no—
113C4×C8—21no—
114C4×C8—21no—
115C4×C8—41no—
116C4×C8C2×D411yescharacteristic
117C4×C8C2×D411yescharacteristic
118C4×C8#4111yescharacteristic
119C4×C8#4111yescharacteristic
120#41—162no—
121C2×Q8—41no—
122C2×Q8—81no—
123C2×Q8—81no—
124C2×Q8—162no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—81no—
129C22×C4—42no—
130C22×C4—81no—
131C22×C4#12111yescharacteristic
132C22×C4#13511yescharacteristic
133C22×C4AGL(1,ℤ/8)11yescharacteristic
134D8—41no—
135D8—41no—
136D8—81no—
137M16—22no—
138M16—41no—
139M16—42no—
140M16—81no—
141M16—82no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—22no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—81no—
148C2×C8—81no—
149C2×C8#12011yescharacteristic
150C2×C8#12311yescharacteristic
151C2×C8#12311yescharacteristic
152C2×C8#12411yescharacteristic
153C4⋊C4—41no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—41no—
157C4⋊C4—81no—
158C4⋊C4—81no—
159C22⋊C4—81no—
160C22⋊C4—81no—
161C42#12111yescharacteristic
162C42#12111yescharacteristic
163C42C2×D811yescharacteristic
164C42AGL(1,ℤ/8)11yescharacteristic
165C16—41no—
166C16—41no—
167C16—81no—
168C23—81no—
169C23#46311yescharacteristic
170Q8—161no—
171Q8—162no—
172Q8—162no—
173D4—81no—
174D4—81no—
175D4—81no—
176D4—81no—
177D4—161no—
178D4—162no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—81no—
185C2×C4—82no—
186C2×C4—162no—
187C2×C4—162no—
188C2×C4#39411yescharacteristic
189C2×C4#45911yescharacteristic
190C2×C4#45911yescharacteristic
191C2×C4#46311yescharacteristic
192C2×C4#46611yescharacteristic
193C2×C4#46911yescharacteristic
194C8—21no—
195C8—21no—
196C8—21no—
197C8—42no—
198C8—42no—
199C8—82no—
200C8—161no—
201C8#45411yescharacteristic
202C8#45611yescharacteristic
203C22—21no—
204C22—21no—
205C22—21no—
206C22—161no—
207C22—161no—
208C22—161no—
209C22#205411yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—161no—
214C4—162no—
215C4#203811yescharacteristic
216C4#204911yescharacteristic
217C4#221211yescharacteristic
218C4#221411yescharacteristic
219C4#222112yes—
220C2—41no—
221C2—321no—
222C2#1050311yescharacteristic
223C2#1343911yescharacteristic
224C2#1344611yescharacteristic
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. #9801C2semidirect
7. #7298C2semidirect

Maximal subgroups

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

subgroupindex
#337922
#132792
#120182
#119862
#98012
#97242
#72982