Finite Group Databasebeta

← #145487

Subgroups — #145487

Order 512.

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

normal subgroups only

This group has 253 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#33795C211yesmaximal, characteristic
2#13279C211yesmaximal, characteristic
3#12018C211yesmaximal, characteristic
4#11988C211yesmaximal, characteristic
5#9801C211yesmaximal, characteristic
6#9725C211yesmaximal, characteristic
7#7298C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9C2×#500C2211yescharacteristic
10#2254—21no—
11#2241—21no—
12#2235—22no—
13#2225—21no—
14#2225—21no—
15#2162C2211yescharacteristic
16#2150C2211yescharacteristic
17#1997—21no—
18C4×#108C2211yescharacteristic
19#1567C2211yescharacteristic
20#1549—21no—
21#1526—21no—
22#1485—21no—
23C8×M16C2211yescharacteristic
24#1387—22no—
25#1370—21no—
26#1367—21no—
27#1364—21no—
28#1362—21no—
29#533—21no—
30C2×C4⋊Q8—21no—
31C2×M32—21no—
32#500—21no—
33#500—21no—
34#500—21no—
35#500—21no—
36#500D411yescharacteristic
37#494D411yescharacteristic
38#487—21no—
39#467—21no—
40#467—21no—
41C4×Q16—21no—
42C4×Q16—21no—
43#431D411yescharacteristic
44C2×#108—21no—
45C2×#107—21no—
46C2×#107D412yes—
47#423—21no—
48C4×M16—21no—
49C4×M16D411yescharacteristic
50C2×C4×C8C2311yescharacteristic
51#367—22no—
52#367—22no—
53C4⋊C16—21no—
54C4⋊C16—21no—
55C4⋊C16D411yescharacteristic
56C4⋊C16D411yescharacteristic
57#358—22no—
58#358—22no—
59C22⋊C16—21no—
60C4×C16—21no—
61#343—42no—
62#341—21no—
63#335—21no—
64#335—21no—
65#332—42no—
66#330—42no—
67Q8⋊C8—42no—
68#322—21no—
69C82—21no—
70C22×Q8—41no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×M16—21no—
74C2×M16—41no—
75C2×M16—41no—
76C22×C8C2×D411yescharacteristic
77C22×C8#4111yescharacteristic
78C4⋊Q8—21no—
79C4⋊Q8—21no—
80C4⋊Q8—41no—
81C4⋊Q8—41no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84#125—41no—
85C22⋊Q8—41no—
86C22⋊Q8—41no—
87C4×Q8—41no—
88C4×Q8—41no—
89#117—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#103—41no—
109#103—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
120C2×Q8—41no—
121C2×Q8—41no—
122C2×Q8—81no—
123C2×Q8—81no—
124C2×Q8—162no—
125C22×C4—42no—
126C22×C4#12111yescharacteristic
127C22×C4#13511yescharacteristic
128C22×C4AGL(1,ℤ/8)11yescharacteristic
129Q16—41no—
130Q16—41no—
131Q16—81no—
132M16—22no—
133M16—41no—
134M16—42no—
135M16—81no—
136M16—82no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—22no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—81no—
143C2×C8—81no—
144C2×C8#12011yescharacteristic
145C2×C8#12311yescharacteristic
146C2×C8#12311yescharacteristic
147C2×C8#12411yescharacteristic
148C4⋊C4—41no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—81no—
155C4⋊C4—81no—
156C4⋊C4—81no—
157C4⋊C4—81no—
158C22⋊C4—81no—
159C42—81no—
160C42#12111yescharacteristic
161C42#12111yescharacteristic
162C42C2×D811yescharacteristic
163C42AGL(1,ℤ/8)11yescharacteristic
164C16—41no—
165C16—41no—
166C16—81no—
167C23#46311yescharacteristic
168Q8—81no—
169Q8—81no—
170Q8—81no—
171Q8—81no—
172Q8—162no—
173Q8—162no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4—82no—
182C2×C4—162no—
183C2×C4#39411yescharacteristic
184C2×C4#45911yescharacteristic
185C2×C4#45911yescharacteristic
186C2×C4#46311yescharacteristic
187C2×C4#46611yescharacteristic
188C2×C4#46911yescharacteristic
189C8—21no—
190C8—21no—
191C8—21no—
192C8—42no—
193C8—42no—
194C8—82no—
195C8—161no—
196C8#45411yescharacteristic
197C8#45611yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22#205411yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—161no—
206C4—162no—
207C4—161no—
208C4#203811yescharacteristic
209C4#204911yescharacteristic
210C4#221211yescharacteristic
211C4#221411yescharacteristic
212C4#222312yes—
213C2—41no—
214C2#1050311yescharacteristic
215C2#1344011yescharacteristic
216C2#1344611yescharacteristic
subgroupG/H
center, socleC22#2054
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#463
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#337952
#132792
#120182
#119882
#98012
#97252
#72982