Finite Group Databasebeta

← #499689

Subgroups — #499689

Order 512.

510 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 130 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#21736C211yesmaximal, characteristic
2#21691C211yesmaximal, characteristic
3#13543C212yesmaximal
4#13488C212yesmaximal
5#13415C212yesmaximal
6#13368C212yesmaximal
7#12648C214yesmaximal
8#12595C211yesmaximal, characteristic
9#3162C2211yescharacteristic
10C2×#494C2211yescharacteristic
11#2961—21no—
12C2×#441—21no—
13C2×#441C2212yes—
14C4×C4⋊Q8—21no—
15C2×#362C2214yes—
16#2141C2214yes—
17C2×#347C2212yes—
18#1994—21no—
19#1971—21no—
20#1971C2212yes—
21#1956—21no—
22#1946—21no—
23#1945C2212yes—
24#1849C2211yescharacteristic
25#1844C2211yescharacteristic
26#1773C2212yes—
27#1772—21no—
28C4×#97C2211yescharacteristic
29#1386C2214yes—
30#1383C2214yes—
31#1377C2212yes—
32#1376C2212yes—
33C22×Q16—21no—
34C22×Q16C2312yes—
35#533—21no—
36C2×C4⋊Q8—21no—
37C2×C4×Q8—22no—
38C2×M32C2314yes—
39#501—24no—
40#494—24no—
41#446—44no—
42#441—28no—
43#441—44no—
44C2×#108—21no—
45C2×#108C2312yes—
46#427—21no—
47#427C2312yes—
48C2×#106—21no—
49C2×#106C2312yes—
50#424—22no—
51#423—21no—
52C2×#103—22no—
53C2×#103—22no—
54#405—22no—
55C2×#97D411yescharacteristic
56C2×#97D411yescharacteristic
57C2×#97C2311yescharacteristic
58C2×#97C2311yescharacteristic
59C2×C4×C8D411yescharacteristic
60C2×C4×C8C2311yescharacteristic
61#391—22no—
62#389—42no—
63#384—21no—
64C4×C4⋊C4—42no—
65C43D411yescharacteristic
66#362—28no—
67#358—28no—
68#347—24no—
69#339—22no—
70#339—22no—
71C22×Q8—22no—
72C2×Q16—24no—
73C2×Q16—24no—
74C2×Q16—24no—
75C2×Q16—22no—
76C2×Q16—42no—
77C2×Q16—42no—
78C2×Q16#4114yes—
79C2×M16—22no—
80C22×C8C2×D411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C22×C8C2411yescharacteristic
84C4⋊Q8—22no—
85C4⋊Q8—24no—
86C4⋊Q8—42no—
87#125—22no—
88C22⋊Q8—42no—
89C4×Q8—42no—
90C4×Q8—44no—
91C4×Q8—44no—
92#117—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C42—22no—
98C2×C42—42no—
99C2×C42C2×D411yescharacteristic
100C2×C42C2×D411yescharacteristic
101C2×C42C2×D411yescharacteristic
102M32—24no—
103M32—24no—
104C2×C16—24no—
105#108—24no—
106#108—42no—
107#107—22no—
108#107#4114yes—
109#106—22no—
110#106—22no—
111#106—24no—
112#106—42no—
113C4⋊C8—42no—
114C4⋊C8—44no—
115#103—44no—
116#103—44no—
117#97—22no—
118#97—22no—
119#97—22no—
120#97—42no—
121#97#4114yes—
122C4×C8—22no—
123C4×C8—22no—
124C4×C8—42no—
125#95—44no—
126C2×Q8—24no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—42no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4AGL(1,ℤ/8)12yes—
138C22×C4C22×D411yescharacteristic
139Q16—28no—
140Q16—48no—
141Q16—44no—
142M16—42no—
143M16—42no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—42no—
151C2×C8#12412yes—
152C2×C8AGL(1,ℤ/8)12yes—
153C2×C8AGL(1,ℤ/8)12yes—
154C2×C8C2×#4112yes—
155C2×C8C2×#4112yes—
156C2×C8C2×#4112yes—
157C4⋊C4—24no—
158C4⋊C4—48no—
159C4⋊C4—44no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C22⋊C4—42no—
166C42—21no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—44no—
171C42—42no—
172C42—84no—
173C42#12412yes—
174C42#13712yes—
175C16—44no—
176C23C2×AGL(1,ℤ/8)11yescharacteristic
177Q8—44no—
178Q8—44no—
179Q8—82no—
180Q8—82no—
181Q8—162no—
182C2×C4—24no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—84no—
193C2×C4—82no—
194C2×C4#52511yescharacteristic
195C2×C4#52511yescharacteristic
196C2×C4C2×#12411yescharacteristic
197C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
198C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
199C2×C4C2×#13711yescharacteristic
200C8—22no—
201C8—22no—
202C8—22no—
203C8—82no—
204C8#52914yes—
205C22—22no—
206C22#310911yescharacteristic
207C22#311011yescharacteristic
208C22#313811yescharacteristic
209C4—24no—
210C4—42no—
211C4—42no—
212C4—84no—
213C4—84no—
214C4—162no—
215C4#239811yescharacteristic
216C4#310911yescharacteristic
217C4#311211yescharacteristic
218C4#314111yescharacteristic
219C2—21no—
220C2—21no—
221C2#2382211yescharacteristic
222C2#3368912yes—
subgroupG/H
center, socleC22#3109
commutator, Frattini, cosocleC22×C8C24
prime radicalC23C2×AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#217362
#216912
#135432
#134882
#134152
#133682
#126482
#125952