Finite Group Databasebeta

← #145612

Subgroups — #145612

Order 512.

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

normal subgroups only

This group has 274 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#19633C211yesmaximal, characteristic
2#11988C211yesmaximal, characteristic
3#11971C211yesmaximal, characteristic
4#11941C211yesmaximal, characteristic
5#11650C211yesmaximal, characteristic
6#9634C211yesmaximal, characteristic
7#7212C211yesmaximal, characteristic
8#3209—21no—
9#3170C2211yescharacteristic
10#3165—21no—
11C2×#500C2211yescharacteristic
12#2162C2211yescharacteristic
13#2150C2211yescharacteristic
14#1723—22no—
15#1713—21no—
16#1707—21no—
17#1574—21no—
18#1567C2211yescharacteristic
19#1564C2211yescharacteristic
20C8×M16C2211yescharacteristic
21#1365—22no—
22#1364—22no—
23#1364—21no—
24#1362—21no—
25#1332—22no—
26#1299—21no—
27#533—21no—
28C2×C4⋊Q8—21no—
29C2×M32—21no—
30#501—42no—
31#500—22no—
32#500—21no—
33#500—21no—
34#500—21no—
35#500D412yes—
36#500D411yescharacteristic
37#498—22no—
38#494—22no—
39#494D411yescharacteristic
40#487—22no—
41#487—21no—
42#467—21no—
43#467—21no—
44#462—41no—
45#462—41no—
46#451—42no—
47C4×Q16—21no—
48C4×Q16—21no—
49#431—21no—
50#431D411yescharacteristic
51#428—42no—
52C2×#107—21no—
53C2×#107—21no—
54#423—21no—
55C4×M16D411yescharacteristic
56C2×C4×C8C2311yescharacteristic
57C4⋊C16—21no—
58C4⋊C16—21no—
59C4⋊C16—21no—
60C22⋊C16—21no—
61C4×C16—21no—
62#335—21no—
63#335—21no—
64#335—41no—
65#335D412yes—
66#334—21no—
67#334—41no—
68#322—21no—
69C82—21no—
70C22×Q8—41no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×Q16—42no—
74C2×Q16—42no—
75C2×M16—21no—
76C2×M16—41no—
77C22×C8#4111yescharacteristic
78C22×C8#4111yescharacteristic
79C4⋊Q8—22no—
80C4⋊Q8—21no—
81C4⋊Q8—21no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84C4⋊Q8—41no—
85#125—41no—
86C22⋊Q8—41no—
87C22⋊Q8—41no—
88C4×Q8—41no—
89C4×Q8—41no—
90#117—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C42C2×D411yescharacteristic
94M32—81no—
95C2×C16—41no—
96C2×C16—41no—
97#107—22no—
98#107—21no—
99#107—22no—
100#107—21no—
101#107—44no—
102#107—41no—
103#107—41no—
104#106—44no—
105C4⋊C8—22no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108C4⋊C8D811yescharacteristic
109C4⋊C8D811yescharacteristic
110#103—42no—
111#103—42no—
112#103—41no—
113#103—41no—
114C22⋊C8—21no—
115C22⋊C8—41no—
116#97SD1612yes—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—41no—
120C4×C8C2×D411yescharacteristic
121C4×C8C2×D411yescharacteristic
122C4×C8C2×D411yescharacteristic
123C4×C8#4111yescharacteristic
124C2×Q8—41no—
125C2×Q8—42no—
126C2×Q8—41no—
127C2×Q8—81no—
128C2×Q8—161no—
129C2×Q8—161no—
130C22×C4—81no—
131C22×C4#12411yescharacteristic
132C22×C4AGL(1,ℤ/8)11yescharacteristic
133C22×C4AGL(1,ℤ/8)11yescharacteristic
134Q16—41no—
135Q16—41no—
136Q16—81no—
137Q16—162no—
138Q16—162no—
139M16—24no—
140M16—161no—
141C2×C8—22no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—41no—
147C2×C8—42no—
148C2×C8—81no—
149C2×C8#12111yescharacteristic
150C2×C8#12111yescharacteristic
151C2×C8#12311yescharacteristic
152C2×C8#12311yescharacteristic
153C4⋊C4—41no—
154C4⋊C4—42no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—42no—
158C4⋊C4—41no—
159C4⋊C4—81no—
160C4⋊C4—81no—
161C4⋊C4—81no—
162C4⋊C4—81no—
163C22⋊C4—81no—
164C42—81no—
165C42#12011yescharacteristic
166C42#12111yescharacteristic
167C42C2×D811yescharacteristic
168C42C2×SD1611yescharacteristic
169C16—81no—
170C16—81no—
171C23#49011yescharacteristic
172Q8—81no—
173Q8—81no—
174Q8—81no—
175Q8—82no—
176Q8—81no—
177Q8—82no—
178Q8—162no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—82no—
187C2×C4—162no—
188C2×C4#39411yescharacteristic
189C2×C4#44711yescharacteristic
190C2×C4#45011yescharacteristic
191C2×C4#45911yescharacteristic
192C2×C4#46011yescharacteristic
193C2×C4#48611yescharacteristic
194C8—24no—
195C8—21no—
196C8—21no—
197C8—21no—
198C8—41no—
199C8—41no—
200C8—82no—
201C8—161no—
202C8#47111yescharacteristic
203C8#47311yescharacteristic
204C22—21no—
205C22—21no—
206C22—21no—
207C22#204111yescharacteristic
208C4—21no—
209C4—21no—
210C4—21no—
211C4—161no—
212C4—162no—
213C4—161no—
214C4#204911yescharacteristic
215C4#205211yescharacteristic
216C4#219211yescharacteristic
217C4#219411yescharacteristic
218C4#221412yes—
219C2—41no—
220C2#1055011yescharacteristic
221C2#1341111yescharacteristic
222C2#1341511yescharacteristic
subgroupG/H
center, socleC22#2041
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#490
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
#196332
#119882
#119712
#119412
#116502
#96342
#72122