Finite Group Databasebeta

← #145608

Subgroups — #145608

Order 512.

278 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 72 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#19636C211yesmaximal, characteristic
2#11983C211yesmaximal, characteristic
3#11971C211yesmaximal, characteristic
4C2×#1364C211yesmaximal, characteristic
5#11646C211yesmaximal, characteristic
6#9624C211yesmaximal, characteristic
7#7209C211yesmaximal, characteristic
8#3229—21no—
9#3170C2211yescharacteristic
10C2×#500C2211yescharacteristic
11#3139—21no—
12#2162C2211yescharacteristic
13C2×C4⋊C16C2211yescharacteristic
14#1950—21no—
15#1718—22no—
16#1710—21no—
17#1707—21no—
18#1567C2211yescharacteristic
19C2×#335C2211yescharacteristic
20C8×M16C2211yescharacteristic
21#1365—21no—
22#1364—22no—
23#1364—22no—
24#1362—21no—
25#1332—22no—
26#1301—21no—
27#533—21no—
28C2×C4⋊Q8—21no—
29C22×C16—21no—
30#500—21no—
31#500—21no—
32#500—21no—
33#500D412yes—
34#500D411yescharacteristic
35#498—22no—
36#494D411yescharacteristic
37#491—42no—
38#487—22no—
39#487—22no—
40#487—22no—
41#487—21no—
42#483—42no—
43#475—42no—
44#467—21no—
45#467—21no—
46C4×Q16—21no—
47C4×Q16—21no—
48#431D411yescharacteristic
49C2×#108—41no—
50C2×#107—21no—
51C2×#107—21no—
52#423—21no—
53C2×C4⋊C8—21no—
54#419—41no—
55#419—41no—
56C4×M16D411yescharacteristic
57C2×C4×C8C2311yescharacteristic
58C4⋊C16—21no—
59C4⋊C16—21no—
60C4⋊C16—21no—
61C22⋊C16—21no—
62C4×C16—21no—
63#341—41no—
64#335—21no—
65#335—21no—
66#335—21no—
67#335D412yes—
68#322—21no—
69C82—21no—
70C22×Q8—41no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×M16—41no—
74C2×M16D811yescharacteristic
75C2×M16D811yescharacteristic
76C22×C8—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
94C2×C16—41no—
95C2×C16—41no—
96C2×C16—81no—
97#108—82no—
98#107—22no—
99#107—21no—
100#107—22no—
101#107—21no—
102#107—41no—
103#107—41no—
104#106—44no—
105C4⋊C8—22no—
106C4⋊C8—21no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110#103—42no—
111#103—42no—
112#103—42no—
113#103—42no—
114#103—44no—
115#103—41no—
116#103—41no—
117C22⋊C8SD1612yes—
118#97—21no—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8C2×D411yescharacteristic
122C4×C8C2×D411yescharacteristic
123C4×C8C2×D411yescharacteristic
124C4×C8#4111yescharacteristic
125C2×Q8—41no—
126C2×Q8—42no—
127C2×Q8—41no—
128C2×Q8—81no—
129C2×Q8—81no—
130C2×Q8—81no—
131C2×Q8—81no—
132C2×Q8—81no—
133C22×C4—81no—
134C22×C4#12411yescharacteristic
135C22×C4C2×D811yescharacteristic
136C22×C4C2×SD1611yescharacteristic
137Q16—41no—
138Q16—41no—
139Q16—81no—
140M16—21no—
141M16—21no—
142M16—21no—
143M16—21no—
144M16—81no—
145M16—81no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—22no—
151C2×C8—41no—
152C2×C8—42no—
153C2×C8—81no—
154C2×C8—81no—
155C2×C8—81no—
156C2×C8#12111yescharacteristic
157C2×C8#12111yescharacteristic
158C2×C8#12311yescharacteristic
159C2×C8#12311yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—42no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—41no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C22⋊C4—81no—
171C42—81no—
172C42#12011yescharacteristic
173C42#12111yescharacteristic
174C42AGL(1,ℤ/8)11yescharacteristic
175C42AGL(1,ℤ/8)11yescharacteristic
176C16—81no—
177C16—81no—
178C23#48611yescharacteristic
179Q8—82no—
180Q8—81no—
181Q8—81no—
182Q8—82no—
183Q8—82no—
184Q8—161no—
185Q8—161no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—82no—
192C2×C4—161no—
193C2×C4—161no—
194C2×C4#39411yescharacteristic
195C2×C4#44711yescharacteristic
196C2×C4#45011yescharacteristic
197C2×C4#45911yescharacteristic
198C2×C4#46011yescharacteristic
199C2×C4#49011yescharacteristic
200C2×C4#50811yescharacteristic
201C2×C4#50811yescharacteristic
202C2×C4#50911yescharacteristic
203C2×C4#51011yescharacteristic
204C8—21no—
205C8—21no—
206C8—21no—
207C8—42no—
208C8—41no—
209C8—41no—
210C8—82no—
211C8—161no—
212C8#47111yescharacteristic
213C8#47311yescharacteristic
214C22—21no—
215C22—21no—
216C22#204111yescharacteristic
217C22#224211yescharacteristic
218C22#224311yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—161no—
223C4—162no—
224C4—161no—
225C4#204911yescharacteristic
226C4#205211yescharacteristic
227C4#218611yescharacteristic
228C4#218911yescharacteristic
229C4#221011yescharacteristic
230C4#221111yescharacteristic
231C2—21no—
232C2—21no—
233C2#1055011yescharacteristic
234C2#1340811yescharacteristic
235C2#1341811yescharacteristic
subgroupG/H
center, socleC22#2041
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#486
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
#196362
#119832
#119712
C2×#13642
#116462
#96242
#72092