Finite Group Databasebeta

← #136558

Subgroups — #136558

Order 512.

257 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 86 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#18068C211yesmaximal, characteristic
2#12032C211yesmaximal, characteristic
3#11949C211yesmaximal, characteristic
4#11672C211yesmaximal, characteristic
5#9624C211yesmaximal, characteristic
6#7163C211yesmaximal, characteristic
7#7154C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9#2962—21no—
10#2162C2211yescharacteristic
11C2×C4×C16C2211yescharacteristic
12#1950—21no—
13#1591—21no—
14#1584—22no—
15C8×Q16—21no—
16#1567C2211yescharacteristic
17C2×#335C2211yescharacteristic
18C2×#335C2211yescharacteristic
19#1481—21no—
20C8×M16C2211yescharacteristic
21#1371—21no—
22#1371—21no—
23#1365C411yescharacteristic
24#1365C411yescharacteristic
25#1362C411yescharacteristic
26#1362C411yescharacteristic
27#1332—22no—
28#1301—21no—
29#533—21no—
30C22×C16—21no—
31#500C2×C411yescharacteristic
32#494C2×C411yescharacteristic
33#487—21no—
34#467—21no—
35#467Q812yes—
36#446—22no—
37#446—42no—
38C8×Q8—22no—
39C4×Q16—21no—
40C4×Q16D412yes—
41#431D411yescharacteristic
42C2×#108—41no—
43#424—41no—
44#423—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47#405—41no—
48C4×M16D411yescharacteristic
49C2×C4×C8C2311yescharacteristic
50C4⋊C16C2×C411yescharacteristic
51C4⋊C16C2×C411yescharacteristic
52C22⋊C16—21no—
53C4×C16—21no—
54C4×C16—21no—
55C4×C16—21no—
56#341—41no—
57#335—21no—
58#335—21no—
59#335—21no—
60#335—21no—
61#335C2×C411yescharacteristic
62#335C2×C411yescharacteristic
63#335D411yescharacteristic
64#335D411yescharacteristic
65Q8⋊C8—22no—
66Q8⋊C8—22no—
67#322—21no—
68C82—21no—
69C2×Q16—21no—
70C2×Q16C4⋊C412yes—
71C2×M16—41no—
72C2×M16D811yescharacteristic
73C2×M16D811yescharacteristic
74C22×C8—41no—
75C22×C8—41no—
76C22×C8C2×Q811yescharacteristic
77C22×C8#4111yescharacteristic
78C4⋊Q8—21no—
79C4⋊Q8—21no—
80#125—41no—
81C22⋊Q8—22no—
82C22⋊Q8—41no—
83C4×Q8—22no—
84C4×Q8—41no—
85#117—41no—
86C2×C42C2×D411yescharacteristic
87C2×C16—22no—
88C2×C16—21no—
89C2×C16—21no—
90C2×C16—41no—
91C2×C16—41no—
92#108—82no—
93#107—21no—
94#107C4⋊C412yes—
95C4⋊C8—21no—
96C4⋊C8—21no—
97C4⋊C8—21no—
98C4⋊C8—21no—
99C4⋊C8—21no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106#103—22no—
107#103—22no—
108C22⋊C8—21no—
109#97—21no—
110#97—42no—
111C4×C8—21no—
112C4×C8—21no—
113C4×C8—42no—
114C4×C8C22×C411yescharacteristic
115C4×C8C2×D411yescharacteristic
116C4×C8C2×D411yescharacteristic
117C4×C8#4111yescharacteristic
118C2×Q8—22no—
119C2×Q8—41no—
120C22×C4C22⋊Q811yescharacteristic
121C22×C4C2×D811yescharacteristic
122C22×C4#13711yescharacteristic
123Q16—42no—
124Q16—41no—
125Q16—41no—
126M16—21no—
127M16—21no—
128M16—21no—
129M16—81no—
130M16—81no—
131M16Q3212yes—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—81no—
143C2×C8—81no—
144C2×C8—81no—
145C2×C8—81no—
146C2×C8—81no—
147C2×C8C2×C4⋊C411yescharacteristic
148C2×C8C4×D411yescharacteristic
149C2×C8C22⋊Q811yescharacteristic
150C2×C8#12311yescharacteristic
151C4⋊C4—22no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C22⋊C4—42no—
158C42—42no—
159C42C4×D411yescharacteristic
160C42#12011yescharacteristic
161C42#13511yescharacteristic
162C42AGL(1,ℤ/8)11yescharacteristic
163C16—42no—
164C16—41no—
165C16—41no—
166C23#47411yescharacteristic
167Q8—42no—
168Q8—42no—
169Q8—161no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4#38511yescharacteristic
176C2×C4C4×D811yescharacteristic
177C2×C4#44011yescharacteristic
178C2×C4#44711yescharacteristic
179C2×C4#45211yescharacteristic
180C2×C4#47811yescharacteristic
181C2×C4C2×Q3211yescharacteristic
182C2×C4#50811yescharacteristic
183C2×C4#50911yescharacteristic
184C2×C4#51011yescharacteristic
185C8—21no—
186C8—21no—
187C8—21no—
188C8—22no—
189C8—41no—
190C8—81no—
191C8—81no—
192C8—81no—
193C8—81no—
194C8—81no—
195C8—161no—
196C8#44311yescharacteristic
197C8#44411yescharacteristic
198C22—21no—
199C22—21no—
200C22#186211yescharacteristic
201C22#222711yescharacteristic
202C22#222911yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—82no—
207C4—82no—
208C4#185811yescharacteristic
209C4#186611yescharacteristic
210C4C4×Q3211yescharacteristic
211C4#217511yescharacteristic
212C4#218611yescharacteristic
213C4#218811yescharacteristic
214C2—21no—
215C2—21no—
216C2#893011yescharacteristic
217C2#1293611yescharacteristic
218C2#1294211yescharacteristic
subgroupG/H
centerC2×C4#447
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
socleC22#1862
prime radicalC23#474
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
#180682
#120322
#119492
#116722
#96242
#71632
#71542