Finite Group Databasebeta

← #153343

Subgroups — #153343

Order 512.

222 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C2×#1424C211yesmaximal, characteristic
2C2×C4⋊C32C211yesmaximal, characteristic
3C2×#1416C211yesmaximal, characteristic
4C2×#1416C211yesmaximal, characteristic
5#13236C211yesmaximal, characteristic
6#13165C211yesmaximal, characteristic
7#7471C211yesmaximal, characteristic
8C22×Q32C2211yescharacteristic
9C22×C32C2211yescharacteristic
10C22×C32C2211yescharacteristic
11C2×#366C2211yescharacteristic
12C2×#366C2211yescharacteristic
13C2×#366C2211yescharacteristic
14C2×#358—21no—
15C2×C4×C16C2211yescharacteristic
16#1969—21no—
17#1942—21no—
18#1424—22no—
19C4⋊C32—22no—
20#1416—22no—
21#1416—22no—
22#1380—21no—
23C22×Q16—21no—
24C2×Q32—21no—
25C2×Q32—21no—
26C2×Q32—21no—
27C2×Q32—21no—
28C2×Q32—21no—
29C2×Q32Q812yes—
30C22×C16D411yescharacteristic
31C22×C16C2311yescharacteristic
32C2×#107—21no—
33C2×#107—21no—
34C2×#107—21no—
35C2×#103—41no—
36C2×C4×C8D411yescharacteristic
37#391—41no—
38#389—41no—
39C2×C32—21no—
40C2×C32—21no—
41C2×C32—21no—
42C2×C32—21no—
43C2×C32—21no—
44C2×C32D412yes—
45#366—21no—
46#366—21no—
47#366—21no—
48#366—21no—
49#366—21no—
50#366D412yes—
51#358—42no—
52C4×C16—21no—
53C4×C16—21no—
54#340—41no—
55C22×Q8—41no—
56C2×Q16—22no—
57C2×Q16—41no—
58C2×Q16—41no—
59C2×Q16—41no—
60C2×Q16—41no—
61C2×Q16—41no—
62C22×C8D811yescharacteristic
63C22×C8C2×D411yescharacteristic
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—81no—
68C2×C42D811yescharacteristic
69Q32—22no—
70Q32—22no—
71Q32—41no—
72Q32—41no—
73C2×C16—21no—
74C2×C16—21no—
75C2×C16—21no—
76C2×C16C2×D411yescharacteristic
77C2×C16C2×D411yescharacteristic
78C2×C16C2×Q811yescharacteristic
79C2×C16#4111yescharacteristic
80C2×C16#4111yescharacteristic
81C2×C16#4111yescharacteristic
82#107—22no—
83#107—41no—
84#107—41no—
85#107—41no—
86#107—41no—
87#107—41no—
88#103—82no—
89C4×C8—21no—
90C4×C8—21no—
91#95—81no—
92#95—81no—
93#95—81no—
94C32—22no—
95C32—41no—
96C2×Q8—42no—
97C2×Q8—81no—
98C2×Q8—81no—
99C2×Q8—81no—
100C2×Q8—81no—
101C2×Q8—81no—
102C22×C4—81no—
103C22×C4—81no—
104C22×C4—81no—
105C22×C4—81no—
106C22×C4D1611yescharacteristic
107C22×C4D1611yescharacteristic
108C22×C4C2×D811yescharacteristic
109Q16—42no—
110Q16—42no—
111Q16—81no—
112Q16—81no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8#12111yescharacteristic
117C2×C8#12111yescharacteristic
118C2×C8C22⋊Q811yescharacteristic
119C2×C8#13511yescharacteristic
120C2×C8AGL(1,ℤ/8)11yescharacteristic
121C2×C8#13711yescharacteristic
122C4⋊C4—42no—
123C4⋊C4—81no—
124C4⋊C4—81no—
125C4⋊C4—81no—
126C4⋊C4—81no—
127C4⋊C4—81no—
128C4⋊C4—162no—
129C42—21no—
130C42—21no—
131C16—41no—
132C16#12111yescharacteristic
133C16C22⋊Q811yescharacteristic
134C16C22⋊Q811yescharacteristic
135C16#12611yescharacteristic
136C23C2×D1611yescharacteristic
137Q8—82no—
138Q8—82no—
139Q8—161no—
140Q8—161no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—82no—
147C2×C4—82no—
148C2×C4—161no—
149C2×C4—161no—
150C2×C4—161no—
151C2×C4—161no—
152C2×C4—161no—
153C2×C4—161no—
154C2×C4—161no—
155C2×C4—161no—
156C2×C4—161no—
157C2×C4—161no—
158C2×C4SD6412yes—
159C2×C4#45911yescharacteristic
160C2×C4#46611yescharacteristic
161C2×C4#47411yescharacteristic
162C2×C4#50811yescharacteristic
163C2×C4#50911yescharacteristic
164C2×C4#51011yescharacteristic
165C8—41no—
166C8#39611yescharacteristic
167C8#46411yescharacteristic
168C8#46811yescharacteristic
169C8#47811yescharacteristic
170C22#220611yescharacteristic
171C22#221511yescharacteristic
172C22#222411yescharacteristic
173C22C2×SD6411yescharacteristic
174C22#226211yescharacteristic
175C22#226411yescharacteristic
176C22#226411yescharacteristic
177C4—22no—
178C4—41no—
179C4—162no—
180C4—162no—
181C4—321no—
182C4—321no—
183C4#206711yescharacteristic
184C4#221111yescharacteristic
185C4#222011yescharacteristic
186C4#222811yescharacteristic
187C2#1347411yescharacteristic
188C2#1368711yescharacteristic
189C2#1369011yescharacteristic
190C2#1369611yescharacteristic
191C2#1369811yescharacteristic
192C2#1370411yescharacteristic
193C2#1370811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×D16
commutator, Frattini, cosocleC22×C16C23
LCSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G
LECSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#14242
C2×C4⋊C322
C2×#14162
#132362
#131652
#74712