Finite Group Databasebeta

← #140333

Subgroups — #140333

Order 512.

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

normal subgroups only

This group has 208 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#11954C211yesmaximal, characteristic
2#11681C211yesmaximal, characteristic
3#11414C211yesmaximal, characteristic
4#9727C211yesmaximal, characteristic
5#9625C211yesmaximal, characteristic
6#7300C211yesmaximal, characteristic
7#7176C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9#2163C2211yescharacteristic
10C4×M32C2211yescharacteristic
11#1950—21no—
12C4×#108C2211yescharacteristic
13#1566C2211yescharacteristic
14C2×#335C2211yescharacteristic
15#1526—21no—
16#1485—21no—
17#1451C2211yescharacteristic
18#1364C411yescharacteristic
19#1364C411yescharacteristic
20#1362C411yescharacteristic
21#1362C411yescharacteristic
22#1334—21no—
23#1334—21no—
24#1313—21no—
25#1313—21no—
26#1301—21no—
27#533—21no—
28C2×M32—21no—
29#500C2×C411yescharacteristic
30#494C2×C411yescharacteristic
31#487—21no—
32#467—21no—
33#467—21no—
34C4×Q16—21no—
35C4×Q16—21no—
36#431D411yescharacteristic
37C2×#108—21no—
38C2×#108—41no—
39#423—21no—
40C2×C4⋊C8—21no—
41C4×M16—21no—
42C4×M16D411yescharacteristic
43C2×C4×C8C2311yescharacteristic
44C4⋊C16C2×C411yescharacteristic
45C4⋊C16C2×C411yescharacteristic
46C22⋊C16—21no—
47#346—21no—
48#346D411yescharacteristic
49#346D411yescharacteristic
50C4×C16—21no—
51#341—21no—
52#341—41no—
53#335—21no—
54#335C2×C411yescharacteristic
55#335C2×C411yescharacteristic
56#334—21no—
57#334—21no—
58#330—42no—
59Q8⋊C8—42no—
60#322—21no—
61#322—21no—
62C2×Q16—21no—
63C2×Q16—21no—
64C2×M16—21no—
65C2×M16—41no—
66C2×M16—41no—
67C22×C8—41no—
68C22×C8#4111yescharacteristic
69C22×C8#4111yescharacteristic
70C4⋊Q8—21no—
71C4⋊Q8—21no—
72#125—41no—
73C22⋊Q8—41no—
74C22⋊Q8—41no—
75C4×Q8—41no—
76C4×Q8—41no—
77#117—41no—
78C2×C42C2×D411yescharacteristic
79M32—42no—
80C2×C16—21no—
81C2×C16—21no—
82C2×C16—21no—
83C2×C16—21no—
84#108—22no—
85#108—41no—
86#108—82no—
87#107—21no—
88#107—21no—
89C4⋊C8—21no—
90C4⋊C8—21no—
91C4⋊C8—21no—
92C4⋊C8—41no—
93C4⋊C8—41no—
94C4⋊C8—41no—
95#103—41no—
96#103—41no—
97C22⋊C8—21no—
98#97—21no—
99#97—41no—
100C4×C8—21no—
101C4×C8—21no—
102C4×C8—41no—
103C4×C8C22×C411yescharacteristic
104C4×C8C2×D411yescharacteristic
105C4×C8#4111yescharacteristic
106C4×C8#4111yescharacteristic
107C2×Q8—41no—
108C2×Q8—41no—
109C22×C4#12411yescharacteristic
110C22×C4AGL(1,ℤ/8)11yescharacteristic
111C22×C4#13711yescharacteristic
112Q16—41no—
113Q16—41no—
114Q16—81no—
115M16—42no—
116M16—41no—
117M16—41no—
118M16—81no—
119M16—81no—
120M16—81no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—81no—
129C2×C8—81no—
130C2×C8—81no—
131C2×C8—81no—
132C2×C8#11711yescharacteristic
133C2×C8C4×D411yescharacteristic
134C2×C8#12311yescharacteristic
135C2×C8#12611yescharacteristic
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—81no—
141C4⋊C4—81no—
142C22⋊C4—81no—
143C42—81no—
144C42C4×D411yescharacteristic
145C42#12111yescharacteristic
146C42AGL(1,ℤ/8)11yescharacteristic
147C42AGL(1,ℤ/8)11yescharacteristic
148C16—22no—
149C16—41no—
150C16—41no—
151C16—41no—
152C23#48911yescharacteristic
153Q8—81no—
154Q8—81no—
155Q8—81no—
156Q8—81no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4#38811yescharacteristic
165C2×C4#44011yescharacteristic
166C2×C4#44211yescharacteristic
167C2×C4#46811yescharacteristic
168C2×C4#46911yescharacteristic
169C2×C4#49011yescharacteristic
170C8—21no—
171C8—21no—
172C8—21no—
173C8—81no—
174C8—81no—
175C8—82no—
176C8—81no—
177C8—81no—
178C8—161no—
179C8#44311yescharacteristic
180C8#44411yescharacteristic
181C22—21no—
182C22—21no—
183C22—21no—
184C22#193611yescharacteristic
185C4—21no—
186C4—21no—
187C4—21no—
188C4—21no—
189C4—161no—
190C4—161no—
191C4#192911yescharacteristic
192C4#193211yescharacteristic
193C4#218011yescharacteristic
194C4AGL(1,ℤ/16)11yescharacteristic
195C2—41no—
196C2#959211yescharacteristic
197C2#1315311yescharacteristic
198C2#1315711yescharacteristic
subgroupG/H
center, socleC22#1936
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#489
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ 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
#119542
#116812
#114142
#97272
#96252
#73002
#71762