Finite Group Databasebeta

← #153315

Subgroups — #153315

Order 512.

260 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C22×SD64C211yesmaximal, characteristic
2#13628C211yesmaximal, characteristic, complemented
3C2×#1420C211yesmaximal, characteristic, complemented
4C2×#1419C211yesmaximal, characteristic
5#13245C211yesmaximal, characteristic
6#13244C211yesmaximal, characteristic, complemented
7#7472C211yesmaximal, characteristic, complemented
8C22×Q32C2211yescharacteristic
9C22×D16C2211yescharacteristic
10C2×SD64—22no—
11C2×SD64—22no—
12C2×SD64—22no—
13C2×M64C2211yescharacteristic
14C22×C32C2211yescharacteristic
15C2×#368C2211yescharacteristic
16C2×#368C2211yescharacteristic
17C2×#362—21no—
18C2×#361—21no—
19#2108C2211yescharacteristic
20#1979—21no—
21#1978—21no—
22C4⋊C32—22no—
23#1420—22no—
24#1419—22no—
25C22×Q16—21no—
26C22×D8—21no—
27C2×Q32—21no—
28C2×Q32—21no—
29C2×Q32—21no—
30C2×Q32—21no—
31C2×Q32—21no—
32C2×Q32D412yes—
33C2×D16—21no—
34C2×D16—21no—
35C2×D16—21no—
36C2×D16—21no—
37C2×D16—21no—
38C2×D16D412yes—
39C2×M32D411yescharacteristic
40C22×C16C2311yescharacteristic
41C2×#108—21no—
42C2×#108—21no—
43#419—41no—
44#418—41no—
45C2×#101—41no—
46C2×#100—41no—
47#405D411yescharacteristic
48SD64—28no—
49M64—21no—
50M64—21no—
51C2×C32—21no—
52C2×C32—21no—
53C2×C32—21no—
54C2×C32D412yes—
55#368—21no—
56#368—21no—
57#368—21no—
58#368—21no—
59#362—42no—
60#361—42no—
61#346—21no—
62C4×C16—21no—
63C22×Q8—41no—
64C22×D4—41no—
65C2×Q16—22no—
66C2×Q16—41no—
67C2×Q16—41no—
68C2×Q16—41no—
69C2×Q16—41no—
70C2×Q16—41no—
71C2×D8—22no—
72C2×D8—41no—
73C2×D8—41no—
74C2×D8—41no—
75C2×D8—41no—
76C2×D8—41no—
77C2×M16—41no—
78C2×M16—41no—
79C2×M16D811yescharacteristic
80C22×C8C2×D411yescharacteristic
81#117D811yescharacteristic
82Q32—22no—
83Q32—22no—
84Q32—41no—
85Q32—41no—
86D16—22no—
87D16—22no—
88D16—41no—
89D16—41no—
90M32—21no—
91M32—21no—
92C2×C16—21no—
93C2×C16C2×D411yescharacteristic
94C2×C16C2×D411yescharacteristic
95C2×C16C2×D411yescharacteristic
96C2×C16#4111yescharacteristic
97C2×C16#4111yescharacteristic
98C2×C16#4111yescharacteristic
99#108—41no—
100#108—41no—
101#108—41no—
102#108—41no—
103#103—82no—
104#102—82no—
105#101—82no—
106#100—82no—
107#97—21no—
108C4×C8—21no—
109C32—22no—
110C32—41no—
111C24—81no—
112C2×Q8—42no—
113C2×Q8—81no—
114C2×Q8—81no—
115C2×Q8—81no—
116C2×Q8—81no—
117C2×Q8—81no—
118C2×D4—42no—
119C2×D4—81no—
120C2×D4—81no—
121C2×D4—81no—
122C2×D4—81no—
123C2×D4—81no—
124C22×C4—81no—
125C22×C4C2×D811yescharacteristic
126Q16—42no—
127Q16—42no—
128Q16—81no—
129Q16—81no—
130D8—42no—
131D8—42no—
132D8—81no—
133D8—81no—
134M16—21no—
135M16—21no—
136M16—81no—
137M16—81no—
138M16—81no—
139M16—81no—
140C2×C8—21no—
141C2×C8—81no—
142C2×C8—81no—
143C2×C8#12111yescharacteristic
144C2×C8#12111yescharacteristic
145C2×C8#12111yescharacteristic
146C2×C8#13511yescharacteristic
147C2×C8AGL(1,ℤ/8)11yescharacteristic
148C2×C8AGL(1,ℤ/8)11yescharacteristic
149C4⋊C4D1612yes—
150C22⋊C4—21no—
151C42—21no—
152C16—41no—
153C16#12011yescharacteristic
154C16#12311yescharacteristic
155C16#12311yescharacteristic
156C16#12411yescharacteristic
157C23—82no—
158C23—161no—
159C23—161no—
160C23—322no—
161C23#50811yescharacteristic
162Q8—82no—
163Q8—82no—
164Q8—161no—
165Q8—161no—
166D4—82no—
167D4—82no—
168D4—161no—
169D4—161no—
170C2×C4—22no—
171C2×C4—82no—
172C2×C4—161no—
173C2×C4—161no—
174C2×C4#45911yescharacteristic
175C2×C4#45911yescharacteristic
176C2×C4#46611yescharacteristic
177C2×C4C2×D1611yescharacteristic
178C2×C4#50911yescharacteristic
179C2×C4#50911yescharacteristic
180C8—41no—
181C8—161no—
182C8—161no—
183C8#39411yescharacteristic
184C8#46311yescharacteristic
185C8#46311yescharacteristic
186C8#46911yescharacteristic
187C22—22no—
188C22—162no—
189C22—322no—
190C22—322no—
191C22—322no—
192C22#221011yescharacteristic
193C22#221011yescharacteristic
194C22#221511yescharacteristic
195C4—42no—
196C4—162no—
197C4#205411yescharacteristic
198C4#220611yescharacteristic
199C4#220611yescharacteristic
200C4#221811yescharacteristic
201C2—21no—
202C2—21no—
203C2—322no—
204C2#1344811yescharacteristic
205C2#1370212yes—
subgroupG/H
center, socleC22#2215
commutator, Frattini, cosocleC22×C16C23
prime radicalC22×D16C22
LCSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G
LECSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #13628C2semidirect
3. C2×#1420C2semidirect
6. #13244C2semidirect
7. #7472C2semidirect

Maximal subgroups

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

subgroupindex
C22×SD642
#136282
C2×#14202
C2×#14192
#132452
#132442
#74722