Finite Group Databasebeta

← #152540

Subgroups — #152540

Order 512.

304 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 80 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×#2264C211yesmaximal, characteristic
2#33465C211yesmaximal, characteristic
3#13625C211yesmaximal, characteristic
4C2×#1420C211yesmaximal, characteristic, complemented
5#13244C211yesmaximal, characteristic, complemented
6#7479C211yesmaximal, characteristic, complemented
7#7469C211yesmaximal, characteristic, complemented
8C2×#508C2211yescharacteristic
9C22×Q32C2211yescharacteristic
10#2957—21no—
11#2264—22no—
12#2264C414yescomplemented
13C2×Q64—21no—
14C2×SD64—21no—
15C2×M64C2211yescharacteristic
16C2×M64C2211yescharacteristic
17#2179—22no—
18#2178—22no—
19C2×#368C2211yescharacteristic
20C2×#368C2211yescharacteristic
21C2×#362—21no—
22#2143—21no—
23#2108C2211yescharacteristic
24#1979—21no—
25#1420—21no—
26#1420—22no—
27#1419—21no—
28C2×#135—21no—
29C22×Q16—21no—
30#568—41no—
31#508—21no—
32#508—21no—
33#508C2×C412yes—
34#508D412yes—
35C2×Q32—21no—
36C2×Q32—21no—
37C2×Q32—21no—
38C2×Q32—21no—
39C2×Q32—21no—
40C2×Q32C2×C412yes—
41C2×Q32D411yescharacteristic
42C2×SD32—21no—
43C2×D16D411yescharacteristic
44C2×M32D411yescharacteristic
45C22×C16C2311yescharacteristic
46#444—42no—
47#443—24no—
48C2×#108—21no—
49C2×#108—21no—
50#421—41no—
51#419—41no—
52C2×#101—41no—
53#405D411yescharacteristic
54Q64—24no—
55SD64—24no—
56M64—21no—
57M64—21no—
58M64—21no—
59M64C2×C412yes—
60C2×C32—21no—
61C2×C32—21no—
62#368—21no—
63#368—21no—
64#368—21no—
65#368D412yes—
66#362—42no—
67#359—24no—
68#346—21no—
69C4×C16—21no—
70C2×#41—41no—
71C22×Q8—41no—
72#135—22no—
73#135—22no—
74#135—41no—
75#135—41no—
76C2×Q16—22no—
77C2×Q16—21no—
78C2×Q16—41no—
79C2×Q16—41no—
80C2×Q16—41no—
81C2×Q16—41no—
82C2×Q16—41no—
83C2×SD16—41no—
84C2×D8—21no—
85#131—44no—
86#131—82no—
87C2×M16—41no—
88C2×M16—41no—
89C2×M16—82no—
90C2×M16D811yescharacteristic
91C22×C8C2×D411yescharacteristic
92#117D811yescharacteristic
93Q32—22no—
94Q32—22no—
95Q32—21no—
96Q32—41no—
97Q32—41no—
98Q32C22⋊C412yes—
99SD32—22no—
100SD32—22no—
101D16—21no—
102D16C22⋊C412yes—
103M32—21no—
104M32—21no—
105C2×C16—21no—
106C2×C16C22×C411yescharacteristic
107C2×C16C2×D411yescharacteristic
108C2×C16C2×D411yescharacteristic
109C2×C16C2×D411yescharacteristic
110C2×C16#4111yescharacteristic
111C2×C16#4111yescharacteristic
112#108—22no—
113#108—41no—
114#108—41no—
115#108—41no—
116#104—44no—
117#103—82no—
118#101—82no—
119#97—21no—
120C4×C8—21no—
121C32—22no—
122C32—41no—
123#41—42no—
124#41—42no—
125#41—81no—
126#41—81no—
127C2×Q8—42no—
128C2×Q8—41no—
129C2×Q8—81no—
130C2×Q8—81no—
131C2×Q8—81no—
132C2×Q8—81no—
133C2×Q8—81no—
134C2×D4—41no—
135C2×D4—81no—
136C22×C4—81no—
137C22×C4—81no—
138C22×C4C2×D811yescharacteristic
139Q16—22no—
140Q16—42no—
141Q16—42no—
142Q16—41no—
143Q16—81no—
144Q16—81no—
145SD16—42no—
146SD16—42no—
147D8—22no—
148D8—41no—
149M16—21no—
150M16—42no—
151M16—84no—
152M16—81no—
153M16—81no—
154M16—81no—
155M16D1612yes—
156C2×C8—21no—
157C2×C8—84no—
158C2×C8—81no—
159C2×C8—81no—
160C2×C8C4×D411yescharacteristic
161C2×C8#12011yescharacteristic
162C2×C8#12111yescharacteristic
163C2×C8#13511yescharacteristic
164C2×C8AGL(1,ℤ/8)11yescharacteristic
165C2×C8AGL(1,ℤ/8)11yescharacteristic
166C4⋊C4—21no—
167C22⋊C4—21no—
168C42—21no—
169C16—41no—
170C16C2×C22⋊C411yescharacteristic
171C16C4×D411yescharacteristic
172C16#12111yescharacteristic
173C16#12311yescharacteristic
174C23—81no—
175C23#50811yescharacteristic
176Q8—42no—
177Q8—81no—
178Q8—82no—
179Q8—82no—
180Q8—161no—
181Q8—161no—
182D4—42no—
183D4—82no—
184D4—82no—
185D4—81no—
186C2×C4—41no—
187C2×C4—81no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—161no—
192C2×C4—161no—
193C2×C4C4×D811yescharacteristic
194C2×C4#44711yescharacteristic
195C2×C4#45911yescharacteristic
196C2×C4C2×D1611yescharacteristic
197C2×C4#50911yescharacteristic
198C2×C4#50911yescharacteristic
199C8—22no—
200C8—82no—
201C8—161no—
202C8#38611yescharacteristic
203C8#44211yescharacteristic
204C8#44911yescharacteristic
205C8#46311yescharacteristic
206C22—41no—
207C22—82no—
208C22—321no—
209C22C4×D1611yescharacteristic
210C22#218611yescharacteristic
211C22#221011yescharacteristic
212C4—81no—
213C4—82no—
214C4—162no—
215C4#187911yescharacteristic
216C4#217711yescharacteristic
217C4#218411yescharacteristic
218C4#220611yescharacteristic
219C2—21no—
220C2—21no—
221C2—21no—
222C2—162no—
223C2#1301211yescharacteristic
subgroupG/H
center, socleC2#13012
commutatorC2×C16C22×C4
Frattini, cosocleC22×C16C23
prime radicalC2×#508C22
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ 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
4. C2×#1420C2semidirect
5. #13244C2semidirect
6. #7479C2semidirect
7. #7469C2semidirect
12. #2264C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22642
#334652
#136252
C2×#14202
#132442
#74792
#74692