Finite Group Databasebeta

← #140203

Subgroups — #140203

Order 512.

236 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 84 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#13197C211yesmaximal, characteristic
2#13164C211yesmaximal, characteristic, complemented
3#12642C211yesmaximal, characteristic
4#11604C211yesmaximal, characteristic
5#11406C211yesmaximal, characteristic, complemented
6#7181C211yesmaximal, characteristic, complemented
7#7161C211yesmaximal, characteristic, complemented
8C2×#443C2211yescharacteristic
9#2153—21no—
10C2×#364C2211yescharacteristic
11C2×#364C2211yescharacteristic
12C2×#359C411yescharacteristic
13#2138C411yescharacteristic
14#2117—21no—
15#2115—21no—
16C4×M32C2211yescharacteristic
17C2×#346C2211yescharacteristic
18#1772C2211yescharacteristic
19#1450C2211yescharacteristic
20#1382C411yescharacteristic
21#1381C411yescharacteristic
22#1378—21no—
23#1336—22no—
24#1313—22no—
25C2×#135C2×C411yescharacteristic
26C2×#131—21no—
27C2×M32—21no—
28C2×M32—21no—
29C2×M32D411yescharacteristic
30C2×M32D411yescharacteristic
31C22×C16C2×C411yescharacteristic
32C22×C16C2×C411yescharacteristic
33#443—21no—
34#443—21no—
35#443—21no—
36#443—21no—
37#431—21no—
38C2×#108C2×C411yescharacteristic
39C2×#108C2×C411yescharacteristic
40#427C2×C411yescharacteristic
41C2×#104—21no—
42#405—21no—
43C2×C4×C8C2311yescharacteristic
44#364—21no—
45#364—21no—
46#364—21no—
47#364—21no—
48#364—42no—
49#359—22no—
50#358—21no—
51#357—21no—
52#350—42no—
53#349—42no—
54#346—21no—
55#346—21no—
56#346—21no—
57C4×C16—21no—
58#339—21no—
59#322D412yes—
60C82—21no—
61C2×#41—21no—
62#135—21no—
63#135—21no—
64#135—21no—
65#135—21no—
66C2×Q16M1611yescharacteristic
67C2×SD16—21no—
68C2×D8M1611yescharacteristic
69#131—41no—
70#131—41no—
71#131—41no—
72#131—41no—
73C2×M16—21no—
74C2×M16—21no—
75C2×M16—41no—
76C22×C8—41no—
77C22×C8C22×C411yescharacteristic
78C22×C8C2×D411yescharacteristic
79#117—21no—
80C2×C42#4111yescharacteristic
81M32—21no—
82M32—21no—
83M32—41no—
84M32—41no—
85M32—41no—
86M32—41no—
87M32D812yes—
88M32SD1612yes—
89C2×C16—21no—
90C2×C16—21no—
91C2×C16—21no—
92C2×C16—21no—
93C2×C16—21no—
94C2×C16—21no—
95C2×C16—21no—
96C2×C16—21no—
97C2×C16—41no—
98C2×C16—41no—
99#108—21no—
100#108—21no—
101#108—21no—
102#108M1612yes—
103#107—21no—
104#106—21no—
105C4⋊C8—41no—
106#104—41no—
107#104—41no—
108C22⋊C8—41no—
109#97—41no—
110C4×C8—22no—
111C4×C8—41no—
112C4×C8C2×D411yescharacteristic
113C4×C8#4111yescharacteristic
114C4×C8#4111yescharacteristic
115C4×C8#4111yescharacteristic
116#41—41no—
117#41—41no—
118#41—41no—
119#41—41no—
120C2×Q8—21no—
121C2×D4—21no—
122C2×D4—41no—
123C22×C4—21no—
124C22×C4—41no—
125C22×C4C4×D411yescharacteristic
126Q16—21no—
127Q16—21no—
128SD16—41no—
129SD16—41no—
130D8—21no—
131D8—21no—
132M16—22no—
133M16—41no—
134M16—41no—
135M16—41no—
136M16—81no—
137M16—81no—
138C2×C8—41no—
139C2×C8—42no—
140C2×C8—41no—
141C2×C8—81no—
142C2×C8—81no—
143C2×C8#11711yescharacteristic
144C2×C8C4×D411yescharacteristic
145C2×C8#12111yescharacteristic
146C2×C8#12411yescharacteristic
147C2×C8C2×M1611yescharacteristic
148C2×C8C2×M1611yescharacteristic
149C2×C8#13111yescharacteristic
150C2×C8#13111yescharacteristic
151C2×C8C2×D811yescharacteristic
152C2×C8C2×SD1611yescharacteristic
153C2×C8#13511yescharacteristic
154C2×C8#13511yescharacteristic
155C4⋊C4—41no—
156C22⋊C4—41no—
157C42—21no—
158C42—41no—
159C42#12311yescharacteristic
160C42#12611yescharacteristic
161C16—22no—
162C16—22no—
163C16—41no—
164C16—41no—
165C16—81no—
166C16—81no—
167C23—41no—
168C23#44411yescharacteristic
169Q8—41no—
170Q8—41no—
171D4—41no—
172D4—41no—
173D4—81no—
174D4—81no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4#38811yescharacteristic
183C2×C4#43511yescharacteristic
184C2×C4#43511yescharacteristic
185C2×C4C4×D811yescharacteristic
186C2×C4C4×SD1611yescharacteristic
187C2×C4#44411yescharacteristic
188C8—42no—
189C8—82no—
190C8—81no—
191C8#43211yescharacteristic
192C8#43311yescharacteristic
193C8#43511yescharacteristic
194C8#43611yescharacteristic
195C8#46511yescharacteristic
196C8#46611yescharacteristic
197C8#48611yescharacteristic
198C8#48811yescharacteristic
199C22—41no—
200C22—81no—
201C22—161no—
202C22#158111yescharacteristic
203C22#158311yescharacteristic
204C22#193811yescharacteristic
205C4—41no—
206C4—41no—
207C4—81no—
208C4—161no—
209C4#158111yescharacteristic
210C4#158211yescharacteristic
211C4#192411yescharacteristic
212C4#193411yescharacteristic
213C2—21no—
214C2—21no—
215C2—21no—
216C2—161no—
217C2#955811yescharacteristic
subgroupG/H
centerC4#1934
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
socleC2#9558
prime radicalC2×#135C2×C4
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C8 ◁ C4×M32 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13164C2semidirect
5. #11406C2semidirect
6. #7181C2semidirect
7. #7161C2semidirect

Maximal subgroups

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

subgroupindex
#131972
#131642
#126422
#116042
#114062
#71812
#71612