Finite Group Databasebeta

← #139156

Subgroups — #139156

Order 512.

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

normal subgroups only

This group has 315 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
1#33494C211yesmaximal, characteristic, complemented
2#19337C211yesmaximal, characteristic
3#11858C211yesmaximal, characteristic, complemented
4#11652C211yesmaximal, characteristic, complemented
5#8680C211yesmaximal, characteristic, complemented
6#7295C211yesmaximal, characteristic, complemented
7#7221C211yesmaximal, characteristic, complemented
8#3253—21no—
9#3169C2211yescharacteristic
10#2931C2211yescharacteristic
11#2168—22no—
12#2168—22no—
13#2163C2211yescharacteristic
14#2163C2211yescharacteristic
15#2113—22no—
16#1835C2211yescharacteristic
17#1695C412yescomplemented
18#1687C412yescomplemented
19#1677—22no—
20#1566C2211yescharacteristic
21#1564C2211yescharacteristic
22#1560—21no—
23#1517—21no—
24#1347—22no—
25#1333—22no—
26#1286—21no—
27C2×#131—21no—
28#532—21no—
29#532D411yescharacteristic
30C2×M32—22no—
31C22×C16—22no—
32#500C2×C411yescharacteristic
33#493C2×C411yescharacteristic
34#486—21no—
35#476—22no—
36#476—41no—
37#469—42no—
38#466—21no—
39#466—21no—
40#459—22no—
41#459—41no—
42C4×D8—21no—
43C4×D8—21no—
44#436—21no—
45#436C2×C412yes—
46C8×D4—21no—
47C8×D4D412yes—
48#431—21no—
49#423—21no—
50#423D411yescharacteristic
51#408—21no—
52C2×C4×C8C2311yescharacteristic
53#382—21no—
54C4⋊C16—21no—
55C4⋊C16—21no—
56C22⋊C16—24no—
57C22⋊C16—21no—
58C22⋊C16D412yes—
59#346—21no—
60#346D412yes—
61#340—21no—
62#340—21no—
63#335—21no—
64#334—21no—
65#334—21no—
66#334C2×C412yes—
67#333—22no—
68#331—22no—
69#327—42no—
70C2×#41—21no—
71C2×#41—41no—
72C2×D8—21no—
73C2×D8—21no—
74C2×D8—42no—
75#131—81no—
76#131—81no—
77C2×M16—21no—
78C2×M16—22no—
79C2×M16—41no—
80C22×C8—22no—
81C22×C8C2×D411yescharacteristic
82C22×C8#4111yescharacteristic
83C4⋊Q8—21no—
84C4⋊Q8SD1611yescharacteristic
85#127—21no—
86#127SD1611yescharacteristic
87#124—21no—
88#124—41no—
89#121—21no—
90#121—41no—
91#121—41no—
92#121D812yes—
93C4×D4—21no—
94C4×D4—41no—
95C4×D4—41no—
96C4×D4C22⋊C412yes—
97C2×C4⋊C4—41no—
98C2×C42C2×D411yescharacteristic
99M32—82no—
100C2×C16—22no—
101C2×C16—22no—
102C2×C16—82no—
103#107—21no—
104#107—21no—
105#106—42no—
106C4⋊C8—22no—
107C4⋊C8—21no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110C4⋊C8C22⋊C412yes—
111#102—42no—
112#102—41no—
113#102—41no—
114#102—42no—
115C22⋊C8—22no—
116C22⋊C8—22no—
117C22⋊C8—41no—
118C4×C8—41no—
119C4×C8C22×C411yescharacteristic
120C4×C8C2×D411yescharacteristic
121C4×C8C2×D411yescharacteristic
122C4×C8#4111yescharacteristic
123#95—41no—
124#95—41no—
125#41—81no—
126#41—81no—
127#41—162no—
128C2×Q8—21no—
129C2×Q8—41no—
130C2×D4—22no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—81no—
137C2×D4#10212yes—
138C22×C4—22no—
139C22×C4—81no—
140C22×C4—81no—
141C22×C4#12111yescharacteristic
142C22×C4C2×D811yescharacteristic
143C22×C4#13511yescharacteristic
144D8—42no—
145D8—81no—
146D8—162no—
147M16—82no—
148M16—81no—
149M16—161no—
150C2×C8—22no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—81no—
157C2×C8C4×D411yescharacteristic
158C2×C8C4×D411yescharacteristic
159C2×C8#12011yescharacteristic
160C2×C8#12311yescharacteristic
161C4⋊C4—21no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C4⋊C4#10212yes—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C22⋊C4—81no—
170C22⋊C4—81no—
171C42C2×C22⋊C411yescharacteristic
172C42#12111yescharacteristic
173C42C2×SD1611yescharacteristic
174C42#13511yescharacteristic
175C16—42no—
176C16—42no—
177C23—22no—
178C23—81no—
179C23#46611yescharacteristic
180Q8—81no—
181Q8—161no—
182D4—42no—
183D4—81no—
184D4—81no—
185D4—81no—
186D4—81no—
187D4—82no—
188D4—81no—
189D4—81no—
190D4—161no—
191D4—162no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—81no—
200C2×C4—162no—
201C2×C4—161no—
202C2×C4—161no—
203C2×C4—161no—
204C2×C4#38611yescharacteristic
205C2×C4C2×#10211yescharacteristic
206C2×C4#41611yescharacteristic
207C2×C4#46511yescharacteristic
208C2×C4#46511yescharacteristic
209C2×C4#46611yescharacteristic
210C8—21no—
211C8—21no—
212C8—21no—
213C8—21no—
214C8—42no—
215C8—82no—
216C8—161no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—161no—
224C22—161no—
225C22—161no—
226C22#189311yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—21no—
232C4—42no—
233C4—161no—
234C4—321no—
235C4#188111yescharacteristic
236C4#190411yescharacteristic
237C2—41no—
238C2—82no—
239C2—321no—
240C2#930511yescharacteristic
241C2#1306811yescharacteristic
242C2#1307711yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#19337C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33494C2semidirect
3. #11858C2semidirect
4. #11652C2semidirect
5. #8680C2semidirect
6. #7295C2semidirect
7. #7221C2semidirect
17. #1695C4semidirect
18. #1687C4semidirect

Maximal subgroups

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

subgroupindex
#334942
#193372
#118582
#116522
#86802
#72952
#72212