Finite Group Databasebeta

← #138479

Subgroups — #138479

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#22747C211yesmaximal, characteristic, complemented
2#19289C211yesmaximal, characteristic
3#11864C211yesmaximal, characteristic
4#11751C211yesmaximal, characteristic
5#11644C211yesmaximal, characteristic, complemented
6#7303C211yesmaximal, characteristic, complemented
7#7224C211yesmaximal, characteristic, complemented
8#3169C2211yescharacteristic
9#3112—21no—
10Q8×M16—21no—
11C2×#446C2211yescharacteristic
12#2163C2211yescharacteristic
13#2163C2211yescharacteristic
14#1950—21no—
15#1950—22no—
16#1838—21no—
17#1837C2211yescharacteristic
18#1778—21no—
19C4×#108C2211yescharacteristic
20#1698—22no—
21#1696C412yes—
22#1688C412yes—
23#1568C2211yescharacteristic
24#1551—21no—
25#1349—22no—
26#1349—21no—
27#1349—21no—
28#1338—22no—
29#532—21no—
30C2×C4×Q8D411yescharacteristic
31#500C2×C411yescharacteristic
32#493C2×C411yescharacteristic
33#486—21no—
34#480—41no—
35#477—22no—
36#466—21no—
37#466—21no—
38#460—22no—
39#448—41no—
40#446—21no—
41#446—21no—
42#446—41no—
43#446—42no—
44#446C2×C412yes—
45#446D412yes—
46C8×Q8—41no—
47#440—42no—
48C4×D8—21no—
49C4×D8—21no—
50#432—21no—
51C2×#108—22no—
52C2×#108—21no—
53#423—22no—
54#423—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8D411yescharacteristic
57C4×M16—22no—
58C4×M16—21no—
59C2×#97—21no—
60C2×C4×C8C2311yescharacteristic
61C4⋊C16—21no—
62C4⋊C16—21no—
63C22⋊C16—21no—
64C22⋊C16—21no—
65#346—21no—
66#346—21no—
67#341—22no—
68#341—21no—
69#341D412yes—
70#341D411yescharacteristic
71#341D411yescharacteristic
72#335—21no—
73#334C2×C412yes—
74#333—22no—
75#331—22no—
76C2×#41—41no—
77C22×Q8SD1611yescharacteristic
78C2×SD16—42no—
79C2×D8—21no—
80C2×D8—21no—
81C2×M16—21no—
82C2×M16—21no—
83C2×M16—22no—
84C22×C8—21no—
85C22×C8C2×D411yescharacteristic
86C22×C8C2×D411yescharacteristic
87C4⋊Q8—21no—
88#127—21no—
89#124—41no—
90#121—41no—
91#121—41no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8C22⋊C412yes—
95C4×Q8D812yes—
96C4×D4—41no—
97C4×D4—41no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4SD1611yescharacteristic
100C2×C42—21no—
101C2×C42C2×D411yescharacteristic
102C2×C16—41no—
103C2×C16—41no—
104#108—42no—
105#108—44no—
106#107—21no—
107#107—21no—
108#107—42no—
109C4⋊C8—22no—
110C4⋊C8—22no—
111C4⋊C8—22no—
112C4⋊C8—21no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8C22⋊C412yes—
118#103—42no—
119#102—41no—
120#102—41no—
121#102—42no—
122C22⋊C8—41no—
123#97—22no—
124#97—22no—
125#97—42no—
126#97—41no—
127#97—41no—
128C4×C8—42no—
129C4×C8—41no—
130C4×C8C22×C411yescharacteristic
131C4×C8C2×D411yescharacteristic
132C4×C8#4111yescharacteristic
133C4×C8#4111yescharacteristic
134#41—162no—
135C2×Q8—21no—
136C2×Q8—41no—
137C2×Q8—81no—
138C2×Q8#10212yes—
139C2×D4—41no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4—81no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—81no—
146C22×C4#12011yescharacteristic
147C22×C4C2×SD1611yescharacteristic
148C22×C4#13711yescharacteristic
149SD16—84no—
150D8—81no—
151D8—81no—
152M16—22no—
153M16—41no—
154M16—41no—
155M16—42no—
156M16—41no—
157M16—42no—
158C2×C8—22no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—42no—
162C2×C8—41no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—41no—
166C2×C8C4×D411yescharacteristic
167C2×C8C4×D411yescharacteristic
168C2×C8#12111yescharacteristic
169C2×C8#12111yescharacteristic
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—21no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4#10212yes—
176C22⋊C4—81no—
177C22⋊C4—81no—
178C42—22no—
179C42—22no—
180C42C2×C22⋊C411yescharacteristic
181C42#12311yescharacteristic
182C42C2×D811yescharacteristic
183C42AGL(1,ℤ/8)11yescharacteristic
184C16—81no—
185C16—81no—
186C23—81no—
187C23#44811yescharacteristic
188Q8—42no—
189Q8—42no—
190Q8—161no—
191D4—82no—
192D4—81no—
193D4—81no—
194D4—161no—
195D4—162no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—162no—
206C2×C4#38611yescharacteristic
207C2×C4C2×#10211yescharacteristic
208C2×C4#41711yescharacteristic
209C2×C4#44711yescharacteristic
210C2×C4#48011yescharacteristic
211C2×C4#48111yescharacteristic
212C8—21no—
213C8—21no—
214C8—21no—
215C8—21no—
216C8—42no—
217C8—42no—
218C8—82no—
219C8—81no—
220C8—82no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—161no—
225C22—161no—
226C22—161no—
227C22#187411yescharacteristic
228C4—21no—
229C4—21no—
230C4—21no—
231C4—21no—
232C4—21no—
233C4—42no—
234C4—42no—
235C4—161no—
236C4#190511yescharacteristic
237C4#190611yescharacteristic
238C2—41no—
239C2—321no—
240C2#918011yescharacteristic
241C2#1299811yescharacteristic
242C2#1300711yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#3169C22
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#446 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22747C2semidirect
5. #11644C2semidirect
6. #7303C2semidirect
7. #7224C2semidirect

Maximal subgroups

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

subgroupindex
#227472
#192892
#118642
#117512
#116442
#73032
#72242