Finite Group Databasebeta

← #145336

Subgroups — #145336

Order 512.

258 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 58 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#19264C211yesmaximal, characteristic
2#12031C211yesmaximal, characteristic, complemented
3#11951C211yesmaximal, characteristic
4#11864C211yesmaximal, characteristic
5#11494C211yesmaximal, characteristic, complemented
6#8661C211yesmaximal, characteristic, complemented
7#7224C211yesmaximal, characteristic, complemented
8#3169C2211yescharacteristic
9#3112—21no—
10C2×C8×Q8C2211yescharacteristic
11#2163C2211yescharacteristic
12#2163C2211yescharacteristic
13#1837C2211yescharacteristic
14#1678—22no—
15#1676—21no—
16#1668—21no—
17#1568C2211yescharacteristic
18#1564C2211yescharacteristic
19#1551—21no—
20#1487—21no—
21#1368—22no—
22#1364—21no—
23#1361—21no—
24#1349—22no—
25#1322—22no—
26#1301—21no—
27#532—21no—
28C2×C4×Q8D411yescharacteristic
29#500D411yescharacteristic
30#493D411yescharacteristic
31#486—21no—
32#480—41no—
33#477—22no—
34#466—21no—
35#466—21no—
36#460—22no—
37#448—41no—
38C8×Q8—21no—
39C8×Q8—21no—
40C8×Q8—21no—
41C8×Q8—21no—
42#440—42no—
43C4×D8—21no—
44C4×D8—21no—
45#432—21no—
46#431—21no—
47#423—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8D411yescharacteristic
50C2×C4×C8—21no—
51C2×C4×C8C2311yescharacteristic
52C4⋊C16—21no—
53C4⋊C16—21no—
54C22⋊C16—21no—
55C22⋊C16D412yes—
56#346—21no—
57#346—21no—
58#341—21no—
59#341—21no—
60#335—21no—
61#335—21no—
62#334—21no—
63#334D412yes—
64#333—22no—
65#331—22no—
66Q8⋊C8—42no—
67C2×#41—41no—
68C22×Q8SD1611yescharacteristic
69C2×SD16—42no—
70C2×D8—21no—
71C2×D8—21no—
72C2×M16—41no—
73C22×C8—21no—
74C22×C8C2×D411yescharacteristic
75C22×C8#4111yescharacteristic
76C4⋊Q8—21no—
77#127—21no—
78#124—41no—
79#121—41no—
80#121—41no—
81C4×Q8—21no—
82C4×Q8—21no—
83C4×Q8—21no—
84C4×Q8—21no—
85C4×D4—41no—
86C4×D4—41no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4SD1611yescharacteristic
89C2×C42—21no—
90C2×C42C2×D411yescharacteristic
91C2×C16—22no—
92C2×C16—41no—
93#107—21no—
94#107—21no—
95#107—42no—
96C4⋊C8—21no—
97C4⋊C8—22no—
98C4⋊C8—21no—
99C4⋊C8—41no—
100C4⋊C8—41no—
101C4⋊C8—41no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104#103—42no—
105#102—42no—
106#102—41no—
107#102—41no—
108C22⋊C8—41no—
109C22⋊C8—41no—
110#97—41no—
111C4×C8—41no—
112C4×C8—41no—
113C4×C8—41no—
114C4×C8C2×D411yescharacteristic
115C4×C8C2×D411yescharacteristic
116C4×C8#4111yescharacteristic
117C4×C8#4111yescharacteristic
118#41—162no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—42no—
122C2×Q8—41no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4—41no—
126C2×D4—81no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—81no—
130C22×C4#12111yescharacteristic
131C22×C4C2×SD1611yescharacteristic
132C22×C4#13511yescharacteristic
133SD16—84no—
134D8—81no—
135D8—81no—
136M16—81no—
137M16—81no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—42no—
142C2×C8—81no—
143C2×C8—81no—
144C2×C8—81no—
145C2×C8#12111yescharacteristic
146C2×C8#12111yescharacteristic
147C2×C8#12311yescharacteristic
148C2×C8#12411yescharacteristic
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C22⋊C4—81no—
156C22⋊C4—81no—
157C42—41no—
158C42—41no—
159C42#12011yescharacteristic
160C42#12311yescharacteristic
161C42AGL(1,ℤ/8)11yescharacteristic
162C42AGL(1,ℤ/8)11yescharacteristic
163C16—42no—
164C16—81no—
165C23—81no—
166C23#46511yescharacteristic
167Q8—22no—
168Q8—22no—
169Q8—41no—
170Q8—41no—
171Q8—161no—
172D4—82no—
173D4—81no—
174D4—81no—
175D4—161no—
176D4—162no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—162no—
188C2×C4#39411yescharacteristic
189C2×C4#44911yescharacteristic
190C2×C4#45011yescharacteristic
191C2×C4#46911yescharacteristic
192C2×C4#48111yescharacteristic
193C2×C4#48211yescharacteristic
194C8—21no—
195C8—21no—
196C8—21no—
197C8—21no—
198C8—82no—
199C8—81no—
200C8—161no—
201C8—161no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—161no—
206C22—161no—
207C22—161no—
208C22#203411yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—81no—
216C4—161no—
217C4#204911yescharacteristic
218C4#205311yescharacteristic
219C2—41no—
220C2—321no—
221C2#1046711yescharacteristic
222C2#1339511yescharacteristic
223C2#1340211yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radical#3169C22
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12031C2semidirect
5. #11494C2semidirect
6. #8661C2semidirect
7. #7224C2semidirect

Maximal subgroups

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

subgroupindex
#192642
#120312
#119512
#118642
#114942
#86612
#72242