Finite Group Databasebeta

← #145228

Subgroups — #145228

Order 512.

271 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 60 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#19114C211yesmaximal, characteristic
2#12031C211yesmaximal, characteristic, complemented
3#11963C211yesmaximal, characteristic
4#11960C211yesmaximal, characteristic
5#11639C211yesmaximal, characteristic, complemented
6#8621C211yesmaximal, characteristic, complemented
7#7289C211yesmaximal, characteristic, complemented
8#3147C2211yescharacteristic
9#3103—21no—
10C2×C8×Q8C2211yescharacteristic
11#2163C2211yescharacteristic
12#2162C2211yescharacteristic
13#1670—21no—
14#1668—21no—
15#1666—22no—
16#1568C2211yescharacteristic
17#1568C2211yescharacteristic
18#1564C2211yescharacteristic
19#1540—21no—
20#1467—21no—
21#1368—22no—
22#1362—21no—
23#1361—21no—
24#1360—22no—
25Q8⋊C16—22no—
26#1290—21no—
27C2×#135—21no—
28C2×#135—21no—
29#532—21no—
30C2×C4×Q8D411yescharacteristic
31#495—21no—
32#495—21no—
33#494D411yescharacteristic
34#493D411yescharacteristic
35#492—21no—
36#481—41no—
37#475—22no—
38#460—22no—
39#448—41no—
40C8×Q8—21no—
41C8×Q8—21no—
42C8×Q8—21no—
43C8×Q8—21no—
44C4×SD16—42no—
45#432—21no—
46#432—21no—
47#431—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8D411yescharacteristic
50C2×C4×C8—21no—
51C2×C4×C8C2311yescharacteristic
52C4⋊C16—21no—
53C4⋊C16D412yes—
54C22⋊C16—21no—
55C22⋊C16—21no—
56#346—21no—
57C4×C16—21no—
58#335—21no—
59#335—21no—
60#335—21no—
61#334—21no—
62#334—21no—
63#334D412yes—
64#332—22no—
65#331—22no—
66#324—42no—
67C2×#41—41no—
68C22×Q8SD1611yescharacteristic
69#135—82no—
70#135—82no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×SD16—42no—
74C2×SD16—41no—
75C2×SD16—41no—
76C2×D8—21no—
77C2×D8—21no—
78C22×C8—21no—
79C22×C8#4111yescharacteristic
80C22×C8#4111yescharacteristic
81C4⋊Q8—21no—
82#127—21no—
83#124—41no—
84#121—41no—
85#121—41no—
86C4×Q8—21no—
87C4×Q8—21no—
88C4×Q8—21no—
89C4×Q8—21no—
90C4×D4—41no—
91C4×D4—41no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4SD1611yescharacteristic
94C2×C42—21no—
95C2×C42C2×D411yescharacteristic
96C2×C16—22no—
97C2×C16—41no—
98#106—42no—
99C4⋊C8—21no—
100C4⋊C8—22no—
101C4⋊C8—21no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106#103—42no—
107#102—42no—
108C22⋊C8—41no—
109C22⋊C8—41no—
110C22⋊C8—41no—
111#97—41no—
112#97—41no—
113C4×C8—41no—
114C4×C8—41no—
115C4×C8—41no—
116C4×C8C2×D411yescharacteristic
117C4×C8C2×D411yescharacteristic
118C4×C8C2×D411yescharacteristic
119C4×C8#4111yescharacteristic
120#41—82no—
121#41—82no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—42no—
125C2×Q8—41no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—81no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—81no—
133C22×C4#12311yescharacteristic
134C22×C4C2×SD1611yescharacteristic
135C22×C4AGL(1,ℤ/8)11yescharacteristic
136Q16—81no—
137Q16—81no—
138SD16—82no—
139SD16—84no—
140SD16—82no—
141D8—81no—
142D8—81no—
143C2×C8—22no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—42no—
147C2×C8—81no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8#12111yescharacteristic
151C2×C8#12111yescharacteristic
152C2×C8#12311yescharacteristic
153C2×C8#12411yescharacteristic
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C22⋊C4—81no—
161C22⋊C4—81no—
162C42—41no—
163C42—41no—
164C42#12011yescharacteristic
165C42#12111yescharacteristic
166C42#13511yescharacteristic
167C42AGL(1,ℤ/8)11yescharacteristic
168C16—42no—
169C16—81no—
170C23—81no—
171C23#48111yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—81no—
177Q8—81no—
178D4—82no—
179D4—81no—
180D4—81no—
181D4—161no—
182D4—162no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—162no—
194C2×C4#39411yescharacteristic
195C2×C4#44911yescharacteristic
196C2×C4#45011yescharacteristic
197C2×C4#46511yescharacteristic
198C2×C4#46911yescharacteristic
199C2×C4#48211yescharacteristic
200C8—21no—
201C8—21no—
202C8—21no—
203C8—82no—
204C8—81no—
205C8—161no—
206C8—161no—
207C8#47112yes—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—161no—
212C22—161no—
213C22—161no—
214C22#203411yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—42no—
221C4—81no—
222C4—161no—
223C4#204611yescharacteristic
224C4#204811yescharacteristic
225C2—41no—
226C2—321no—
227C2#1044311yescharacteristic
228C2#1339511yescharacteristic
229C2#1340111yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radical#3147C22
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. #11639C2semidirect
6. #8621C2semidirect
7. #7289C2semidirect

Maximal subgroups

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

subgroupindex
#191142
#120312
#119632
#119602
#116392
#86212
#72892