Finite Group Databasebeta

← #18541

Subgroups — #18541

Order 256.

443 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#557C211yesmaximal, characteristic
2#3165C211yesmaximal, characteristic
3#3113C211yesmaximal, characteristic
4#3113C211yesmaximal, characteristic
5#1653C214yesmaximal, complemented
6#1648C214yesmaximal, complemented
7#1556C211yesmaximal, characteristic
8#1491C211yesmaximal, characteristic
9#1491C211yesmaximal, characteristic
10#557—22no—
11#557—22no—
12#557—22no—
13#557C2214yes—
14#533C2211yescharacteristic
15C2×C4⋊Q8—21no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125—22no—
18C2×#125—21no—
19C2×C4×Q8—21no—
20C2×C4×Q8—21no—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23#501—22no—
24#500C2212yes—
25#498C2212yes—
26#483—21no—
27#483—21no—
28#475C2212yes—
29#475C2212yes—
30#462C2212yes—
31#462C2212yes—
32#451—21no—
33#451—21no—
34#441—22no—
35#441—22no—
36#432C2211yescharacteristic
37#432C2211yescharacteristic
38#428—22no—
39C4×M16C2211yescharacteristic
40#332C2214yes—
41Q8⋊C8C2214yes—
42Q8⋊C8C2214yes—
43C22×Q8—21no—
44C22×Q8D411yescharacteristic
45C22×Q8D411yescharacteristic
46C2×Q16—22no—
47C2×Q16—22no—
48C2×M16—21no—
49C4⋊Q8—24no—
50C4⋊Q8—24no—
51C4⋊Q8—21no—
52C4⋊Q8—21no—
53C4⋊Q8D412yes—
54C4⋊Q8C2311yescharacteristic
55C4⋊Q8C2311yescharacteristic
56C4⋊Q8C2312yes—
57#125—28no—
58#125—24no—
59#125—28no—
60#125—24no—
61#125—21no—
62C22⋊Q8—21no—
63C22⋊Q8—21no—
64C4×Q8—24no—
65C4×Q8—24no—
66C4×Q8—24no—
67C4×Q8—24no—
68C4×Q8—21no—
69C4×Q8—21no—
70C4×Q8—21no—
71C4×Q8—21no—
72C4×Q8—21no—
73C4×Q8—21no—
74C4×Q8D412yes—
75C4×Q8D412yes—
76C4×Q8C2312yes—
77C4×Q8C2312yes—
78#117—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4D412yes—
88C2×C42—22no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42C2311yescharacteristic
92#107—24no—
93#106—24no—
94#106—22no—
95#106—22no—
96C4⋊C8C2312yes—
97C4⋊C8C2312yes—
98#103—22no—
99#103—22no—
100#103—22no—
101#103—22no—
102C22⋊C8—21no—
103C22⋊C8—21no—
104#97—21no—
105#97—21no—
106#97—21no—
107C4×C8C2312yes—
108C2×Q8—24no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—42no—
116C2×Q8—44no—
117C2×Q8C2×D412yes—
118C2×Q8C2×D412yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127Q16—44no—
128Q16—44no—
129M16—24no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C4⋊C4—28no—
134C4⋊C4—28no—
135C4⋊C4—28no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4C2×D412yes—
152C4⋊C4C2×D412yes—
153C4⋊C4C2×D414yes—
154C22⋊C4—41no—
155C42—28no—
156C42—22no—
157C42—22no—
158C42—22no—
159C42—22no—
160C42—41no—
161C42C2×D411yescharacteristic
162C42C2×D411yescharacteristic
163C42C2×D411yescharacteristic
164C42C2411yescharacteristic
165C23#12011yescharacteristic
166Q8—24no—
167Q8—22no—
168Q8—22no—
169Q8—22no—
170Q8—42no—
171Q8—44no—
172Q8—44no—
173Q8—42no—
174Q8—42no—
175Q8#13714yes—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—42no—
186C2×C4—44no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4#12011yescharacteristic
192C2×C4#12011yescharacteristic
193C2×C4#12011yescharacteristic
194C2×C4C22×D411yescharacteristic
195C2×C4C22×D411yescharacteristic
196C2×C4C22×D411yescharacteristic
197C8—24no—
198C8—42no—
199C8—42no—
200C22—21no—
201C22—21no—
202C22#45612yes—
203C22C2×#12011yescharacteristic
204C4—24no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—42no—
210C4—44no—
211C4—42no—
212C4—42no—
213C4—81no—
214C4—81no—
215C4C2×#13712yes—
216C4#57512yes—
217C2—22no—
218C2#300411yescharacteristic
219C2#300511yescharacteristic
220C2C2×#45611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC23#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #1653C2semidirect
6. #1648C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
#31652
#31132
#16532
#16482
#15562
#14912