Finite Group Databasebeta

← #18779

Subgroups — #18779

Order 256.

439 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 129 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#3230C211yesmaximal, characteristic
3#3104C211yesmaximal, characteristic
4#3078C211yesmaximal, characteristic
5#1648C214yesmaximal, complemented
6#1636C214yesmaximal, complemented
7#1549C211yesmaximal, characteristic
8#1487C211yesmaximal, characteristic
9#1479C211yesmaximal, 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#500C2212yes—
24#487C2212yes—
25#484—21no—
26#484—22no—
27#479—22no—
28#477—22no—
29#477C2212yes—
30#477C2212yes—
31#475C2212yes—
32#462C2212yes—
33#451—21no—
34C4×Q16—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37#428—21no—
38#423C2211yescharacteristic
39#419—21no—
40#332C2214yes—
41Q8⋊C8C2214yes—
42Q8⋊C8C2214yes—
43C22×Q8—21no—
44C22×Q8D411yescharacteristic
45C22×Q8D411yescharacteristic
46C2×Q16—22no—
47C2×M16—21no—
48C4⋊Q8—24no—
49C4⋊Q8—24no—
50C4⋊Q8—21no—
51C4⋊Q8—21no—
52C4⋊Q8D412yes—
53C4⋊Q8C2311yescharacteristic
54C4⋊Q8C2311yescharacteristic
55C4⋊Q8C2312yes—
56#125—28no—
57#125—24no—
58#125—28no—
59#125—24no—
60#125—21no—
61C22⋊Q8—21no—
62C22⋊Q8—21no—
63C4×Q8—24no—
64C4×Q8—24no—
65C4×Q8—24no—
66C4×Q8—24no—
67C4×Q8—21no—
68C4×Q8—21no—
69C4×Q8—21no—
70C4×Q8—21no—
71C4×Q8—21no—
72C4×Q8—21no—
73C4×Q8D412yes—
74C4×Q8D412yes—
75C4×Q8C2312yes—
76C4×Q8C2312yes—
77#117—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D412yes—
87C2×C42—22no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42C2311yescharacteristic
91#107—22no—
92#107—22no—
93#107—24no—
94#106—22no—
95C4⋊C8—21no—
96C4⋊C8—21no—
97C4⋊C8C2312yes—
98C4⋊C8C2312yes—
99#103—22no—
100#103—22no—
101#103—22no—
102#103—22no—
103#103—24no—
104C22⋊C8—21no—
105C22⋊C8—21no—
106C4×C8—21no—
107C4×C8C2312yes—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—42no—
116C2×Q8—44no—
117C2×Q8Q1614yescomplemented
118C2×Q8C2×D412yes—
119C2×Q8C2×D412yes—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128Q16—44no—
129M16—42no—
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×D414yes—
153C4⋊C4C2×D412yes—
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—22no—
167Q8—24no—
168Q8—22no—
169Q8—22no—
170Q8—42no—
171Q8—44no—
172Q8—42no—
173Q8—44no—
174Q8—42no—
175Q8C2×Q1614yes—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4#12011yescharacteristic
191C2×C4#12011yescharacteristic
192C2×C4#12011yescharacteristic
193C2×C4C2×Q1612yes—
194C2×C4#13712yes—
195C2×C4C22×D411yescharacteristic
196C2×C4C22×D411yescharacteristic
197C2×C4C22×D411yescharacteristic
198C8—42no—
199C8—42no—
200C8—42no—
201C22—21no—
202C22—21no—
203C22#45112yes—
204C22C2×#12011yescharacteristic
205C4—24no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—42no—
210C4—44no—
211C4—42no—
212C4—42no—
213C4—81no—
214C4—81no—
215C4#45112yes—
216C4C22×Q1611yescharacteristic
217C4C2×#13711yescharacteristic
218C4#57512yes—
219C2—22no—
220C2C2×#45111yescharacteristic
221C2#300511yescharacteristic
222C2#302011yescharacteristic
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. #1648C2semidirect
6. #1636C2semidirect
117. C2×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
#32302
#31042
#30782
#16482
#16362
#15492
#14872
#14792