Finite Group Databasebeta

← #18769

Subgroups — #18769

Order 256.

461 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 461 conjugacy classes of subgroups — too many to draw. Only its 133 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×Q82C211yesmaximal, characteristic
2#3209C211yesmaximal, characteristic
3#3104C211yesmaximal, characteristic
4#3078C211yesmaximal, characteristic
5#1645C214yesmaximal, complemented
6#1633C214yesmaximal, complemented
7#1548C211yesmaximal, characteristic
8#1487C211yesmaximal, characteristic
9#1479C211yesmaximal, characteristic
10Q82—22no—
11Q82—22no—
12Q82—22no—
13Q82C2214yes—
14#533C2211yescharacteristic
15C2×C4⋊Q8—22no—
16C2×C4⋊Q8—21no—
17C2×C4⋊Q8—21no—
18C2×C4⋊Q8C2211yescharacteristic
19C2×C4×Q8—21no—
20C2×C4×Q8—21no—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23#494C2212yes—
24#487C2212yes—
25#484—21no—
26#479—22no—
27#477C2212yes—
28#477C2212yes—
29#475C2212yes—
30#462—22no—
31#462C2212yes—
32#451—21no—
33#451—22no—
34C4×Q16—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37#428—21no—
38#423C2211yescharacteristic
39#419—21no—
40#333C2214yes—
41Q8⋊C8C2214yes—
42Q8⋊C8C2214yes—
43C22×Q8—22no—
44C22×Q8D411yescharacteristic
45C22×Q8D411yescharacteristic
46C22×Q8D412yes—
47C2×Q16—22no—
48C2×Q16—24no—
49C2×M16—21no—
50C4⋊Q8—28no—
51C4⋊Q8—24no—
52C4⋊Q8—24no—
53C4⋊Q8—28no—
54C4⋊Q8—24no—
55C4⋊Q8—24no—
56C4⋊Q8—21no—
57C4⋊Q8—21no—
58C4⋊Q8D412yes—
59C4⋊Q8C2311yescharacteristic
60C4⋊Q8C2311yescharacteristic
61C4⋊Q8C2312yes—
62#125—21no—
63C22⋊Q8—21no—
64C22⋊Q8—21no—
65C4×Q8—24no—
66C4×Q8—24no—
67C4×Q8—24no—
68C4×Q8—24no—
69C4×Q8—21no—
70C4×Q8—21no—
71C4×Q8—21no—
72C4×Q8—21no—
73C4×Q8—21no—
74C4×Q8—21no—
75C4×Q8D412yes—
76C4×Q8D412yes—
77C4×Q8C2312yes—
78C4×Q8C2312yes—
79#117—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4D411yescharacteristic
88C2×C42—22no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42C2311yescharacteristic
92#107—22no—
93#107—22no—
94#106—22no—
95C4⋊C8—21no—
96C4⋊C8—21no—
97C4⋊C8C2312yes—
98C4⋊C8C2312yes—
99#103—22no—
100#103—22no—
101#103—22no—
102#103—24no—
103#103—22no—
104C22⋊C8—21no—
105C22⋊C8—21no—
106C4×C8—21no—
107C4×C8C2312yes—
108C2×Q8—24no—
109C2×Q8—28no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—82no—
117C2×Q8—84no—
118C2×Q8Q1614yescomplemented
119C2×Q8C2×D412yes—
120C2×Q8C2×D412yes—
121C2×Q8C2×D414yes—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130Q16—44no—
131Q16—84no—
132M16—42no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C4⋊C4—28no—
137C4⋊C4—28no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4C2×D412yes—
155C4⋊C4C2×D412yes—
156C22⋊C4—41no—
157C42—28no—
158C42—22no—
159C42—22no—
160C42—22no—
161C42—22no—
162C42—41no—
163C42C2×D411yescharacteristic
164C42C2×D411yescharacteristic
165C42C2×D411yescharacteristic
166C42C2411yescharacteristic
167C23#12011yescharacteristic
168Q8—24no—
169Q8—22no—
170Q8—22no—
171Q8—24no—
172Q8—44no—
173Q8—42no—
174Q8—42no—
175Q8—82no—
176Q8—88no—
177Q8C2×Q1614yes—
178Q8#13714yes—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—44no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4#12011yescharacteristic
194C2×C4#12011yescharacteristic
195C2×C4#12011yescharacteristic
196C2×C4C2×Q1612yes—
197C2×C4#13712yes—
198C2×C4C22×D411yescharacteristic
199C2×C4C22×D411yescharacteristic
200C2×C4C22×D411yescharacteristic
201C8—42no—
202C8—42no—
203C8—42no—
204C22—21no—
205C22—21no—
206C22#45112yes—
207C22C2×#12011yescharacteristic
208C4—24no—
209C4—24no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—44no—
214C4—42no—
215C4—42no—
216C4—81no—
217C4—81no—
218C4#45112yes—
219C4C22×Q1611yescharacteristic
220C4C2×#13711yescharacteristic
221C4C2×#13712yes—
222C2—22no—
223C2C2×#45111yescharacteristic
224C2#300411yescharacteristic
225C2#301911yescharacteristic
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. #1645C2semidirect
6. #1633C2semidirect
118. C2×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×Q822
#32092
#31042
#30782
#16452
#16332
#15482
#14872
#14792