Finite Group Databasebeta

← #18125

Subgroups — #18125

Order 256.

464 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 127 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#3458C211yesmaximal, characteristic
2#3104C212yesmaximal, complemented
3#2396C211yesmaximal, characteristic
4#1617C212yesmaximal, complemented
5#1614C212yesmaximal
6#1605C212yesmaximal, complemented
7#1602C212yesmaximal
8#1503C212yesmaximal, complemented
9#1458C211yesmaximal, characteristic
10#544—22no—
11#544—22no—
12#544—21no—
13#544C2212yes—
14#543—21no—
15#543C2212yes—
16#533—24no—
17#533—21no—
18#533C2212yes—
19C2×C4×Q8C2211yescharacteristic
20C2×#117—21no—
21C2×#117C2211yescharacteristic
22#484C2214yes—
23#477—22no—
24#462—22no—
25#451C2214yes—
26C4×Q16—24no—
27#431C2212yes—
28#400C2211yescharacteristic
29#399C2211yescharacteristic
30#398—21no—
31#396—21no—
32#395C2211yescharacteristic
33#393C2211yescharacteristic
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2211yescharacteristic
36C4×C22⋊C4—21no—
37#328C2214yes—
38#328C2214yes—
39#324C2212yes—
40#323C2212yes—
41C22×Q8C2311yescharacteristic
42C23×C4D411yescharacteristic
43C2×Q16—24no—
44C4⋊Q8—24no—
45C4⋊Q8—21no—
46C4⋊Q8—21no—
47C4⋊Q8—24no—
48C4⋊Q8D412yes—
49C4⋊Q8D412yes—
50#125—24no—
51#125—22no—
52#125—24no—
53#125—22no—
54C22⋊Q8—24no—
55C22⋊Q8—24no—
56C22⋊Q8—24no—
57C22⋊Q8—22no—
58C22⋊Q8—22no—
59C22⋊Q8—22no—
60C22⋊Q8—24no—
61C22⋊Q8C2314yes—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8—22no—
65C4×Q8—22no—
66C4×Q8—22no—
67C4×Q8D414yes—
68#117—22no—
69#117—22no—
70#117—22no—
71#117—22no—
72#117—22no—
73#117—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4D411yescharacteristic
87C2×C22⋊C4C2311yescharacteristic
88C2×C42—22no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42D411yescharacteristic
92C2×C42C2311yescharacteristic
93#107—24no—
94C4⋊C8—22no—
95#103—24no—
96#103—24no—
97C22⋊C8C2314yes—
98C4×C8—22no—
99#95—21no—
100#95—21no—
101#95—21no—
102#95—21no—
103#95C2311yescharacteristic
104#95C2311yescharacteristic
105C24C2×D411yescharacteristic
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—42no—
111C2×Q8C2×D414yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4Q1614yescomplemented
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2411yescharacteristic
128Q16—48no—
129C2×C8—24no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—24no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—24no—
141C4⋊C4—22no—
142C4⋊C4—24no—
143C4⋊C4—22no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4C2×D414yes—
149C22⋊C4—24no—
150C22⋊C4—24no—
151C22⋊C4—22no—
152C22⋊C4—22no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C42—24no—
156C42—21no—
157C42—24no—
158C42—21no—
159C42—41no—
160C42—41no—
161C42—42no—
162C42C2×D412yes—
163C42C2×D412yes—
164C23—21no—
165C23—21no—
166C23—41no—
167C23—41no—
168C23C2×Q1612yes—
169C23C22×D411yescharacteristic
170Q8—24no—
171Q8—44no—
172Q8—42no—
173Q8—82no—
174Q8—82no—
175Q8—82no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4#12012yes—
203C2×C4#12011yescharacteristic
204C2×C4#12011yescharacteristic
205C2×C4C2×Q1612yes—
206C2×C4C2×Q1612yes—
207C2×C4#13712yes—
208C2×C4C22×D412yes—
209C8—44no—
210C22—21no—
211C22—21no—
212C22—22no—
213C22—42no—
214C22—41no—
215C22—41no—
216C22—81no—
217C22C2×#12011yescharacteristic
218C22C22×Q1611yescharacteristic
219C22C2×#13711yescharacteristic
220C4—22no—
221C4—22no—
222C4—22no—
223C4—42no—
224C4—41no—
225C4—44no—
226C4—42no—
227C4—41no—
228C4—44no—
229C4—81no—
230C4—81no—
231C4#45112yes—
232C4#45112yes—
233C2—21no—
234C2—21no—
235C2—42no—
236C2C2×#45111yescharacteristic
237C2#301811yescharacteristic
238C2#302511yescharacteristic
subgroupG/H
centerC2×C4#120
commutator, Frattini, cosocleC22×C4C24
socleC22C2×#120
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3104C2semidirect
4. #1617C2semidirect
6. #1605C2semidirect
8. #1503C2semidirect
121. C22×C4Q16semidirect

Maximal subgroups

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

subgroupindex
#34582
#31042
#23962
#16172
#16142
#16052
#16022
#15032
#14582