Finite Group Databasebeta

← #8450

Subgroups — #8450

Order 256.

424 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 103 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×#554C211yesmaximal, characteristic
2#1956C211yesmaximal, characteristic, complemented
3#1803C211yesmaximal, characteristic, complemented
4#1802C211yesmaximal, characteristic
5C2×#333C211yesmaximal, characteristic, complemented
6C2×Q8⋊C8C211yesmaximal, characteristic, complemented
7C2×D4⋊C8C211yesmaximal, characteristic
8#554—22no—
9#554—22no—
10#554—22no—
11#554C414yescomplemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125—21no—
14C2×C22⋊Q8—21no—
15C2×C22⋊Q8—22no—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4—21no—
18C2×C4×D4C2211yescharacteristic
19C22×C4⋊C4—21no—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×#103—21no—
23C2×#103—22no—
24C2×#102—21no—
25C2×C4×C8C2211yescharacteristic
26#384C2211yescharacteristic
27#340—21no—
28#340—21no—
29#333—22no—
30Q8⋊C8—22no—
31D4⋊C8—22no—
32C22×Q8D411yescharacteristic
33C22×Q8D412yes—
34C22×D4D411yescharacteristic
35C23×C4—22no—
36C23×C4—21no—
37C22×C8—21no—
38C22×C8—21no—
39C22×C8—21no—
40C4⋊Q8—21no—
41C4⋊Q8—21no—
42C4⋊Q8C2×C412yes—
43C4⋊Q8D412yes—
44#125—24no—
45#125—24no—
46C22⋊Q8—24no—
47C22⋊Q8—28no—
48C22⋊Q8—24no—
49C22⋊Q8—28no—
50C4×Q8—21no—
51C4×Q8—21no—
52C4×Q8C2×C412yes—
53C4×Q8D412yes—
54C4×D4—24no—
55C4×D4—24no—
56C4×D4—21no—
57C4×D4—21no—
58C4×D4C2×C412yes—
59C4×D4D412yes—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4D411yescharacteristic
71C2×C4⋊C4D411yescharacteristic
72C2×C22⋊C4—22no—
73C2×C22⋊C4—21no—
74C2×C42—21no—
75C2×C42C2311yescharacteristic
76C4⋊C8—21no—
77C4⋊C8—21no—
78C4⋊C8—21no—
79C4⋊C8—21no—
80#103—44no—
81#103—42no—
82#102—42no—
83C4×C8—21no—
84C4×C8—21no—
85C24—21no—
86C2×Q8—21no—
87C2×Q8—22no—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—42no—
93C2×Q8—42no—
94C2×Q8C22⋊C412yes—
95C2×Q8C22⋊C414yes—
96C2×D4—21no—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4C22⋊C412yes—
100C2×D4SD1612yes—
101C2×D4Q1612yes—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—28no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—82no—
115C22×C4—82no—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C2×C8—22no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—41no—
128C4⋊C4—28no—
129C4⋊C4—28no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—42no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4C22⋊C412yes—
144C4⋊C4C22⋊C412yes—
145C22⋊C4—28no—
146C22⋊C4—22no—
147C22⋊C4—22no—
148C42—22no—
149C42—22no—
150C42C22×C411yescharacteristic
151C42C2×D411yescharacteristic
152C42C2×D411yescharacteristic
153C42C2×D411yescharacteristic
154C23—22no—
155C23—22no—
156C23—22no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23#12011yescharacteristic
162Q8—22no—
163Q8—22no—
164Q8—44no—
165Q8—41no—
166Q8—41no—
167Q8—82no—
168D4—22no—
169D4—22no—
170D4—22no—
171D4#10314yescomplemented
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—88no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4C2×C22⋊C411yescharacteristic
195C2×C4C2×C22⋊C411yescharacteristic
196C2×C4C2×C22⋊C411yescharacteristic
197C2×C4#12011yescharacteristic
198C2×C4#12011yescharacteristic
199C2×C4#12011yescharacteristic
200C2×C4C2×SD1611yescharacteristic
201C2×C4C2×Q1611yescharacteristic
202C2×C4#13511yescharacteristic
203C2×C4#13511yescharacteristic
204C2×C4AGL(1,ℤ/8)11yescharacteristic
205C2×C4#13711yescharacteristic
206C2×C4#13711yescharacteristic
207C2×C4#13711yescharacteristic
208C8—42no—
209C8—81no—
210C8—81no—
211C22—24no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22#37911yescharacteristic
219C22#44811yescharacteristic
220C22#44911yescharacteristic
221C22#45111yescharacteristic
222C22#45211yescharacteristic
223C22#45311yescharacteristic
224C22#45511yescharacteristic
225C4—21no—
226C4—21no—
227C4—42no—
228C4—44no—
229C4—42no—
230C4—42no—
231C4—161no—
232C4C2×#10311yescharacteristic
233C4#41611yescharacteristic
234C4#41711yescharacteristic
235C4#41911yescharacteristic
236C4#44811yescharacteristic
237C4#44911yescharacteristic
238C4#45111yescharacteristic
239C4#45211yescharacteristic
240C2—44no—
241C2#162211yescharacteristic
242C2#163211yescharacteristic
243C2#164311yescharacteristic
244C2#164411yescharacteristic
245C2#178711yescharacteristic
246C2#178811yescharacteristic
247C2#179211yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1956C2semidirect
3. #1803C2semidirect
5. C2×#333C2semidirect
6. C2×Q8⋊C8C2semidirect
11. #554C4semidirect
171. D4#103semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
#19562
#18032
#18022
C2×#3332
C2×Q8⋊C82
C2×D4⋊C82