Finite Group Databasebeta

← #8573

Subgroups — #8573

Order 256.

349 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 349 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C4×#121C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic, complemented
3C2×#327C212yesmaximal
4#1434C211yesmaximal, characteristic, complemented
5#1294C212yesmaximal, complemented
6C2×#121C2211yescharacteristic
7C2×C4×D4—21no—
8C2×C4×D4C411yescharacteristic, complemented
9C22×C42C2211yescharacteristic
10C2×C4⋊C8—21no—
11C2×C22⋊C8C2212yes—
12#390C411yescharacteristic, complemented
13#389C411yescharacteristic, complemented
14#388—21no—
15#386—21no—
16#380C2211yescharacteristic
17C4×C4⋊C4C411yescharacteristic, complemented
18C4×C22⋊C4—21no—
19C2×#95—21no—
20C2×#95—21no—
21C2×#95C2212yes—
22#336—21no—
23#327—24no—
24C22×D4—21no—
25C22×D4C2×C411yescharacteristic
26C23×C4—22no—
27C23×C4—21no—
28C23×C4D411yescharacteristic
29C23×C4D411yescharacteristic
30C23×C4C2311yescharacteristic
31C22×C8—22no—
32#121—21no—
33#121—21no—
34#121D412yes—
35#121Q812yes—
36C4×D4—42no—
37C4×D4—44no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C4⋊C4C2×C411yescharacteristic
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C42—21no—
48C2×C42—42no—
49C2×C42C2×C411yescharacteristic
50C2×C42C2×C411yescharacteristic
51C2×C42D411yescharacteristic
52C2×C42D411yescharacteristic
53C4⋊C8—42no—
54C22⋊C8—22no—
55C22⋊C8—22no—
56#95—24no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—22no—
61#95—22no—
62#95—42no—
63C24—21no—
64C24C2×D411yescharacteristic
65C2×D4—22no—
66C2×D4—22no—
67C2×D4—21no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4—42no—
71C2×D4—42no—
72C2×D4C4⋊C412yes—
73C22×C4—21no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—24no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—41no—
82C22×C4—41no—
83C22×C4—41no—
84C22×C4—41no—
85C22×C4—42no—
86C22×C4—42no—
87C22×C4—42no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4C22⋊C411yescharacteristic
95C22×C4C22⋊C411yescharacteristic
96C22×C4C22⋊C411yescharacteristic
97C22×C4C22⋊C411yescharacteristic
98C22×C4C22×C411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×Q811yescharacteristic
102C22×C4#4111yescharacteristic
103C22×C4#4111yescharacteristic
104C2×C8—42no—
105C2×C8—42no—
106C2×C8—42no—
107C4⋊C4—21no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—42no—
111C4⋊C4C4⋊C412yes—
112C22⋊C4—22no—
113C22⋊C4—22no—
114C22⋊C4—42no—
115C22⋊C4—42no—
116C22⋊C4—42no—
117C22⋊C4—42no—
118C42—22no—
119C42—22no—
120C42—41no—
121C42—41no—
122C42—42no—
123C23—21no—
124C23—21no—
125C23—21no—
126C23—21no—
127C23—22no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23C2×C22⋊C411yescharacteristic
135C23#12111yescharacteristic
136C23C22⋊Q811yescharacteristic
137C23#13512yes—
138C23AGL(1,ℤ/8)12yes—
139D4—42no—
140D4—42no—
141D4—44no—
142D4—44no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—44no—
163C2×C4—44no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—44no—
167C2×C4—44no—
168C2×C4—44no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—84no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4#9912yes—
181C2×C4#10412yes—
182C2×C4C2×C4⋊C411yescharacteristic
183C2×C4#11711yescharacteristic
184C2×C4#12111yescharacteristic
185C2×C4C22⋊Q811yescharacteristic
186C8—82no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—22no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22#38011yescharacteristic
203C22C2×#9911yescharacteristic
204C22#41011yescharacteristic
205C22#41611yescharacteristic
206C22#41811yescharacteristic
207C22C2×#10411yescharacteristic
208C22#42111yescharacteristic
209C22#46312yes—
210C22#47812yes—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—42no—
218C4—41no—
219C4—41no—
220C4—84no—
221C4—81no—
222C4—81no—
223C2—22no—
224C2—22no—
225C2—82no—
226C2#150011yescharacteristic
227C2#150711yescharacteristic
228C2#151411yescharacteristic
229C2#151611yescharacteristic
230C2#179911yescharacteristic
231C2#180411yescharacteristic
232C2#180611yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1797C2semidirect
4. #1434C2semidirect
5. #1294C2semidirect
8. C2×C4×D4C4semidirect
12. #390C4semidirect
13. #389C4semidirect
17. C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C4×#1212
#17972
C2×#3272
#14342
#12942