Finite Group Databasebeta

← #24280

Subgroups — #24280

Order 256.

664 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22×#135C211yesmaximal, characteristic, complemented
2C2×#473C211yesmaximal, characteristic, complemented
3C2×#472C211yesmaximal, characteristic, complemented
4#3005C212yesmaximal, complemented
5#2929C211yesmaximal, characteristic, complemented
6#2046C212yesmaximal, complemented
7#2019C212yesmaximal, complemented
8#1979C211yesmaximal, characteristic, complemented
9#1978C211yesmaximal, characteristic, complemented
10#1810C211yesmaximal, characteristic, complemented
11#1785C212yesmaximal, complemented
12C22×#41C2212yes—
13C2×#137C2212yes—
14C2×AGL(1,ℤ/8)C2212yes—
15C2×#135—24no—
16C2×#135—24no—
17C2×#135—22no—
18C2×#135—22no—
19C22×Q16C2211yescharacteristic
20C22×SD16C2212yes—
21C22×D8C2211yescharacteristic
22C2×#131C2211yescharacteristic
23C23×C8C2211yescharacteristic
24#525C2211yescharacteristic
25#473—24no—
26#472—24no—
27#452—24no—
28#449—24no—
29#435—22no—
30C8×D4—22no—
31C2×#108C2211yescharacteristic
32C2×#108C2211yescharacteristic
33#424C2211yescharacteristic
34#419C2212yes—
35#418C2212yes—
36C2×#101C2212yes—
37C2×#101C2212yes—
38C2×#100C2212yes—
39C2×#100C2212yes—
40#408C2211yescharacteristic
41#407C2212yes—
42#407C2211yescharacteristic
43#405C2211yescharacteristic
44#341C2212yes—
45C2×#41—24no—
46C2×#41—24no—
47C2×#41—24no—
48C2×#41—24no—
49C2×#41—42no—
50C2×#41—42no—
51C2×#41C2311yescharacteristic
52C22×Q8C2312yes—
53C22×D4—22no—
54C22×D4C2312yes—
55C23×C4—22no—
56C23×C4C2311yescharacteristic
57#137—24no—
58#137—42no—
59AGL(1,ℤ/8)—24no—
60AGL(1,ℤ/8)—42no—
61#135—216no—
62#135—216no—
63C2×Q16—22no—
64C2×Q16—21no—
65C2×Q16—21no—
66C2×Q16—22no—
67C2×Q16D414yescomplemented
68C2×SD16—24no—
69C2×SD16—24no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—24no—
73C2×D8—22no—
74C2×D8—21no—
75C2×D8—21no—
76C2×D8—22no—
77C2×D8D414yescomplemented
78#131—22no—
79#131—22no—
80C2×M16—21no—
81C2×M16C2311yescharacteristic
82C2×M16C2312yes—
83C2×M16C2312yes—
84C2×M16C2311yescharacteristic
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—21no—
88C22×C8D412yes—
89C22×C8D412yes—
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92#123—21no—
93C22⋊Q8D412yes—
94#121D412yes—
95#120—21no—
96C4×D4—22no—
97#117C2311yescharacteristic
98#108—22no—
99#108—22no—
100C4⋊C8—22no—
101#103—24no—
102#102—24no—
103#101—24no—
104#101—42no—
105#100—24no—
106#100—42no—
107C22⋊C8—22no—
108C22⋊C8—24no—
109C22⋊C8—22no—
110#97—21no—
111C4×C8—21no—
112C24—22no—
113C24#4111yescharacteristic
114#41—216no—
115#41—21no—
116#41—21no—
117#41—21no—
118#41—21no—
119#41—48no—
120#41—48no—
121#41—48no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8C2×D414yes—
128C2×Q8C2×D411yescharacteristic
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—44no—
137C2×D4—44no—
138C2×D4—44no—
139C2×D4C2×D414yes—
140C2×D4C2×D411yescharacteristic
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4C2×D412yes—
151C22×C4C2×D412yes—
152C22×C4#4111yescharacteristic
153C22×C4C2411yescharacteristic
154Q16—24no—
155Q16—24no—
156Q16—44no—
157SD16—216no—
158SD16—44no—
159SD16—44no—
160D8—24no—
161D8—24no—
162D8—44no—
163M16—21no—
164M16—22no—
165M16—21no—
166M16—22no—
167M16—21no—
168M16—22no—
169M16—21no—
170M16—22no—
171M16—42no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—24no—
175C2×C8—24no—
176C2×C8—21no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—21no—
180C2×C8—42no—
181C2×C8C2×D412yes—
182C2×C8C2×D411yescharacteristic
183C2×C8C2×D411yescharacteristic
184C2×C8C2×D412yes—
185C2×C8C2×D412yes—
186C4⋊C4—22no—
187C4⋊C4C2×D412yes—
188C22⋊C4—22no—
189C22⋊C4—21no—
190C22⋊C4—22no—
191C42—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—24no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—84no—
200C23#12112yes—
201C23C2×#4111yescharacteristic
202Q8—24no—
203Q8—21no—
204Q8—24no—
205Q8—21no—
206Q8—42no—
207Q8—42no—
208D4—24no—
209D4—21no—
210D4—24no—
211D4—21no—
212D4—44no—
213D4—416no—
214D4—41no—
215D4—41no—
216D4—42no—
217D4—42no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—24no—
227C2×C4—416no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4#12112yes—
233C2×C4C22×D411yescharacteristic
234C2×C4C22×D411yescharacteristic
235C2×C4C22×D411yescharacteristic
236C2×C4C22×D411yescharacteristic
237C2×C4H(5,𝔽2)11yescharacteristic
238C2×C4H(5,𝔽2)11yescharacteristic
239C8—22no—
240C8—41no—
241C8—41no—
242C8—42no—
243C8—42no—
244C8#12012yes—
245C8#12012yes—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—42no—
251C22—42no—
252C22—44no—
253C22—41no—
254C22—42no—
255C22—84no—
256C22—84no—
257C22—84no—
258C22C2×#12111yescharacteristic
259C22#54611yescharacteristic
260C22#54611yescharacteristic
261C4—22no—
262C4—44no—
263C4—41no—
264C4—42no—
265C4C2×#12011yescharacteristic
266C4#53511yescharacteristic
267C4D4211yescharacteristic
268C4D4211yescharacteristic
269C2—21no—
270C2—21no—
271C2—42no—
272C2—81no—
273C2—84no—
274C2#240811yescharacteristic
275C2#306712yes—
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#135C2semidirect
2. C2×#473C2semidirect
3. C2×#472C2semidirect
4. #3005C2semidirect
5. #2929C2semidirect
6. #2046C2semidirect
7. #2019C2semidirect
8. #1979C2semidirect
9. #1978C2semidirect
10. #1810C2semidirect
11. #1785C2semidirect
67. C2×Q16D4semidirect
77. C2×D8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#1352
C2×#4732
C2×#4722
#30052
#29292
#20462
#20192
#19792
#19782
#18102
#17852