Finite Group Databasebeta

← #25882

Subgroups — #25882

Order 256.

667 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 667 conjugacy classes of subgroups — too many to draw. Only its 143 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×#547C211yesmaximal, characteristic
2C2×#487C211yesmaximal, characteristic, complemented
3C2×#474C211yesmaximal, characteristic, complemented
4#2547C211yesmaximal, characteristic, complemented
5#2036C214yesmaximal, complemented
6#1981C211yesmaximal, characteristic, complemented
7#1954C211yesmaximal, characteristic, complemented
8#1788C214yesmaximal, complemented
9C4×#102C211yesmaximal, characteristic, complemented
10C22×#41C2212yes—
11#547—24no—
12#547—24no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124—21no—
16C2×#123—22no—
17C2×C22⋊Q8C2212yes—
18C2×#121—21no—
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C22×C4⋊C4—21no—
22#487—24no—
23#474—24no—
24C2×#107C2212yes—
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2214yes—
27C2×#103C2214yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×C22⋊C8C2214yes—
31C2×C4×C8C2211yescharacteristic
32#397—22no—
33#391—21no—
34#380C2214yes—
35C4×C4⋊C4C2211yescharacteristic
36#340C2212yes—
37C2×#41—24no—
38C2×#41—24no—
39C2×#41—28no—
40C2×#41—44no—
41C22×Q8—21no—
42C22×Q8C2312yes—
43C22×D4—22no—
44C22×D4D411yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4—22no—
48C23×C4C2312yes—
49C22×C8C2312yes—
50C22×C8C2312yes—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C4⋊Q8D414yes—
55#124—44no—
56#123—216no—
57C22⋊Q8—24no—
58C22⋊Q8—24no—
59#121—28no—
60C4×D4—22no—
61C4×D4—22no—
62C4×D4—22no—
63C4×D4D414yes—
64C2×C4⋊C4—28no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4D411yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2311yescharacteristic
73C2×C22⋊C4—22no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4D412yes—
77C2×C42—21no—
78C2×C42C2311yescharacteristic
79#107—24no—
80C4⋊C8—22no—
81#103—28no—
82#103—44no—
83#102—24no—
84#102—24no—
85C22⋊C8—44no—
86C4×C8—22no—
87#95—21no—
88#95—24no—
89C24—21no—
90C24C2×D412yes—
91#41—416no—
92#41—48no—
93#41—48no—
94C2×Q8—22no—
95C2×Q8—24no—
96C2×Q8—24no—
97C2×Q8—42no—
98C2×Q8—42no—
99C2×Q8—42no—
100C2×Q8C2×D414yes—
101C2×D4—28no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4—48no—
108C2×D4C2×D412yes—
109C2×D4#4114yes—
110C22×C4—28no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—24no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—24no—
117C22×C4—24no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—48no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4C2×D412yes—
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2411yescharacteristic
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—24no—
132C4⋊C4—28no—
133C4⋊C4—22no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4C2×D412yes—
143C22⋊C4—28no—
144C22⋊C4—24no—
145C22⋊C4—24no—
146C22⋊C4—48no—
147C42—42no—
148C42C2×D412yes—
149C42C2×D412yes—
150C23—22no—
151C23—24no—
152C23—22no—
153C23—24no—
154C23—22no—
155C23—22no—
156C23—24no—
157C23—41no—
158C23—41no—
159C23—42no—
160C23C22×D411yescharacteristic
161Q8—28no—
162Q8—42no—
163Q8—42no—
164Q8—84no—
165D4—22no—
166D4—24no—
167D4—416no—
168D4—44no—
169D4#12414yescomplemented
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—416no—
176C2×C4—44no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—48no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4#12012yes—
190C2×C4#12012yes—
191C2×C4#13512yes—
192C2×C4AGL(1,ℤ/8)12yes—
193C2×C4C22×D411yescharacteristic
194C2×C4C22×D411yescharacteristic
195C2×C4C2×#4112yes—
196C2×C4H(5,𝔽2)12yes—
197C8—42no—
198C8—42no—
199C22—24no—
200C22—44no—
201C22—42no—
202C22—42no—
203C22—44no—
204C22—44no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—44no—
209C22C2×#12011yescharacteristic
210C22#54612yes—
211C22C2×#13511yescharacteristic
212C22C2×AGL(1,ℤ/8)11yescharacteristic
213C22#57812yes—
214C4—22no—
215C4—42no—
216C4—44no—
217C4—42no—
218C4—82no—
219C4—82no—
220C4#44914yes—
221C4C2×#12411yescharacteristic
222C4#53511yescharacteristic
223C4#54612yes—
224C2—44no—
225C2—82no—
226C2#243511yescharacteristic
227C2C2×#44911yescharacteristic
228C2#301111yescharacteristic
229C2#330212yes—
230C2#333012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#487C2semidirect
3. C2×#474C2semidirect
4. #2547C2semidirect
5. #2036C2semidirect
6. #1981C2semidirect
7. #1954C2semidirect
8. #1788C2semidirect
9. C4×#102C2semidirect
169. D4#124semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4872
C2×#4742
#25472
#20362
#19812
#19542
#17882
C4×#1022