Finite Group Databasebeta

← #25933

Subgroups — #25933

Order 256.

683 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 683 conjugacy classes of subgroups — too many to draw. Only its 151 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×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#486C211yesmaximal, characteristic, complemented
3C2×#459C211yesmaximal, characteristic
4#2548C211yesmaximal, characteristic, complemented
5#2023C214yesmaximal, complemented
6#1988C211yesmaximal, characteristic, complemented
7#1984C211yesmaximal, characteristic, complemented
8#1879C214yesmaximal, complemented
9C4×#102C211yesmaximal, characteristic, complemented
10C23×Q8—21no—
11C22×D8C2212yes—
12D4×Q8—24no—
13D4×Q8—24no—
14C2×C4⋊Q8C2212yes—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#127C2211yescharacteristic
17C2×#124—21no—
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8—22no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2212yes—
22C2×C4×D4C2211yescharacteristic
23#486—24no—
24#459—24no—
25C2×C4⋊C8C2211yescharacteristic
26C2×#102C2214yes—
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×C4×C8C2211yescharacteristic
30#394—22no—
31#390—21no—
32#390C2214yes—
33C4×C4⋊C4C2211yescharacteristic
34#340C2214yes—
35#340C2212yes—
36#336C2214yes—
37C22×Q8—28no—
38C22×Q8—22no—
39C22×Q8—21no—
40C22×Q8—44no—
41C22×Q8—44no—
42C22×Q8D411yescharacteristic
43C22×D4—21no—
44C22×D4—21no—
45C22×D4C2312yes—
46C22×D4C2311yescharacteristic
47C23×C4—22no—
48C23×C4—21no—
49C2×D8—24no—
50C2×D8—28no—
51C22×C8C2312yes—
52C22×C8C2312yes—
53C4⋊Q8—24no—
54C4⋊Q8—24no—
55C4⋊Q8—22no—
56C4⋊Q8D414yes—
57#127—22no—
58#127—22no—
59#124—44no—
60C22⋊Q8—28no—
61C22⋊Q8—216no—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×D4—24no—
65C4×D4—24no—
66C4×D4—22no—
67C4×D4D414yes—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4D411yescharacteristic
71C2×C4⋊C4D412yes—
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2311yescharacteristic
75C2×C22⋊C4—22no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—24no—
79C2×C42—21no—
80C2×C42C2312yes—
81C2×C42C2311yescharacteristic
82C4⋊C8—22no—
83#102—24no—
84#102—24no—
85#102—28no—
86C4×C8—22no—
87#95—21no—
88C24—21no—
89C24—22no—
90C2×Q8—28no—
91C2×Q8—22no—
92C2×Q8—24no—
93C2×Q8—22no—
94C2×Q8—44no—
95C2×Q8—432no—
96C2×Q8—44no—
97C2×Q8—42no—
98C2×Q8—42no—
99C2×Q8D814yescomplemented
100C2×D4—21no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—24no—
104C2×D4—22no—
105C2×D4—24no—
106C2×D4—42no—
107C2×D4—42no—
108C2×D4—42no—
109C2×D4—42no—
110C2×D4—82no—
111C2×D4C2×D412yes—
112C2×D4#4114yes—
113C22×C4—28no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—44no—
119C22×C4—44no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4C2×D412yes—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D412yes—
125C22×C4C2×D411yescharacteristic
126C22×C4C2411yescharacteristic
127D8—48no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—24no—
132C2×C8—22no—
133C4⋊C4—28no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4C2×D414yes—
142C4⋊C4C2×D412yes—
143C22⋊C4—28no—
144C22⋊C4—24no—
145C22⋊C4—48no—
146C22⋊C4—84no—
147C42—22no—
148C42—22no—
149C42—42no—
150C42C2×D412yes—
151C42C2×D412yes—
152C23—22no—
153C23—24no—
154C23—41no—
155C23—41no—
156C23—42no—
157C23—42no—
158C23—44no—
159C23—42no—
160C23—42no—
161C23—44no—
162C23C22×D411yescharacteristic
163Q8—28no—
164Q8—416no—
165Q8—48no—
166D4—22no—
167D4—24no—
168D4—42no—
169D4—42no—
170D4—44no—
171D4—84no—
172D4—84no—
173D4#12414yescomplemented
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—416no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—48no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4#12012yes—
191C2×C4#12012yes—
192C2×C4C2×D814yes—
193C2×C4C2×D812yes—
194C2×C4AGL(1,ℤ/8)12yes—
195C2×C4C22×D411yescharacteristic
196C2×C4C22×D411yescharacteristic
197C2×C4C2×#4112yes—
198C2×C42−1+412yes—
199C8—42no—
200C8—42no—
201C22—24no—
202C22—44no—
203C22—44no—
204C22—42no—
205C22—42no—
206C22—84no—
207C22—84no—
208C22—82no—
209C22—82no—
210C22—82no—
211C22C2×#12011yescharacteristic
212C22#54812yes—
213C22C22×D811yescharacteristic
214C22C2×AGL(1,ℤ/8)11yescharacteristic
215C22#57712yes—
216C4—24no—
217C4—22no—
218C4—44no—
219C4—42no—
220C4—42no—
221C4—82no—
222C4#44714yes—
223C4C2×#12411yescharacteristic
224C4#53611yescharacteristic
225C4#54812yes—
226C2—44no—
227C2—162no—
228C2#243611yescharacteristic
229C2C2×#44711yescharacteristic
230C2#301011yescharacteristic
231C2#335812yes—
232C2#337312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×D4×Q8C2semidirect
2. C2×#486C2semidirect
4. #2548C2semidirect
5. #2023C2semidirect
6. #1988C2semidirect
7. #1984C2semidirect
8. #1879C2semidirect
9. C4×#102C2semidirect
99. C2×Q8D8semidirect
173. D4#124semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
C2×#4862
C2×#4592
#25482
#20232
#19882
#19842
#18792
C4×#1022