Finite Group Databasebeta

← #25578

Subgroups — #25578

Order 256.

613 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 613 conjugacy classes of subgroups — too many to draw. Only its 131 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×#555C211yesmaximal, characteristic, complemented
2C2×#489C211yesmaximal, characteristic, complemented
3C2×#463C211yesmaximal, characteristic, complemented
4#2540C211yesmaximal, characteristic, complemented
5#2000C212yesmaximal, complemented
6#1999C212yesmaximal, complemented
7#1988C211yesmaximal, characteristic, complemented
8#1955C211yesmaximal, characteristic, complemented
9#1875C212yesmaximal, complemented
10#1873C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C22×D8C2211yescharacteristic
14#555—24no—
15#555—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#124—22no—
18C2×#124C2211yescharacteristic
19C2×C22⋊Q8C2212yes—
20C2×#121—21no—
21C2×#121C2212yes—
22C2×#121C2212yes—
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×#117—21no—
26C2×#117C2212yes—
27#489—24no—
28#463—24no—
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2211yescharacteristic
32C2×#102C2212yes—
33C2×#102C2211yescharacteristic
34C2×#102C2212yes—
35C2×C22⋊C8C2212yes—
36C2×C22⋊C8C2212yes—
37C2×#97C2211yescharacteristic
38#399—21no—
39#392—21no—
40#390—22no—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—21no—
46C22×Q8C2311yescharacteristic
47C22×D4—22no—
48C22×D4—22no—
49C22×D4D411yescharacteristic
50C22×D4C2311yescharacteristic
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C23×C4C2312yes—
54C2×SD16—22no—
55C2×SD16—42no—
56C2×D8—22no—
57C2×D8—42no—
58C22×C8C2311yescharacteristic
59C22×C8C2311yescharacteristic
60C22×C8C2312yes—
61C4⋊Q8—22no—
62C4⋊Q8D414yescomplemented
63#124—216no—
64#124—22no—
65#124—22no—
66C22⋊Q8—24no—
67C22⋊Q8—42no—
68#121—28no—
69#121—24no—
70#121—24no—
71#121—24no—
72#121—42no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4D414yescomplemented
77#117—28no—
78#117—24no—
79#117—24no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2312yes—
85C2×C4⋊C4C2312yes—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4D412yes—
94C2×C42—22no—
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C4⋊C8—22no—
98#103—22no—
99#103—24no—
100#102—22no—
101#102—24no—
102#102—24no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105#97—22no—
106#95—21no—
107C24—21no—
108C24—21no—
109C24C2×D412yes—
110C2×Q8—24no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—81no—
116C2×Q8—81no—
117C2×D4—28no—
118C2×D4—22no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—42no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4—82no—
130C2×D4—82no—
131C2×D4—82no—
132C2×D4C2×D412yes—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—44no—
145C22×C4C2×D412yes—
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2411yescharacteristic
149SD16—44no—
150D8—44no—
151C2×C8—22no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C4⋊C4—24no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—44no—
164C4⋊C4C2×D412yes—
165C4⋊C4C2×D414yes—
166C4⋊C4#4114yes—
167C22⋊C4—28no—
168C22⋊C4—28no—
169C22⋊C4—24no—
170C22⋊C4—24no—
171C22⋊C4—44no—
172C22⋊C4—44no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C42—28no—
176C42—81no—
177C42C2×D412yes—
178C42C2×D412yes—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—42no—
188C23—42no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—42no—
193C23—42no—
194C23—41no—
195C23—41no—
196C23C22×D411yescharacteristic
197Q8—41no—
198Q8—41no—
199Q8—42no—
200Q8—84no—
201D4—22no—
202D4—22no—
203D4—44no—
204D4—41no—
205D4—41no—
206D4—42no—
207D4—42no—
208D4—88no—
209D4—88no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—48no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4#12012yes—
232C2×C4#12012yes—
233C2×C4C22×D411yescharacteristic
234C2×C4C22×D411yescharacteristic
235C2×C4C2×#4112yes—
236C2×C4H(5,𝔽2)12yes—
237C8—41no—
238C8—41no—
239C8—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22—82no—
252C22—81no—
253C22—81no—
254C22—81no—
255C22—82no—
256C22—82no—
257C22C2×#12011yescharacteristic
258C22#54612yes—
259C22#57512yes—
260C22#57611yescharacteristic
261C22#57711yescharacteristic
262C4—22no—
263C4—22no—
264C4—42no—
265C4—42no—
266C4—44no—
267C4—44no—
268C4—81no—
269C4#53211yescharacteristic
270C4#53511yescharacteristic
271C4#55012yes—
272C2—82no—
273C2—82no—
274C2—161no—
275C2#243111yescharacteristic
276C2#300511yescharacteristic
277C2#301211yescharacteristic
278C2#331912yes—
279C2#332112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
1. C2×#555C2semidirect
2. C2×#489C2semidirect
3. C2×#463C2semidirect
4. #2540C2semidirect
5. #2000C2semidirect
6. #1999C2semidirect
7. #1988C2semidirect
8. #1955C2semidirect
9. #1875C2semidirect
10. #1873C2semidirect
11. #1762C2semidirect
62. C4⋊Q8D4semidirect
76. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5552
C2×#4892
C2×#4632
#25402
#20002
#19992
#19882
#19552
#18752
#18732
#17622