Finite Group Databasebeta

← #25711

Subgroups — #25711

Order 256.

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

normal subgroups only

This group has 508 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×#556C211yesmaximal, characteristic
2C2×#501C211yesmaximal, characteristic, complemented
3C2×#479C211yesmaximal, characteristic, complemented
4#2562C211yesmaximal, characteristic
5#2036C212yesmaximal, complemented
6#2035C212yesmaximal, complemented
7#1987C211yesmaximal, characteristic, complemented
8#1955C211yesmaximal, characteristic
9#1854C212yesmaximal, complemented
10#1854C212yesmaximal, complemented
11#1763C211yesmaximal, characteristic, complemented
12#556—24no—
13#556—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#126—22no—
16C2×#125C2211yescharacteristic
17C2×#124—21no—
18C2×C22⋊Q8C2212yes—
19C2×C4×Q8—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×#117C2212yes—
23#501—24no—
24#479—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2212yes—
27C2×#106C2211yescharacteristic
28C2×#106C2212yes—
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2212yes—
33C2×C22⋊C8C2212yes—
34C2×C22⋊C8C2212yes—
35C2×#97C2211yescharacteristic
36#400—21no—
37#394—21no—
38#388—22no—
39#380C2212yes—
40#380C2212yes—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—21no—
46C22×Q8C2311yescharacteristic
47C22×D4D411yescharacteristic
48C23×C4C2312yes—
49C22×C8C2311yescharacteristic
50C22×C8C2311yescharacteristic
51C22×C8C2312yes—
52C4⋊Q8—22no—
53C4⋊Q8D414yes—
54#126—216no—
55#125—22no—
56#125—22no—
57#124—28no—
58C22⋊Q8—24no—
59C22⋊Q8—24no—
60C4×Q8—28no—
61C4×Q8—22no—
62C4×Q8D414yes—
63C4×D4—22no—
64C4×D4—22no—
65#117—24no—
66#117—24no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4D411yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2312yes—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4D412yes—
83C2×C42—22no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42C2311yescharacteristic
87#107—22no—
88#107—24no—
89#106—22no—
90#106—24no—
91C4⋊C8—22no—
92#103—22no—
93#103—22no—
94#103—24no—
95C22⋊C8—24no—
96C22⋊C8—24no—
97#97—22no—
98#95—21no—
99#95—22no—
100#95—22no—
101C24C2×D412yes—
102C2×Q8—24no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—44no—
107C2×Q8C2×D412yes—
108C2×D4—22no—
109C2×D4—42no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4C2×D412yes—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2411yescharacteristic
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8#4114yes—
133C4⋊C4—28no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4C2×D412yes—
148C4⋊C4C2×D414yes—
149C22⋊C4—28no—
150C22⋊C4—28no—
151C22⋊C4—24no—
152C22⋊C4—44no—
153C22⋊C4—82no—
154C42—28no—
155C42—24no—
156C42—81no—
157C42C2×D412yes—
158C42C2×D412yes—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—22no—
163C23—22no—
164C23—22no—
165C23—22no—
166C23C22×D411yescharacteristic
167Q8—22no—
168Q8—22no—
169Q8—48no—
170Q8—42no—
171D4—44no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—42no—
182C2×C4—44no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4#12012yes—
198C2×C4#12012yes—
199C2×C4C22×D411yescharacteristic
200C2×C4C22×D411yescharacteristic
201C2×C4C2×#4112yes—
202C2×C42−1+412yes—
203C8—24no—
204C8—41no—
205C8—41no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22C2×#12011yescharacteristic
214C22#54712yes—
215C22#57512yes—
216C22#57711yescharacteristic
217C22#57811yescharacteristic
218C4—22no—
219C4—22no—
220C4—42no—
221C4—42no—
222C4—44no—
223C4—42no—
224C4—82no—
225C4—81no—
226C4—81no—
227C4#53211yescharacteristic
228C4#53611yescharacteristic
229C4#54812yes—
230C2—82no—
231C2#243211yescharacteristic
232C2#300511yescharacteristic
233C2#301311yescharacteristic
234C2#334912yes—
235C2#335012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
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×#501C2semidirect
3. C2×#479C2semidirect
5. #2036C2semidirect
6. #2035C2semidirect
7. #1987C2semidirect
9. #1854C2semidirect
10. #1854C2semidirect
11. #1763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#5012
C2×#4792
#25622
#20362
#20352
#19872
#19552
#18542
#17632