Finite Group Databasebeta

← #25656

Subgroups — #25656

Order 256.

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

normal subgroups only

This group has 506 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×#491C211yesmaximal, characteristic, complemented
3C2×#478C211yesmaximal, characteristic
4#2584C211yesmaximal, characteristic, complemented
5#2042C212yesmaximal
6#2042C212yesmaximal
7#1985C211yesmaximal, characteristic, complemented
8#1981C211yesmaximal, characteristic, complemented
9#1865C212yesmaximal, complemented
10#1863C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic
12#556—24no—
13#556—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#126—22no—
16C2×#125C2211yescharacteristic
17C2×#124C2212yes—
18C2×C22⋊Q8—21no—
19C2×C4×Q8C2212yes—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×#117—21no—
23#491—24no—
24#478—24no—
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2212yes—
35C2×#97C2211yescharacteristic
36#398—21no—
37#395—21no—
38#389—22no—
39#389C2212yes—
40#389C2212yes—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8D411yescharacteristic
46C22×Q8C2312yes—
47C22×D4C2311yescharacteristic
48C23×C4—21no—
49C22×C8C2312yes—
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C4⋊Q8—22no—
53C4⋊Q8D414yes—
54#126—216no—
55#125—22no—
56#125—22no—
57#124—24no—
58#124—24no—
59C22⋊Q8—28no—
60C4×Q8—24no—
61C4×Q8—24no—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×D4—22no—
65C4×D4D414yes—
66#117—28no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4D411yescharacteristic
75C2×C4⋊C4D412yes—
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2312yes—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—22no—
83C2×C42—22no—
84C2×C42—21no—
85C2×C42C2312yes—
86C2×C42C2311yescharacteristic
87#107—22no—
88#106—22no—
89C4⋊C8—22no—
90#103—24no—
91#103—24no—
92#103—24no—
93#103—24no—
94#102—22no—
95#102—22no—
96#102—24no—
97#97—22no—
98#95—21no—
99#95—22no—
100#95—22no—
101C24—21no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—42no—
109C2×Q8C2×D414yes—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4C2×D412yes—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—24no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—44no—
121C22×C4C2×D412yes—
122C22×C4C2×D412yes—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2411yescharacteristic
126C2×C8—22no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C4⋊C4—28no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C4⋊C4—22no—
137C4⋊C4—24no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—44no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—82no—
148C4⋊C4C2×D412yes—
149C4⋊C4#4114yes—
150C22⋊C4—28no—
151C22⋊C4—28no—
152C22⋊C4—24no—
153C42—28no—
154C42—22no—
155C42—22no—
156C42—81no—
157C42C2×D412yes—
158C42C2×D412yes—
159C23—22no—
160C23—42no—
161C23—42no—
162C23—41no—
163C23—41no—
164C23C22×D411yescharacteristic
165Q8—24no—
166Q8—24no—
167Q8—44no—
168Q8—42no—
169Q8—42no—
170D4—22no—
171D4—22no—
172D4—42no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—48no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4#12012yes—
197C2×C4#12012yes—
198C2×C4C22×D411yescharacteristic
199C2×C4C22×D411yescharacteristic
200C2×C4C2×#4112yes—
201C2×C42−1+412yes—
202C8—41no—
203C8—41no—
204C8—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—82no—
209C22—82no—
210C22C2×#12011yescharacteristic
211C22#54812yes—
212C22#57512yes—
213C22#57711yescharacteristic
214C22#57811yescharacteristic
215C4—22no—
216C4—22no—
217C4—44no—
218C4—44no—
219C4—42no—
220C4—42no—
221C4—42no—
222C4—81no—
223C4—81no—
224C4#53211yescharacteristic
225C4#53611yescharacteristic
226C4#54712yes—
227C2—82no—
228C2#243211yescharacteristic
229C2#300511yescharacteristic
230C2#301311yescharacteristic
231C2#337612yes—
232C2#337712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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
2. C2×#491C2semidirect
4. #2584C2semidirect
7. #1985C2semidirect
8. #1981C2semidirect
9. #1865C2semidirect
10. #1863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#4912
C2×#4782
#25842
#20422
#19852
#19812
#18652
#18632
#17622