Finite Group Databasebeta

← #18781

Subgroups — #18781

Order 256.

455 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 121 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
2#3229C211yesmaximal, characteristic, complemented
3#3104C211yesmaximal, characteristic, complemented
4#3077C211yesmaximal, characteristic
5#1652C214yesmaximal, complemented
6#1638C214yesmaximal, complemented
7#1546C211yesmaximal, characteristic, complemented
8#1487C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic
10#556—22no—
11#556—22no—
12#556—22no—
13#556C2214yes—
14#533C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#126—22no—
17C2×#124—21no—
18C2×C22⋊Q8—21no—
19C2×C4×Q8—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×#117—21no—
23#498C2212yescomplemented
24#487C2212yescomplemented
25#484—21no—
26#483—22no—
27#478—22no—
28#477C2212yescomplemented
29#476C2212yes—
30#475—22no—
31#474C2212yes—
32#462C2212yescomplemented
33#451—21no—
34C4×Q16—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37#428—21no—
38#423C2211yescharacteristic
39#418—21no—
40#332C2214yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yes—
43C22×Q8—21no—
44C22×Q8D411yescharacteristic
45C22×D4D411yescharacteristic
46C23×C4—21no—
47C2×Q16—22no—
48C2×M16—21no—
49C4⋊Q8—21no—
50C4⋊Q8—21no—
51C4⋊Q8D412yes—
52C4⋊Q8C2311yescharacteristic
53C4⋊Q8C2311yescharacteristic
54C4⋊Q8C2312yes—
55#126—28no—
56#126—28no—
57#125—21no—
58#124—24no—
59#124—24no—
60C22⋊Q8—24no—
61C22⋊Q8—24no—
62C22⋊Q8—21no—
63C22⋊Q8—21no—
64C4×Q8—24no—
65C4×Q8—24no—
66C4×Q8—21no—
67C4×Q8—21no—
68C4×Q8—21no—
69C4×Q8—21no—
70C4×Q8D412yes—
71C4×Q8C2312yes—
72C4×D4—21no—
73C4×D4—21no—
74C4×D4D412yes—
75C4×D4C2312yes—
76#117—24no—
77#117—24no—
78#117—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D412yes—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—21no—
88C2×C42—22no—
89C2×C42—21no—
90C2×C42C2311yescharacteristic
91#107—22no—
92#107—22no—
93#106—22no—
94#106—24no—
95C4⋊C8—21no—
96C4⋊C8—21no—
97C4⋊C8C2312yes—
98C4⋊C8C2312yes—
99#103—22no—
100#103—22no—
101#103—24no—
102#102—22no—
103#102—22no—
104C22⋊C8—21no—
105C22⋊C8—21no—
106C4×C8—21no—
107C4×C8C2312yes—
108C24—21no—
109C2×Q8—21no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8C2×D412yes—
118C2×D4—21no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4C2×D412yes—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—44no—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132Q16—44no—
133M16—42no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C4⋊C4—28no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4C2×D414yes—
152C4⋊C4C2×D412yes—
153C4⋊C4C2×D412yes—
154C22⋊C4—28no—
155C22⋊C4—28no—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—41no—
159C42—28no—
160C42—22no—
161C42—22no—
162C42—41no—
163C42C2×D411yescharacteristic
164C42C2×D411yescharacteristic
165C42C2×D411yescharacteristic
166C42C2411yescharacteristic
167C23—22no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23#12011yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—42no—
175Q8—42no—
176Q8—42no—
177Q8—42no—
178Q8—44no—
179Q8—42no—
180D4—22no—
181D4—22no—
182D4—42no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—41no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4#12011yescharacteristic
198C2×C4#12011yescharacteristic
199C2×C4#12011yescharacteristic
200C2×C4#13512yes—
201C2×C4#13712yes—
202C2×C4C22×D411yescharacteristic
203C2×C4C22×D411yescharacteristic
204C2×C4C22×D411yescharacteristic
205C8—42no—
206C8—42no—
207C8—42no—
208C22—21no—
209C22—21no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—84no—
214C22#45212yes—
215C22C2×#12011yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—44no—
220C4—42no—
221C4—42no—
222C4—42no—
223C4—81no—
224C4—81no—
225C4#45212yes—
226C4C2×#13511yescharacteristic
227C4C2×#13711yescharacteristic
228C4#57512yes—
229C2—22no—
230C2—82no—
231C2C2×#45211yescharacteristic
232C2#300511yescharacteristic
233C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC22×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3229C2semidirect
3. #3104C2semidirect
5. #1652C2semidirect
6. #1638C2semidirect
7. #1546C2semidirect
8. #1487C2semidirect
23. #498C22semidirect
24. #487C22semidirect
28. #477C22semidirect
32. #462C22semidirect
40. #332C22semidirect
41. Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
#32292
#31042
#30772
#16522
#16382
#15462
#14872
#14782