Finite Group Databasebeta

← #19239

Subgroups — #19239

Order 256.

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

normal subgroups only

This group has 329 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
1#3246C211yesmaximal, characteristic
2#3229C211yesmaximal, characteristic, complemented
3#3152C211yesmaximal, characteristic
4#2962C211yesmaximal, characteristic, complemented
5#1690C212yesmaximal, complemented
6#1686C212yesmaximal, complemented
7#1670C212yesmaximal, complemented
8#1666C212yesmaximal, complemented
9#1565C211yesmaximal, characteristic, complemented
10#1540C211yesmaximal, characteristic
11C2×#332C211yesmaximal, characteristic, complemented
12C2×#137—21no—
13C2×#137—21no—
14#533C2211yescharacteristic
15#532C2211yescharacteristic
16C2×C4⋊Q8C2211yescharacteristic
17#498—21no—
18#497—22no—
19#494C2212yes—
20#492C2212yes—
21#487—21no—
22#483—22no—
23#476—21no—
24#475C2212yes—
25#475C2212yes—
26#475C2212yes—
27#465—22no—
28#461—21no—
29#460C2212yes—
30#446—22no—
31#446C2212yes—
32C8×Q8C2212yes—
33#431C2211yescharacteristic
34#424—21no—
35C2×C4⋊C8C2211yescharacteristic
36C2×C4⋊C8C2211yescharacteristic
37#405—21no—
38C4×M16C2211yescharacteristic
39#335C2212yes—
40#334C2212yes—
41#332C2212yes—
42#332C2212yes—
43#332C2211yescharacteristic
44#332C2211yescharacteristic
45#331C2211yescharacteristic
46#331C2211yescharacteristic
47C2×#41—21no—
48C22×Q8—21no—
49#137—44no—
50#137—44no—
51C2×Q16—22no—
52C2×Q16—22no—
53C2×SD16—22no—
54C2×SD16—22no—
55C2×SD16—22no—
56C2×M16D411yescharacteristic
57C2×M16D411yescharacteristic
58C22×C8—21no—
59C22×C8—21no—
60C4⋊Q8—21no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8D411yescharacteristic
64C4⋊Q8D411yescharacteristic
65C4⋊Q8C2312yes—
66C4⋊Q8C2311yescharacteristic
67#127C2311yescharacteristic
68#125—21no—
69#124—21no—
70C22⋊Q8—21no—
71C22⋊Q8D412yes—
72#121—21no—
73#121—21no—
74C4×Q8—21no—
75C4×Q8C2312yes—
76C4×D4—21no—
77C4×D4—21no—
78#117—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C42C2311yescharacteristic
82#106—22no—
83#106—22no—
84#106—22no—
85C4⋊C8—22no—
86C4⋊C8—22no—
87C4⋊C8—21no—
88C4⋊C8—21no—
89C4⋊C8C2311yescharacteristic
90C4⋊C8C2311yescharacteristic
91C4⋊C8C2311yescharacteristic
92C4⋊C8C2311yescharacteristic
93C4⋊C8C2312yes—
94#103—22no—
95#103—22no—
96#103—22no—
97#102—22no—
98C22⋊C8—21no—
99#97—22no—
100#97D412yes—
101C4×C8—22no—
102C4×C8—21no—
103C4×C8C2312yes—
104#41—42no—
105#41—42no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—22no—
109C2×Q8—41no—
110C2×Q8—42no—
111C2×Q8—42no—
112C2×Q8C2×D412yes—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—41no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4#4111yescharacteristic
122Q16—44no—
123Q16—44no—
124SD16—44no—
125SD16—44no—
126SD16—82no—
127M16—22no—
128M16SD1614yescomplemented
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—22no—
134C2×C8—42no—
135C2×C8—81no—
136C2×C8C2×D412yes—
137C2×C8C2×D412yes—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4C2×D412yes—
150C22⋊C4—22no—
151C22⋊C4—41no—
152C22⋊C4—41no—
153C42—22no—
154C42C2×D411yescharacteristic
155C42C2×D411yescharacteristic
156C42#4111yescharacteristic
157C42C2411yescharacteristic
158C23—41no—
159C23#12111yescharacteristic
160Q8—22no—
161Q8—22no—
162Q8—42no—
163Q8—41no—
164Q8—42no—
165Q8—41no—
166Q8—81no—
167Q8—82no—
168D4—41no—
169D4—41no—
170D4—41no—
171D4—41no—
172D4—81no—
173D4—82no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4#12111yescharacteristic
184C2×C4#12111yescharacteristic
185C2×C4#12111yescharacteristic
186C2×C4C2×SD1612yes—
187C2×C4AGL(1,ℤ/8)12yes—
188C2×C4C22×D411yescharacteristic
189C2×C4C22×D411yescharacteristic
190C2×C4C2×#4111yescharacteristic
191C8—22no—
192C8—41no—
193C8—41no—
194C8—41no—
195C8—42no—
196C8—41no—
197C8C2×SD1614yes—
198C22—21no—
199C22—21no—
200C22—81no—
201C22—81no—
202C22—81no—
203C22#46012yes—
204C22C2×#12111yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—42no—
209C4—42no—
210C4—81no—
211C4—82no—
212C4#46012yes—
213C4C22×SD1611yescharacteristic
214C4C2×AGL(1,ℤ/8)11yescharacteristic
215C4#57711yescharacteristic
216C4#57811yescharacteristic
217C2—22no—
218C2—161no—
219C2C2×#46011yescharacteristic
220C2#306311yescharacteristic
221C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#532C22
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
4. #2962C2semidirect
5. #1690C2semidirect
6. #1686C2semidirect
7. #1670C2semidirect
8. #1666C2semidirect
9. #1565C2semidirect
11. C2×#332C2semidirect
128. M16SD16semidirect

Maximal subgroups

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

subgroupindex
#32462
#32292
#31522
#29622
#16902
#16862
#16702
#16662
#15652
#15402
C2×#3322