Finite Group Databasebeta

← #26273

Subgroups — #26273

Order 256.

374 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 115 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#3156C211yesmaximal, characteristic
2#3097C212yesmaximal, complemented
3#2952C211yesmaximal, characteristic
4#2052C212yesmaximal, complemented
5#2038C212yesmaximal, complemented
6#1979C211yesmaximal, characteristic, complemented
7#1975C211yesmaximal, characteristic
8#1971C211yesmaximal, characteristic, complemented
9#1951C211yesmaximal, characteristic, complemented
10#1882C212yesmaximal, complemented
11#1773C211yesmaximal, characteristic, complemented
12C2×#137C2212yes—
13C2×#135C2211yescharacteristic
14C22×Q16C2211yescharacteristic
15#536C2212yes—
16#520C2212yes—
17#497—22no—
18#495—21no—
19#494—21no—
20#485—22no—
21#485—22no—
22#484—22no—
23#483—22no—
24#442—21no—
25#441—21no—
26C4×SD16—22no—
27C2×#108C2211yescharacteristic
28C2×#108C2211yescharacteristic
29#428C2212yes—
30#427C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×#104C2212yes—
33#419C2212yes—
34#417C2212yes—
35C2×#103C2212yes—
36C2×#101C2212yes—
37#408C2212yes—
38#405C2211yescharacteristic
39#405C2211yescharacteristic
40C2×#97C2211yescharacteristic
41#391C2212yes—
42#343C2212yes—
43#339C2211yescharacteristic
44#339C2211yescharacteristic
45C2×#41C2312yes—
46C22×Q8C2312yes—
47#137—24no—
48#137—42no—
49#135—22no—
50#135—22no—
51C2×Q16—22no—
52C2×Q16—21no—
53C2×Q16—21no—
54C2×Q16—22no—
55C2×Q16—22no—
56C2×Q16D411yescharacteristic
57C2×SD16—22no—
58C2×SD16D412yes—
59C2×D8D411yescharacteristic
60C2×M16C2311yescharacteristic
61C2×M16C2312yes—
62C2×M16C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C4⋊Q8—22no—
66#124—22no—
67#123—42no—
68C22⋊Q8—22no—
69C22⋊Q8—22no—
70C4×Q8—22no—
71C4×D4—22no—
72C4×D4—42no—
73#117—22no—
74#117C2311yescharacteristic
75#117C2311yescharacteristic
76C2×C4⋊C4—22no—
77C2×C4⋊C4C2312yes—
78C2×C42C2311yescharacteristic
79#108—22no—
80#108—22no—
81#107—22no—
82#107D412yes—
83#106—21no—
84#106—21no—
85#106—22no—
86#106D412yes—
87#104—42no—
88#103—22no—
89#103—22no—
90#103—22no—
91#103—22no—
92#103—22no—
93#102—22no—
94#101—42no—
95C22⋊C8—22no—
96C22⋊C8—22no—
97#97—21no—
98#97—21no—
99#97—22no—
100C4×C8—21no—
101C4×C8—21no—
102#95—42no—
103#41—22no—
104#41—22no—
105#41—22no—
106#41—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8C2×D412yes—
114C2×D4—22no—
115C2×D4C2×D412yes—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—42no—
121C22×C4C2411yescharacteristic
122Q16—22no—
123Q16—44no—
124Q16—42no—
125Q16—42no—
126SD16—24no—
127SD16—44no—
128D8—22no—
129M16—21no—
130M16—21no—
131M16—22no—
132M16—21no—
133M16—22no—
134M16#4112yes—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—22no—
140C2×C8C2×D411yescharacteristic
141C2×C8C2×D411yescharacteristic
142C2×C8C2×D411yescharacteristic
143C2×C8C2×D411yescharacteristic
144C2×C8#4112yes—
145C2×C8#4112yes—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4C2×D412yes—
152C4⋊C4C2×D412yes—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—42no—
163C23—22no—
164C23H(5,𝔽2)11yescharacteristic
165Q8—22no—
166Q8—22no—
167Q8—42no—
168Q8—42no—
169Q8—42no—
170Q8—42no—
171D4—22no—
172D4—22no—
173D4—42no—
174D4—42no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—82no—
187C2×C4—162no—
188C2×C4C22×D411yescharacteristic
189C2×C4C22×D411yescharacteristic
190C2×C4C2×#4111yescharacteristic
191C2×C4C2×#4111yescharacteristic
192C2×C4C2×#4111yescharacteristic
193C2×C4H(5,𝔽2)11yescharacteristic
194C8—21no—
195C8—21no—
196C8—22no—
197C8—41no—
198C8—42no—
199C8#12312yes—
200C8#12312yes—
201C22—22no—
202C22—42no—
203C22—42no—
204C22—42no—
205C22#54011yescharacteristic
206C22D4211yescharacteristic
207C22#55911yescharacteristic
208C4—42no—
209C4—42no—
210C4—42no—
211C4—81no—
212C4—82no—
213C4—82no—
214C4C2×#12311yescharacteristic
215C4#53211yescharacteristic
216C4#54611yescharacteristic
217C4#55011yescharacteristic
218C2—21no—
219C2—21no—
220C2—82no—
221C2#244511yescharacteristic
222C2#328912yes—
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#135C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3097C2semidirect
4. #2052C2semidirect
5. #2038C2semidirect
6. #1979C2semidirect
8. #1971C2semidirect
9. #1951C2semidirect
10. #1882C2semidirect
11. #1773C2semidirect

Maximal subgroups

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

subgroupindex
#31562
#30972
#29522
#20522
#20382
#19792
#19752
#19712
#19512
#18822
#17732