Finite Group Databasebeta

← #25619

Subgroups — #25619

Order 256.

526 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 123 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#3512C211yesmaximal, characteristic
2#3152C211yesmaximal, characteristic, complemented
3#3039C211yesmaximal, characteristic, complemented
4#2642C211yesmaximal, characteristic, complemented
5#2007C214yesmaximal, complemented
6#1998C211yesmaximal, characteristic, complemented
7#1963C211yesmaximal, characteristic, complemented
8#1882C214yesmaximal, complemented
9#1759C211yesmaximal, characteristic, complemented
10C2×#137C2214yes—
11C2×#137C2212yes—
12#553—22no—
13#553—24no—
14#552—22no—
15#541—21no—
16#540—21no—
17#537C2212yes—
18#532C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20#529—22no—
21C2×#125C2211yescharacteristic
22#520C2212yes—
23C4×#41C2211yescharacteristic
24#497—22no—
25#494—21no—
26#492—21no—
27#462—22no—
28#461—22no—
29#423C2211yescharacteristic
30C2×#104C2212yes—
31C2×#104C2214yes—
32#419C2212yes—
33C4×M16C2211yescharacteristic
34#400—24no—
35#391C2214yes—
36#376C2211yescharacteristic
37#343C2214yes—
38#339C2212yes—
39C2×#41C2311yescharacteristic
40C2×#41C2312yes—
41C22×Q8C2312yes—
42#137—28no—
43#137—44no—
44#137—44no—
45C2×Q16—22no—
46C2×Q16—24no—
47C2×SD16—22no—
48C2×SD16—24no—
49C2×M16C2312yes—
50C2×M16C2312yes—
51C4⋊Q8—22no—
52C4⋊Q8—21no—
53C4⋊Q8—21no—
54C4⋊Q8D411yescharacteristic
55#127D411yescharacteristic
56#126—24no—
57#126—24no—
58#125—24no—
59#125—21no—
60#125—21no—
61#125—22no—
62#124D412yes—
63#123—24no—
64#123—24no—
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67#121—24no—
68#121—22no—
69#121—22no—
70#121—22no—
71#121—24no—
72C4×Q8—22no—
73C4×Q8D412yes—
74C4×D4—24no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—24no—
78C4×D4—22no—
79C4×D4D412yes—
80#117—24no—
81#117—22no—
82#117—21no—
83#117C2311yescharacteristic
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4D414yes—
87C2×C4⋊C4C2312yes—
88C2×C42—21no—
89C2×C42C2312yes—
90C2×C42C2311yescharacteristic
91C4⋊C8—22no—
92#104—24no—
93#104—44no—
94#103—24no—
95#97—21no—
96C4×C8—21no—
97#95—22no—
98#95—28no—
99#41—22no—
100#41—24no—
101#41—24no—
102#41—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—24no—
106C2×Q8—24no—
107C2×Q8C2×D412yes—
108C2×Q8C2×D411yescharacteristic
109C2×D4—21no—
110C2×D4—22no—
111C2×D4—24no—
112C2×D4—22no—
113C2×D4C2×D412yes—
114C2×D4C2×D411yescharacteristic
115C22×C4—24no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4C2×D414yes—
122C22×C4C2×D412yes—
123C22×C4C2411yescharacteristic
124Q16—48no—
125Q16—44no—
126SD16—48no—
127SD16—44no—
128M16—22no—
129M16—24no—
130M16#4114yes—
131C2×C8—22no—
132C2×C8—22no—
133C4⋊C4—24no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C4⋊C4—24no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4C2×D412yes—
144C22⋊C4—24no—
145C22⋊C4—24no—
146C22⋊C4—24no—
147C22⋊C4—21no—
148C22⋊C4—22no—
149C22⋊C4—22no—
150C22⋊C4—22no—
151C42—24no—
152C42—22no—
153C42—21no—
154C42—22no—
155C42—22no—
156C42C2×D411yescharacteristic
157C42C2×D411yescharacteristic
158C42C2×D412yes—
159C23—22no—
160C23—21no—
161C23C22×D411yescharacteristic
162Q8—21no—
163Q8—21no—
164Q8—24no—
165Q8—42no—
166Q8—42no—
167Q8—44no—
168D4—21no—
169D4—21no—
170D4—24no—
171D4—42no—
172D4—44no—
173D4—82no—
174D4—84no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—44no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—82no—
189C2×C4—84no—
190C2×C4#12012yes—
191C2×C4#12012yes—
192C2×C4C22×D411yescharacteristic
193C2×C4C22×D411yescharacteristic
194C2×C4C2×#4112yes—
195C2×C4H(5,𝔽2)12yes—
196C8—24no—
197C8—42no—
198C22—22no—
199C22—42no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22C2×#12011yescharacteristic
205C22#55012yes—
206C4—22no—
207C4—22no—
208C4—42no—
209C4—41no—
210C4—44no—
211C4—42no—
212C4—82no—
213C4—82no—
214C4#53211yescharacteristic
215C4#53511yescharacteristic
216C4#54612yes—
217C2—22no—
218C2—82no—
219C2—81no—
220C2#243111yescharacteristic
221C2#302212yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3512C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3152C2semidirect
3. #3039C2semidirect
4. #2642C2semidirect
5. #2007C2semidirect
6. #1998C2semidirect
7. #1963C2semidirect
8. #1882C2semidirect
9. #1759C2semidirect

Maximal subgroups

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

subgroupindex
#35122
#31522
#30392
#26422
#20072
#19982
#19632
#18822
#17592