Finite Group Databasebeta

← #28173

Subgroups — #28173

Order 256.

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

normal subgroups only

This group has 433 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
1C2×#464C212yesmaximal
2#2327C211yesmaximal, characteristic, complemented
3#2326C211yesmaximal, characteristic
4#2074C211yesmaximal, characteristic, complemented
5#2073C211yesmaximal, characteristic
6#2016C211yesmaximal, characteristic, complemented
7#2013C211yesmaximal, characteristic
8#1878C212yesmaximal, complemented
9#1877C212yesmaximal
10#1832C211yesmaximal, characteristic, complemented
11#1803C212yesmaximal, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C4×Q8C2212yes—
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#464—28no—
20C2×C4⋊C8C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#102C2212yes—
25#391C2211yescharacteristic
26#391C2212yes—
27#390C2211yescharacteristic
28#388—22no—
29#387—22no—
30#387—22no—
31#385—22no—
32#384C2211yescharacteristic
33#382C2211yescharacteristic
34#376C2211yescharacteristic
35#340C2212yes—
36#340C2212yes—
37#336C2211yescharacteristic
38#336C2211yescharacteristic
39#336C2212yes—
40C22×Q8C2311yescharacteristic
41C22×Q8C2312yes—
42C22×D4C2311yescharacteristic
43C23×C4—21no—
44C2×Q16—24no—
45C2×Q16—24no—
46C2×Q16—42no—
47C2×SD16—24no—
48C2×SD16—24no—
49C2×SD16—42no—
50C22×C8C2312yes—
51C22×C8C2312yes—
52C4⋊Q8—24no—
53#124—22no—
54#124D414yes—
55C4×Q8—24no—
56C4×Q8—24no—
57C4×Q8—42no—
58C4×D4—42no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C22⋊C4—22no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—22no—
69C2×C42—22no—
70C2×C42—21no—
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74C4⋊C8—24no—
75#103—24no—
76#103—42no—
77#103—42no—
78#102—24no—
79#95—22no—
80#95—22no—
81#95—22no—
82#95—22no—
83#95D412yes—
84#95D412yes—
85C24—21no—
86C2×Q8—21no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8C2×D412yes—
95C2×Q8#4114yes—
96C2×D4—22no—
97C2×D4—21no—
98C2×D4—22no—
99C2×D4C2×D412yes—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—42no—
106C22×C4—82no—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D412yes—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D412yes—
111C22×C4C2411yescharacteristic
112Q16—48no—
113SD16—48no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C4⋊C4—28no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—42no—
126C4⋊C4—44no—
127C4⋊C4—82no—
128C22⋊C4—28no—
129C22⋊C4—42no—
130C22⋊C4—82no—
131C42—22no—
132C42—22no—
133C42—42no—
134C42—44no—
135C42C2×D412yes—
136C42#4112yes—
137C23—22no—
138C23—41no—
139C23—41no—
140C23—42no—
141C23—42no—
142C23C22×D411yescharacteristic
143Q8—24no—
144Q8—24no—
145Q8—24no—
146Q8—42no—
147Q8—42no—
148Q8—42no—
149D4—24no—
150D4—42no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—84no—
174C2×C4#12114yes—
175C2×C4C22×D411yescharacteristic
176C2×C4C2×#4112yes—
177C2×C4C2×#4111yescharacteristic
178C2×C4H(5,𝔽2)12yes—
179C8—42no—
180C8—42no—
181C22—44no—
182C22—42no—
183C22—82no—
184C22—81no—
185C22—81no—
186C22C2×#12111yescharacteristic
187C22#52511yescharacteristic
188C22#53511yescharacteristic
189C22#57712yes—
190C22#57812yes—
191C4—24no—
192C4—42no—
193C4—41no—
194C4—41no—
195C4—42no—
196C4—44no—
197C4—81no—
198C4—81no—
199C4—82no—
200C4#54612yes—
201C4#55312yes—
202C2—82no—
203C2#253511yescharacteristic
204C2#304612yes—
205C2#312112yes—
206C2#322211yescharacteristic
207C2#322311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2327C2semidirect
4. #2074C2semidirect
6. #2016C2semidirect
8. #1878C2semidirect
10. #1832C2semidirect
11. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4642
#23272
#23262
#20742
#20732
#20162
#20132
#18782
#18772
#18322
#18032