Finite Group Databasebeta

← #28192

Subgroups — #28192

Order 256.

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

normal subgroups only

This group has 389 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×#446C211yesmaximal, characteristic, complemented
2C2×C4×SD16C211yesmaximal, characteristic
3#2581C211yesmaximal, characteristic, complemented
4#2580C211yesmaximal, characteristic
5#2088C212yesmaximal, complemented
6#2055C212yesmaximal
7#1936C212yesmaximal
8#1935C212yesmaximal, complemented
9#1835C211yesmaximal, characteristic, complemented
10#1805C211yesmaximal, characteristic, complemented
11#1804C211yesmaximal, characteristic
12C22×SD16C2211yescharacteristic
13C2×C4×Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16#446—24no—
17C4×SD16—24no—
18C2×#106C2211yescharacteristic
19C2×C4⋊C8C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×#103C2212yes—
22C2×#103C2211yescharacteristic
23C2×#102C2212yes—
24C2×#102C2211yescharacteristic
25C2×#97C2212yes—
26C2×C4×C8C2211yescharacteristic
27#400—21no—
28#399—21no—
29#395—21no—
30#393—21no—
31#391C2212yes—
32#390C2212yes—
33#389C2212yes—
34#389C2212yes—
35#387—22no—
36#385—22no—
37#382C2211yescharacteristic
38#382C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43C22×Q8D411yescharacteristic
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C2×SD16—22no—
48C2×SD16—22no—
49C2×SD16—22no—
50C22×C8C2312yes—
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53C4×Q8—22no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×D4—22no—
58C4×D4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4D412yes—
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C42—21no—
72C2×C42C2312yes—
73C2×C42C2311yescharacteristic
74#106—22no—
75C4⋊C8—24no—
76C4⋊C8—22no—
77#103—24no—
78#103—22no—
79#102—24no—
80#102—22no—
81#97—24no—
82C4×C8D414yes—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95—22no—
91C24—21no—
92C2×Q8—22no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—42no—
99C2×Q8#4112yes—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4#4112yes—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—42no—
112C22×C4—82no—
113C22×C4C2×D412yes—
114C22×C4C2×D412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2411yescharacteristic
118SD16—28no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8C2×D412yes—
127C2×C8C2×D412yes—
128C4⋊C4—22no—
129C4⋊C4—24no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—44no—
139C4⋊C4—42no—
140C4⋊C4—82no—
141C22⋊C4—44no—
142C22⋊C4—42no—
143C22⋊C4—82no—
144C42—22no—
145C42—22no—
146C42—42no—
147C42C2×D412yes—
148C42#4112yes—
149C23—22no—
150C23—41no—
151C23—41no—
152C23—42no—
153C23—42no—
154C23C22×D411yescharacteristic
155Q8—22no—
156Q8—22no—
157Q8—44no—
158Q8—41no—
159Q8—41no—
160D4—22no—
161D4—22no—
162D4—41no—
163D4—41no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—84no—
186C2×C4#12112yes—
187C2×C4#12112yes—
188C2×C4C22×D411yescharacteristic
189C2×C4C2×#4111yescharacteristic
190C2×C4C2×#4111yescharacteristic
191C2×C4C2×#4111yescharacteristic
192C2×C42−1+411yescharacteristic
193C2×C42−1+411yescharacteristic
194C8—22no—
195C8—22no—
196C8—42no—
197C22—44no—
198C22—42no—
199C22—82no—
200C22—81no—
201C22—81no—
202C22C2×#12111yescharacteristic
203C22#54811yescharacteristic
204C22#54811yescharacteristic
205C22#57512yes—
206C22#57712yes—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—42no—
211C4—42no—
212C4—82no—
213C4—82no—
214C4—81no—
215C4—81no—
216C4#52511yescharacteristic
217C4#53611yescharacteristic
218C4#55211yescharacteristic
219C4#55211yescharacteristic
220C2—82no—
221C2#253611yescharacteristic
222C2#305411yescharacteristic
223C2#305711yescharacteristic
224C2#337112yes—
225C2#337612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#446C2semidirect
3. #2581C2semidirect
5. #2088C2semidirect
8. #1935C2semidirect
9. #1835C2semidirect
10. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4462
C2×C4×SD162
#25812
#25802
#20882
#20552
#19362
#19352
#18352
#18052
#18042