Finite Group Databasebeta

← #28212

Subgroups — #28212

Order 256.

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

normal subgroups only

This group has 387 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×#475C211yesmaximal, characteristic, complemented
2C2×#476C211yesmaximal, characteristic
3#2331C211yesmaximal, characteristic, complemented
4#2326C211yesmaximal, characteristic
5#2084C212yesmaximal, complemented
6#2042C212yesmaximal
7#1927C212yesmaximal, complemented
8#1926C212yesmaximal
9#1832C211yesmaximal, characteristic, complemented
10#1805C211yesmaximal, characteristic, complemented
11#1804C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#124C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16#476—24no—
17#475—24no—
18C2×#106C2212yes—
19C2×#106C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26#391—22no—
27#391C2211yescharacteristic
28#390C2211yescharacteristic
29#389C2212yes—
30#388—22no—
31#387—22no—
32#385—22no—
33#384C2211yescharacteristic
34#382C2211yescharacteristic
35C4×C4⋊C4C2212yes—
36#376C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#336C2212yes—
40#336C2212yes—
41C22×Q8—21no—
42C22×Q8C2311yescharacteristic
43C22×D4C2311yescharacteristic
44C23×C4—21no—
45C22×C8C2312yes—
46C22×C8C2312yes—
47C4⋊Q8—22no—
48C4⋊Q8D414yes—
49#124—22no—
50#124—22no—
51C4×Q8—22no—
52C4×Q8—22no—
53C4×D4—22no—
54C4×D4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—22no—
66C2×C42—21no—
67C2×C42—22no—
68C2×C42C2311yescharacteristic
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71#106—24no—
72#106—24no—
73C4⋊C8—22no—
74C4⋊C8—22no—
75#103—24no—
76#103—24no—
77#102—24no—
78#102—24no—
79#95—22no—
80#95—21no—
81#95—21no—
82#95—22no—
83#95—22no—
84#95D412yes—
85#95D412yes—
86C24—21no—
87C2×Q8—24no—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—82no—
94C2×Q8#4112yes—
95C2×D4—21no—
96C2×D4—21no—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4#4112yes—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—42no—
107C22×C4—82no—
108C22×C4C2×D412yes—
109C22×C4C2×D412yes—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2411yescharacteristic
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C4⋊C4—24no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—42no—
128C4⋊C4—82no—
129C4⋊C4—82no—
130C4⋊C4C2×D414yes—
131C22⋊C4—28no—
132C22⋊C4—42no—
133C22⋊C4—82no—
134C42—21no—
135C42—21no—
136C42—21no—
137C42—21no—
138C42—42no—
139C42—82no—
140C42C2×D412yes—
141C42#4112yes—
142C23—22no—
143C23—41no—
144C23—41no—
145C23—42no—
146C23—42no—
147C23C22×D411yescharacteristic
148Q8—22no—
149Q8—22no—
150Q8—41no—
151Q8—41no—
152Q8—84no—
153D4—22no—
154D4—22no—
155D4—41no—
156D4—41no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—84no—
179C2×C4#12112yes—
180C2×C4#12112yes—
181C2×C4C22×D411yescharacteristic
182C2×C4C2×#4111yescharacteristic
183C2×C4C2×#4111yescharacteristic
184C2×C4C2×#4111yescharacteristic
185C2×C42−1+411yescharacteristic
186C2×C42−1+411yescharacteristic
187C8—42no—
188C8—42no—
189C22—44no—
190C22—42no—
191C22—82no—
192C22—81no—
193C22—81no—
194C22C2×#12111yescharacteristic
195C22#52511yescharacteristic
196C22#53611yescharacteristic
197C22#57712yes—
198C22#57712yes—
199C4—22no—
200C4—22no—
201C4—42no—
202C4—42no—
203C4—44no—
204C4—42no—
205C4—82no—
206C4—81no—
207C4—81no—
208C4#54811yescharacteristic
209C4#54811yescharacteristic
210C4#55211yescharacteristic
211C4#55211yescharacteristic
212C2—82no—
213C2#253611yescharacteristic
214C2#304211yescharacteristic
215C2#304311yescharacteristic
216C2#312512yes—
217C2#324412yes—
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
1. C2×#475C2semidirect
3. #2331C2semidirect
5. #2084C2semidirect
7. #1927C2semidirect
9. #1832C2semidirect
10. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4752
C2×#4762
#23312
#23262
#20842
#20422
#19272
#19262
#18322
#18052
#18042