Finite Group Databasebeta

← #24583

Subgroups — #24583

Order 256.

420 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 131 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×#481C211yesmaximal, characteristic
2C22×#107C211yesmaximal, characteristic, complemented
3#2504C211yesmaximal, characteristic
4#2345C211yesmaximal, characteristic, complemented
5#2085C212yesmaximal, complemented
6#2067C212yesmaximal, complemented
7#1949C212yesmaximal, complemented
8#1925C212yesmaximal, complemented
9#1893C211yesmaximal, characteristic
10#1850C211yesmaximal, characteristic, complemented
11#1824C211yesmaximal, characteristic, complemented
12C23×C8C2211yescharacteristic
13C2×#121C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16#481—24no—
17C2×#107—24no—
18C2×#107C2212yes—
19C2×#107C2212yes—
20C2×#106C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×#102C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2212yes—
26C2×C22⋊C8C2211yescharacteristic
27#397—22no—
28#390C2212yes—
29#389C2212yes—
30#387—22no—
31#386—21no—
32#386—21no—
33#382C2212yes—
34#380—21no—
35#380C2211yescharacteristic
36C4×C4⋊C4C2212yes—
37C4×C22⋊C4—21no—
38C4×C22⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×D4—21no—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C23×C4—21no—
47C23×C4C2311yescharacteristic
48C22×C8—22no—
49C22×C8—24no—
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C22×C8C2312yes—
53#121—22no—
54#121—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4D414yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4D411yescharacteristic
72C2×C22⋊C4D411yescharacteristic
73C2×C42—22no—
74C2×C42C2312yes—
75#107—24no—
76#107—24no—
77#106—22no—
78#106—22no—
79C4⋊C8—24no—
80#102—24no—
81#102—22no—
82#102—22no—
83C22⋊C8—22no—
84#95—22no—
85#95—22no—
86#95—21no—
87#95—22no—
88#95—22no—
89#95D412yes—
90C24—21no—
91C24C2×D411yescharacteristic
92C2×D4—21no—
93C2×D4—22no—
94C2×D4—21no—
95C2×D4—22no—
96C2×D4—42no—
97C2×D4—42no—
98C2×D4—42no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—24no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—42no—
108C22×C4—82no—
109C22×C4—82no—
110C22×C4C2×D412yes—
111C22×C4C2×D412yes—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D412yes—
114C22×C4#4112yes—
115C22×C4C2411yescharacteristic
116C2×C8—28no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8#4114yes—
124C4⋊C4—24no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—42no—
134C4⋊C4—82no—
135C4⋊C4C2×D412yes—
136C4⋊C4C2×D412yes—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—42no—
140C22⋊C4—42no—
141C22⋊C4—82no—
142C22⋊C4—82no—
143C42—22no—
144C42—22no—
145C42—82no—
146C23—22no—
147C23—22no—
148C23—41no—
149C23—42no—
150C23—41no—
151C23—41no—
152C23—42no—
153C23#12112yes—
154C23#13512yes—
155C23#13712yes—
156C23C22×D411yescharacteristic
157D4—41no—
158D4—41no—
159D4—42no—
160D4—84no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
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—42no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—84no—
185C2×C4—84no—
186C2×C4#12112yes—
187C2×C4C22×D411yescharacteristic
188C2×C4C2×#4112yes—
189C2×C4C2×#4111yescharacteristic
190C2×C42−1+412yes—
191C8—24no—
192C8—41no—
193C8—41no—
194C22—24no—
195C22—24no—
196C22—22no—
197C22—22no—
198C22—82no—
199C22—82no—
200C22—81no—
201C22—81no—
202C22—81no—
203C22#46414yes—
204C22C2×#12111yescharacteristic
205C22#54712yes—
206C22C2×#13511yescharacteristic
207C22C2×#13711yescharacteristic
208C22#57612yes—
209C4—22no—
210C4—42no—
211C4—42no—
212C4—42no—
213C4—42no—
214C4—82no—
215C4—81no—
216C4—82no—
217C4#52511yescharacteristic
218C4#53611yescharacteristic
219C4#55212yes—
220C2—24no—
221C2—161no—
222C2#241311yescharacteristic
223C2C2×#46411yescharacteristic
224C2#304111yescharacteristic
225C2#333912yes—
226C2#334612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#107C2semidirect
4. #2345C2semidirect
5. #2085C2semidirect
6. #2067C2semidirect
7. #1949C2semidirect
8. #1925C2semidirect
10. #1850C2semidirect
11. #1824C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C22×#1072
#25042
#23452
#20852
#20672
#19492
#19252
#18932
#18502
#18242