Finite Group Databasebeta

← #24694

Subgroups — #24694

Order 256.

418 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 127 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×#477C211yesmaximal, characteristic, complemented
2C2×#440C211yesmaximal, characteristic
3#2542C211yesmaximal, characteristic
4#2356C211yesmaximal, characteristic, complemented
5#2072C212yesmaximal, complemented
6#2055C212yesmaximal
7#1993C211yesmaximal, characteristic, complemented
8#1947C212yesmaximal, complemented
9#1940C211yesmaximal, characteristic, complemented
10#1925C212yesmaximal
11#1804C211yesmaximal, characteristic
12C22×SD16C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8—22no—
15C2×C4×Q8C2211yescharacteristic
16C2×C4×D4C2211yescharacteristic
17#477—24no—
18#440—24no—
19C2×#107C2212yes—
20C2×#107C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×#103C2212yes—
24C2×#103C2211yescharacteristic
25C2×#102C2212yes—
26C2×#102C2211yescharacteristic
27C2×#97C2211yescharacteristic
28#399—21no—
29#399—21no—
30#391C2212yes—
31#390C2212yes—
32#387—22no—
33#384C2212yes—
34#383—21no—
35#382C2211yescharacteristic
36C4×C4⋊C4C2212yes—
37C4×C4⋊C4C2211yescharacteristic
38#376—21no—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C2×SD16—22no—
48C2×SD16—22no—
49C2×SD16—22no—
50C22×C8C2311yescharacteristic
51C22×C8C2312yes—
52C22×C8C2311yescharacteristic
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C22⋊Q8—48no—
56C4×Q8—22no—
57C4×Q8D414yes—
58C4×D4—22no—
59C4×D4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4D412yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C42—21no—
73C2×C42—22no—
74C2×C42C2312yes—
75C2×C42C2311yescharacteristic
76#107—22no—
77#107—24no—
78C4⋊C8—24no—
79C4⋊C8—22no—
80#103—22no—
81#103—24no—
82#102—24no—
83#102—22no—
84#97—22no—
85#95—21no—
86#95—22no—
87#95—22no—
88#95—21no—
89#95—22no—
90#95D411yescharacteristic
91#95D411yescharacteristic
92C24—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—44no—
99C2×Q8—42no—
100C2×Q8C2×D412yes—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4#4112yes—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—44no—
113C22×C4—42no—
114C22×C4C2×D412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D412yes—
118C22×C4C2411yescharacteristic
119SD16—28no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8#4112yes—
129C4⋊C4—24no—
130C4⋊C4—24no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—21no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—44no—
139C4⋊C4—42no—
140C4⋊C4—44no—
141C4⋊C4C2×D412yes—
142C22⋊C4—44no—
143C22⋊C4—42no—
144C22⋊C4—44no—
145C42—24no—
146C42—22no—
147C42—22no—
148C42—44no—
149C42C2×D412yes—
150C42#4112yes—
151C23—22no—
152C23—41no—
153C23—41no—
154C23—42no—
155C23—42no—
156C23C22×D411yescharacteristic
157Q8—22no—
158Q8—22no—
159Q8—41no—
160Q8—41no—
161Q8—84no—
162D4—22no—
163D4—22no—
164D4—41no—
165D4—41no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—84no—
187C2×C4#12112yes—
188C2×C4#12112yes—
189C2×C4AGL(1,ℤ/8)14yes—
190C2×C4C22×D411yescharacteristic
191C2×C4C2×#4111yescharacteristic
192C2×C4C2×#4111yescharacteristic
193C2×C4C2×#4111yescharacteristic
194C2×C42−1+411yescharacteristic
195C2×C42−1+411yescharacteristic
196C8—22no—
197C8—42no—
198C8—41no—
199C22—44no—
200C22—42no—
201C22—82no—
202C22—81no—
203C22—81no—
204C22C2×#12111yescharacteristic
205C22#54711yescharacteristic
206C22#54811yescharacteristic
207C22C2×AGL(1,ℤ/8)12yes—
208C22#57812yes—
209C4—24no—
210C4—22no—
211C4—22no—
212C4—42no—
213C4—42no—
214C4—42no—
215C4—82no—
216C4—81no—
217C4—82no—
218C4#52511yescharacteristic
219C4#53611yescharacteristic
220C4#55211yescharacteristic
221C4#55611yescharacteristic
222C2—82no—
223C2#241511yescharacteristic
224C2#303811yescharacteristic
225C2#304411yescharacteristic
226C2#334312yes—
227C2#336712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#477C2semidirect
4. #2356C2semidirect
5. #2072C2semidirect
7. #1993C2semidirect
8. #1947C2semidirect
9. #1940C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
C2×#4402
#25422
#23562
#20722
#20552
#19932
#19472
#19402
#19252
#18042