Finite Group Databasebeta

← #24682

Subgroups — #24682

Order 256.

381 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 139 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
2C2×C4×Q16C211yesmaximal, characteristic
3#2567C211yesmaximal, characteristic
4#2355C211yesmaximal, characteristic
5#2072C212yesmaximal
6#2056C212yesmaximal
7#1990C211yesmaximal, characteristic
8#1947C212yesmaximal
9#1939C211yesmaximal, characteristic
10#1928C212yesmaximal
11#1803C211yesmaximal, characteristic
12C22×Q16C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16#477—24no—
17C4×Q16—24no—
18C2×#107C2212yes—
19C2×#107C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×C4×C8C2211yescharacteristic
27#395—22no—
28#391C2212yes—
29#391C2212yes—
30#389—21no—
31#389—21no—
32#387—22no—
33#384C2212yes—
34#384C2211yescharacteristic
35#382—21no—
36C4×C4⋊C4—21no—
37C4×C4⋊C4C2212yes—
38C4×C4⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×Q8C2311yescharacteristic
46C2×Q16—22no—
47C2×Q16—22no—
48C2×Q16—22no—
49C22×C8C2311yescharacteristic
50C22×C8C2312yes—
51C22×C8C2311yescharacteristic
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8D414yes—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4D411yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C42—21no—
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#103—24no—
83#103—22no—
84C4×C8—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95D412yes—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—21no—
101C2×Q8—42no—
102C2×Q8—81no—
103C2×Q8—81no—
104C2×Q8C2×D412yes—
105C2×Q8#4112yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4C2×D412yes—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2411yescharacteristic
117Q16—28no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8#4112yes—
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—82no—
140C4⋊C4—81no—
141C4⋊C4—82no—
142C4⋊C4Q1614yescomplemented
143C4⋊C4C2×D412yes—
144C42—24no—
145C42—22no—
146C42—22no—
147C42—42no—
148C42—82no—
149C42C2×D412yes—
150C42#4112yes—
151C23C22×D411yescharacteristic
152Q8—22no—
153Q8—22no—
154Q8—22no—
155Q8—22no—
156Q8—41no—
157Q8—41no—
158Q8—41no—
159Q8—41no—
160Q8—84no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4#12112yes—
187C2×C4#12112yes—
188C2×C4C2×Q1614yes—
189C2×C4C2×Q1612yes—
190C2×C4#13712yes—
191C2×C4C22×D411yescharacteristic
192C2×C4C2×#4111yescharacteristic
193C2×C4C2×#4111yescharacteristic
194C2×C4C2×#4111yescharacteristic
195C2×C42−1+411yescharacteristic
196C2×C42−1+411yescharacteristic
197C8—22no—
198C8—42no—
199C8—41no—
200C22C2×#12111yescharacteristic
201C22#54711yescharacteristic
202C22#54811yescharacteristic
203C22C22×Q1611yescharacteristic
204C22C2×#13711yescharacteristic
205C22#57611yescharacteristic
206C22#57711yescharacteristic
207C4—24no—
208C4—22no—
209C4—42no—
210C4—42no—
211C4—42no—
212C4—42no—
213C4—82no—
214C4—81no—
215C4—82no—
216C4#46214yes—
217C4#52511yescharacteristic
218C4#53611yescharacteristic
219C4#55211yescharacteristic
220C4#55611yescharacteristic
221C2#241511yescharacteristic
222C2C2×#46211yescharacteristic
223C2#304511yescharacteristic
224C2#333811yescharacteristic
225C2#335611yescharacteristic
226C2#336311yescharacteristic
227C2#337511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
142. C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
C2×C4×Q162
#25672
#23552
#20722
#20562
#19902
#19472
#19392
#19282
#18032