Finite Group Databasebeta

← #27777

Subgroups — #27777

Order 256.

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

normal subgroups only

This group has 379 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×#491C212yesmaximal
2#2552C211yesmaximal, characteristic
3#2546C211yesmaximal, characteristic
4#2043C212yesmaximal
5#2026C212yesmaximal
6#1981C211yesmaximal, characteristic
7#1956C211yesmaximal, characteristic
8#1935C212yesmaximal
9#1803C211yesmaximal, characteristic
10#1770C211yesmaximal, characteristic
11C4×#103C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2212yes—
14C2×#125C2212yes—
15C2×C4×Q8C2211yescharacteristic
16#491—28no—
17C2×C4⋊C8C2211yescharacteristic
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2211yescharacteristic
23C2×#103C2212yes—
24C2×#103C2211yescharacteristic
25C2×#97C2212yes—
26C2×C4×C8C2211yescharacteristic
27#398—22no—
28#398—22no—
29#391C2212yes—
30#389—21no—
31#389—21no—
32#389—21no—
33#389C2212yes—
34#384—21no—
35#384C2211yescharacteristic
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2211yescharacteristic
41#336C2212yes—
42C22×Q8C2312yes—
43C22×Q8C2311yescharacteristic
44C22×C8C2311yescharacteristic
45C22×C8C2312yes—
46C22×C8C2311yescharacteristic
47C4⋊Q8—24no—
48C4⋊Q8—22no—
49C4⋊Q8—22no—
50C4⋊Q8—42no—
51#125—24no—
52#125—42no—
53C4×Q8—22no—
54C4×Q8—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4D411yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C42—21no—
72C2×C42—21no—
73C2×C42C2312yes—
74C2×C42C2311yescharacteristic
75C4⋊C8—22no—
76#103—24no—
77#103—24no—
78#103—24no—
79#103—22no—
80#103—24no—
81#103—42no—
82#103D414yes—
83#97—24no—
84C4×C8—22no—
85#95—21no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95D412yes—
90C2×Q8—21no—
91C2×Q8—22no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—21no—
100C2×Q8—22no—
101C2×Q8C2×D412yes—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4C2×D412yes—
109C22×C4C2×D412yes—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2411yescharacteristic
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—22no—
116C2×C8—21no—
117C2×C8—22no—
118C2×C8—21no—
119C2×C8—22no—
120C2×C8—21no—
121C2×C8C2×D412yes—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—44no—
131C4⋊C4—44no—
132C4⋊C4—44no—
133C4⋊C4—44no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—81no—
139C4⋊C4—82no—
140C4⋊C4C2×D412yes—
141C4⋊C4#4112yes—
142C42—22no—
143C42—22no—
144C42—42no—
145C42—42no—
146C42#4112yes—
147C42#4112yes—
148C23C22×D411yescharacteristic
149Q8—22no—
150Q8—22no—
151Q8—41no—
152Q8—41no—
153Q8—42no—
154Q8—42no—
155Q8—42no—
156Q8—42no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4#12412yes—
184C2×C4#12412yes—
185C2×C4#13712yes—
186C2×C4#13712yes—
187C2×C4C22×D411yescharacteristic
188C2×C4C2×#4111yescharacteristic
189C2×C4C2×#4111yescharacteristic
190C2×C4C2×#4111yescharacteristic
191C2×C4H(5,𝔽2)11yescharacteristic
192C2×C42−1+411yescharacteristic
193C8—22no—
194C8—42no—
195C8—41no—
196C22C2×#12411yescharacteristic
197C22D4211yescharacteristic
198C22#54811yescharacteristic
199C22C2×#13711yescharacteristic
200C22C2×#13711yescharacteristic
201C22#57711yescharacteristic
202C22#57811yescharacteristic
203C4—22no—
204C4—42no—
205C4—42no—
206C4—42no—
207C4—42no—
208C4—42no—
209C4—81no—
210C4—82no—
211C4—82no—
212C4#49114yescomplemented
213C4#53211yescharacteristic
214C4#54211yescharacteristic
215C4#54711yescharacteristic
216C4#55511yescharacteristic
217C2#249111yescharacteristic
218C2C2×#49111yescharacteristic
219C2#313711yescharacteristic
220C2#327611yescharacteristic
221C2#327811yescharacteristic
222C2#337211yescharacteristic
223C2#337411yescharacteristic
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
212. C4#491semidirect

Maximal subgroups

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

subgroupindex
C2×#4912
#25522
#25462
#20432
#20262
#19812
#19562
#19352
#18032
#17702
C4×#1032