Finite Group Databasebeta

← #28760

Subgroups — #28760

Order 256.

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

normal subgroups only

This group has 402 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×#491C211yesmaximal, characteristic, complemented
2C2×#488C211yesmaximal, characteristic
3#2658C211yesmaximal, characteristic
4#2330C211yesmaximal, characteristic, complemented
5#2043C212yesmaximal, complemented
6#2042C212yesmaximal
7#1983C212yesmaximal
8#1981C212yesmaximal, complemented
9#1959C211yesmaximal, characteristic
10#1956C211yesmaximal, characteristic, complemented
11#1768C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#125C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×#124C2211yescharacteristic
19#491—24no—
20#488—24no—
21C2×C4⋊C8C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×#97C2211yescharacteristic
31C2×C4×C8C2211yescharacteristic
32#399—22no—
33#399—22no—
34#389—22no—
35#389C2212yes—
36#387—22no—
37#384C2211yescharacteristic
38C4×C4⋊C4C2212yes—
39#376C2211yescharacteristic
40#336C2212yes—
41C22×Q8C2311yescharacteristic
42C22×Q8C2311yescharacteristic
43C22×Q8C2311yescharacteristic
44C22×D4C2311yescharacteristic
45C22×C8C2312yes—
46C22×C8C2312yes—
47C4⋊Q8—22no—
48C4⋊Q8—22no—
49C4⋊Q8—24no—
50C4⋊Q8—42no—
51#125—22no—
52#125D414yes—
53#124—22no—
54#124—22no—
55#124—24no—
56#124—42no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C42—22no—
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74C4⋊C8—24no—
75#103—24no—
76#103—24no—
77#103—24no—
78#103—42no—
79#103—42no—
80#103—42no—
81#102—24no—
82#102—42no—
83#97—22no—
84C4×C8—22no—
85#95—22no—
86#95—22no—
87#95D412yes—
88#95D412yes—
89C24—21no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—22no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×D4—21no—
103C2×D4—22no—
104C2×D4—21no—
105C2×D4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D412yes—
114C22×C4C2×D412yes—
115C22×C4C2411yescharacteristic
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C4⋊C4—28no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—44no—
129C4⋊C4—44no—
130C4⋊C4—82no—
131C4⋊C4—82no—
132C4⋊C4C2×D414yes—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C22⋊C4—44no—
136C22⋊C4—44no—
137C42—22no—
138C42—82no—
139C42C2×D412yes—
140C42#4112yes—
141C42#4114yes—
142C23—41no—
143C23—41no—
144C23—41no—
145C23—42no—
146C23—42no—
147C23C22×D411yescharacteristic
148Q8—42no—
149Q8—42no—
150Q8—42no—
151Q8—42no—
152Q8—42no—
153Q8—42no—
154D4—42no—
155D4—42no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4#12114yes—
180C2×C4#12414yes—
181C2×C4C22×D411yescharacteristic
182C2×C4C2×#4112yes—
183C2×C4C2×#4111yescharacteristic
184C2×C4H(5,𝔽2)12yes—
185C8—42no—
186C8—42no—
187C22—82no—
188C22—81no—
189C22—81no—
190C22—82no—
191C22—81no—
192C22C2×#12111yescharacteristic
193C22C2×#12411yescharacteristic
194C22#53511yescharacteristic
195C22#57712yes—
196C22#57812yes—
197C4—24no—
198C4—24no—
199C4—41no—
200C4—41no—
201C4—44no—
202C4—82no—
203C4—81no—
204C4—81no—
205C4—81no—
206C4#55012yes—
207C4#55912yes—
208C2—161no—
209C2#254811yescharacteristic
210C2#304612yes—
211C2#313712yes—
212C2#322111yescharacteristic
213C2#322311yescharacteristic
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×#491C2semidirect
4. #2330C2semidirect
5. #2043C2semidirect
8. #1981C2semidirect
10. #1956C2semidirect
11. #1768C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4912
C2×#4882
#26582
#23302
#20432
#20422
#19832
#19812
#19592
#19562
#17682