Finite Group Databasebeta

← #33229

Subgroups — #33229

Order 256.

444 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 143 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×#489C211yesmaximal, characteristic
3C2×#488C211yesmaximal, characteristic
4C2×#487C211yesmaximal, characteristic, complemented
5#2660C211yesmaximal, characteristic, complemented
6#2658C211yesmaximal, characteristic
7#1960C212yesmaximal, complemented
8#1959C212yesmaximal
9#1958C212yesmaximal
10#1956C212yesmaximal, complemented
11#1726C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#125C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×#124C2212yes—
18#491—24no—
19#489—24no—
20#488—24no—
21#487—24no—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2212yes—
32C2×#97C2212yes—
33C2×#97C2211yescharacteristic
34C2×C4×C8C2212yes—
35C2×C4×C8C2211yescharacteristic
36#399—22no—
37#399—22no—
38#398—22no—
39#395—22no—
40#384C2212yes—
41#382C2212yes—
42#376C2211yescharacteristic
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×Q8C2312yes—
46C22×D4C2311yescharacteristic
47C22×C8C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2311yescharacteristic
50C4⋊Q8—22no—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C4⋊Q8—24no—
55C4⋊Q8—24no—
56#125—22no—
57#125—22no—
58#124—22no—
59#124—22no—
60#124—24no—
61#124—24no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2312yes—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C42C2311yescharacteristic
76C2×C42C2312yes—
77#103—24no—
78#103—24no—
79#103—22no—
80#103—22no—
81#103—24no—
82#103—24no—
83#103—24no—
84#102—22no—
85#102—22no—
86#102—24no—
87#97—24no—
88#97D414yes—
89C4×C8—24no—
90C4×C8D414yes—
91#95—24no—
92#95D412yes—
93#95D412yes—
94C24—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—42no—
106C2×Q8—82no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—22no—
110C2×D4—22no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4C2×D414yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2411yescharacteristic
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8C2×D412yes—
125C2×C8C2×D412yes—
126C2×C8C2×D414yes—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—48no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—84no—
140C22⋊C4—48no—
141C22⋊C4—42no—
142C22⋊C4—42no—
143C42C2×D412yes—
144C42C2×D412yes—
145C42#4114yes—
146C42#4114yes—
147C23—42no—
148C23—41no—
149C23—42no—
150C23—42no—
151C23C22×D411yescharacteristic
152Q8—41no—
153Q8—41no—
154Q8—42no—
155Q8—42no—
156Q8—42no—
157Q8—42no—
158Q8—42no—
159Q8—84no—
160D4—41no—
161D4—41no—
162D4—42no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4#12418yes—
179C2×C4#12712yes—
180C2×C4#12712yes—
181C2×C4C22×D411yescharacteristic
182C2×C4C22×D411yescharacteristic
183C2×C4C2×#4112yes—
184C2×C4C2×#4112yes—
185C8—22no—
186C8—24no—
187C8—22no—
188C22—81no—
189C22—84no—
190C22—81no—
191C22—81no—
192C22C2×#12412yes—
193C22C2×#12711yescharacteristic
194C22#57712yes—
195C22#57812yes—
196C4—28no—
197C4—22no—
198C4—22no—
199C4—82no—
200C4—81no—
201C4—81no—
202C4—82no—
203C4—81no—
204C4#53211yescharacteristic
205C4#53212yes—
206C4#53211yescharacteristic
207C2—161no—
208C2#285811yescharacteristic
209C2#313714yes—
210C2#315411yescharacteristic
211C2#315611yescharacteristic
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. C2×#487C2semidirect
5. #2660C2semidirect
7. #1960C2semidirect
10. #1956C2semidirect
11. #1726C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4912
C2×#4892
C2×#4882
C2×#4872
#26602
#26582
#19602
#19592
#19582
#19562
#17262