Finite Group Databasebeta

← #33204

Subgroups — #33204

Order 256.

394 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 155 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×#479C211yesmaximal, characteristic
2C2×#475C211yesmaximal, characteristic
3#2660C211yesmaximal, characteristic
4#2276C211yesmaximal, characteristic
5#1989C211yesmaximal, characteristic
6#1987C211yesmaximal, characteristic
7#1949C212yesmaximal
8#1948C212yesmaximal
9#1843C211yesmaximal, characteristic
10#1805C212yesmaximal
11#1803C212yesmaximal
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×Q8C2212yes—
16#479—24no—
17#475—24no—
18C2×#107C2211yescharacteristic
19C2×#106C2212yes—
20C2×#106C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2212yes—
24C2×C4⋊C8C2212yes—
25C2×#103C2212yes—
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28#398—22no—
29#395—21no—
30#395—21no—
31#384C2212yes—
32#382C2212yes—
33C4×C4⋊C4—21no—
34C4×C4⋊C4—22no—
35C4×C4⋊C4C2212yes—
36C4×C4⋊C4C2211yescharacteristic
37#376—21no—
38#376C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×C8C2312yes—
46C22×C8C2311yescharacteristic
47C22×C8C2311yescharacteristic
48C4⋊Q8—22no—
49C4⋊Q8—22no—
50#125—22no—
51#125—22no—
52C4×Q8—22no—
53C4×Q8—24no—
54C4×Q8D414yes—
55C4×Q8Q818yescomplemented
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C42—21no—
69C2×C42—22no—
70C2×C42—21no—
71C2×C42—22no—
72C2×C42C2311yescharacteristic
73C2×C42C2312yes—
74#107—22no—
75#106—24no—
76#106—22no—
77C4⋊C8—22no—
78C4⋊C8—22no—
79C4⋊C8—24no—
80C4⋊C8—24no—
81#103—24no—
82#103—22no—
83#103—22no—
84#95—21no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95D411yescharacteristic
89#95D411yescharacteristic
90#95D412yes—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—42no—
94C2×Q8—82no—
95C2×Q8C2×D412yes—
96C2×Q8C2×Q814yes—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D412yes—
106C22×C4C2×D412yes—
107C22×C4C2411yescharacteristic
108C2×C8—22no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C4⋊C4—24no—
116C4⋊C4—28no—
117C4⋊C4—21no—
118C4⋊C4—22no—
119C4⋊C4—21no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—82no—
129C4⋊C4—82no—
130C4⋊C4C2×D412yes—
131C4⋊C4C2×Q814yes—
132C42—24no—
133C42—28no—
134C42—82no—
135C42—81no—
136C42C2×D412yes—
137C42C2×Q814yes—
138C42#4112yes—
139C42#4114yes—
140C23C22×D411yescharacteristic
141Q8—22no—
142Q8—24no—
143Q8—84no—
144Q8C4⋊Q814yes—
145C2×C4—22no—
146C2×C4—24no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4#12112yes—
169C2×C4#12112yes—
170C2×C4C22⋊Q814yes—
171C2×C4C22⋊Q814yes—
172C2×C4C22×D411yescharacteristic
173C2×C4C22×Q812yes—
174C2×C4C2×#4112yes—
175C2×C4C2×#4111yescharacteristic
176C8—42no—
177C8—41no—
178C8—41no—
179C22C2×#12111yescharacteristic
180C22C2×C22⋊Q812yes—
181C22#57712yes—
182C22#57711yescharacteristic
183C22#57811yescharacteristic
184C4—24no—
185C4—24no—
186C4—24no—
187C4—22no—
188C4—22no—
189C4—44no—
190C4—42no—
191C4—81no—
192C4—82no—
193C4—81no—
194C4C2×C4⋊Q811yescharacteristic
195C4#53211yescharacteristic
196C4#53312yes—
197C2#285711yescharacteristic
198C2#304311yescharacteristic
199C2#304611yescharacteristic
200C2#308212yes—
201C2#308412yes—
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
55. C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4792
C2×#4752
#26602
#22762
#19892
#19872
#19492
#19482
#18432
#18052
#18032