Finite Group Databasebeta

← #28291

Subgroups — #28291

Order 256.

373 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 135 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×#475C211yesmaximal, characteristic
3#2331C211yesmaximal, characteristic
4#2327C211yesmaximal, characteristic
5#2068C212yesmaximal
6#2066C212yesmaximal
7#1981C212yesmaximal
8#1949C212yesmaximal
9#1843C211yesmaximal, characteristic
10#1805C211yesmaximal, characteristic
11#1805C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16#477—24no—
17#475—24no—
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×C4⋊C8C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28#391—22no—
29#391C2211yescharacteristic
30#391C2211yescharacteristic
31#389C2212yes—
32#387—22no—
33#387—22no—
34#387—22no—
35#382C2211yescharacteristic
36#382C2211yescharacteristic
37C4×C4⋊C4C2212yes—
38#376C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41C22×Q8—21no—
42C22×Q8C2311yescharacteristic
43C22×Q8C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2312yes—
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48C4⋊Q8D414yes—
49C4⋊Q8Q814yescomplemented
50C4×Q8—22no—
51C4×Q8—22no—
52C4×Q8—22no—
53C4×Q8—22no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C42—21no—
64C2×C42—21no—
65C2×C42—22no—
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69#107—24no—
70#106—24no—
71C4⋊C8—22no—
72C4⋊C8—22no—
73C4⋊C8—24no—
74C4⋊C8—42no—
75#103—24no—
76#103—24no—
77#103—42no—
78#103—42no—
79#95—22no—
80#95—21no—
81#95—22no—
82#95—21no—
83#95—22no—
84#95—22no—
85#95—22no—
86#95D412yes—
87#95D412yes—
88C2×Q8—24no—
89C2×Q8—21no—
90C2×Q8—21no—
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—82no—
100C2×Q8C2×Q812yes—
101C2×Q8C2×Q812yes—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4C2×D412yes—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D412yes—
112C22×C4C2411yescharacteristic
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C4⋊C4—24no—
120C4⋊C4—28no—
121C4⋊C4—22no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—82no—
131C4⋊C4C2×D414yes—
132C42—21no—
133C42—21no—
134C42—42no—
135C42—42no—
136C42—82no—
137C42C2×D412yes—
138C42C2×Q812yes—
139C42#4112yes—
140C42#4112yes—
141C23C22×D411yescharacteristic
142Q8—22no—
143Q8—22no—
144Q8—22no—
145Q8—22no—
146Q8—41no—
147Q8—41no—
148Q8—41no—
149Q8—41no—
150Q8—84no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4#12112yes—
175C2×C4#12112yes—
176C2×C4C22⋊Q814yes—
177C2×C4C22×D411yescharacteristic
178C2×C4C22×Q811yescharacteristic
179C2×C4C2×#4111yescharacteristic
180C2×C4C2×#4111yescharacteristic
181C2×C4H(5,𝔽2)11yescharacteristic
182C2×C42−1+411yescharacteristic
183C8—42no—
184C8—42no—
185C22C2×#12111yescharacteristic
186C22C2×C22⋊Q811yescharacteristic
187C22#53711yescharacteristic
188C22#57712yes—
189C22#57812yes—
190C4—24no—
191C4—22no—
192C4—22no—
193C4—42no—
194C4—44no—
195C4—42no—
196C4—42no—
197C4—82no—
198C4—81no—
199C4—81no—
200C4D4×Q811yescharacteristic
201C4#55011yescharacteristic
202C4#55411yescharacteristic
203C4#55611yescharacteristic
204C2#253811yescharacteristic
205C2#304211yescharacteristic
206C2#304411yescharacteristic
207C2#308212yes—
208C2#325812yes—
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
49. C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
C2×#4752
#23312
#23272
#20682
#20662
#19812
#19492
#18432
#18052