Finite Group Databasebeta

← #29310

Subgroups — #29310

Order 256.

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

normal subgroups only

This group has 355 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
1#2628C211yesmaximal, characteristic
2#2317C211yesmaximal, characteristic
3#2085C212yesmaximal, complemented
4#2057C212yesmaximal, complemented
5#1944C212yesmaximal, complemented
6#1942C212yesmaximal, complemented
7#1854C212yesmaximal
8#1852C212yesmaximal
9#1750C211yesmaximal, characteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×#117C2211yescharacteristic
12C2×#117C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C2×#107C2212yes—
15C2×#107C2212yes—
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×C22⋊C8C2212yes—
19C2×#97C2211yescharacteristic
20C2×C4×C8C2211yescharacteristic
21#400—21no—
22#399—21no—
23#395—21no—
24#393—21no—
25#389C2212yes—
26#389C2212yes—
27#389C2212yes—
28#385—21no—
29#385—21no—
30#384—21no—
31#382—21no—
32#382C2211yescharacteristic
33#380C2212yes—
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39C22×Q8—21no—
40C23×C4—21no—
41C23×C4C2311yescharacteristic
42C22×C8C2312yes—
43C22×C8C2312yes—
44C4⋊Q8—41no—
45C4⋊Q8—41no—
46#117—24no—
47#117—41no—
48#117—41no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—41no—
55C2×C4⋊C4—41no—
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4D411yescharacteristic
65C2×C22⋊C4D411yescharacteristic
66C2×C42—21no—
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69#107—24no—
70#107—42no—
71#107—42no—
72#106—24no—
73C22⋊C8Q818yescomplemented
74#97—41no—
75C4×C8—41no—
76#95—22no—
77#95—22no—
78#95—22no—
79#95—21no—
80#95—21no—
81#95—22no—
82C24C2×D411yescharacteristic
83C2×Q8—42no—
84C2×Q8—82no—
85C22×C4—22no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—42no—
93C22×C4—82no—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×Q814yes—
100C22×C4C2411yescharacteristic
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8C2×Q814yes—
106C2×C8C2×Q814yes—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—42no—
117C4⋊C4—42no—
118C4⋊C4—82no—
119C4⋊C4—82no—
120C4⋊C4—82no—
121C4⋊C4#4114yes—
122C22⋊C4—24no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—82no—
126C42—24no—
127C42—21no—
128C42—21no—
129C42—21no—
130C42—21no—
131C23—21no—
132C23—21no—
133C23—22no—
134C23—22no—
135C23C4⋊Q812yes—
136C23C22×D411yescharacteristic
137Q8—84no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—24no—
141C2×C4—24no—
142C2×C4—24no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—84no—
163C2×C4C4⋊Q812yes—
164C2×C4C22×Q812yes—
165C2×C4C2×#4112yes—
166C2×C4H(5,𝔽2)11yescharacteristic
167C2×C4H(5,𝔽2)11yescharacteristic
168C8—24no—
169C8—24no—
170C22—22no—
171C22—24no—
172C22—42no—
173C22—41no—
174C22—41no—
175C22#52511yescharacteristic
176C22C2×C4⋊Q811yescharacteristic
177C22#53411yescharacteristic
178C22#57612yes—
179C22#57812yes—
180C4—22no—
181C4—44no—
182C4—41no—
183C4—41no—
184C4—41no—
185C4—41no—
186C4—82no—
187C4—82no—
188C4—82no—
189C4#55412yes—
190C4#55412yes—
191C2—42no—
192C2#258311yescharacteristic
193C2#312412yes—
194C2#316412yes—
195C2#319911yescharacteristic
196C2#320211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #2085C2semidirect
4. #2057C2semidirect
5. #1944C2semidirect
6. #1942C2semidirect
73. C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#26282
#23172
#20852
#20572
#19442
#19422
#18542
#18522
#17502