Finite Group Databasebeta

← #10186

Subgroups — #10186

Order 256.

264 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 61 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#3229C211yesmaximal, characteristic, complemented
2#3077C211yesmaximal, characteristic
3C2×#461C211yesmaximal, characteristic
4#1546C211yesmaximal, characteristic, complemented
5#1481C211yesmaximal, characteristic, complemented
6#1460C211yesmaximal, characteristic
7#1301C211yesmaximal, characteristic, complemented
8C22×SD16—21no—
9#533C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4×D4C2211yescharacteristic
12#498—21no—
13#487—21no—
14#483—22no—
15#478—22no—
16#476—21no—
17#475—22no—
18#474—21no—
19#461—22no—
20#461—22no—
21#431C2211yescharacteristic
22#431C2211yescharacteristic
23#428—21no—
24#423C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26#418—21no—
27C2×#103—21no—
28#332—22no—
29Q8⋊C8—22no—
30#323—22no—
31C22×Q8—21no—
32C22×D4D411yescharacteristic
33C23×C4—21no—
34C2×SD16—24no—
35C2×SD16—42no—
36C2×SD16—42no—
37C2×M16—21no—
38C22×C8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8—21no—
42C4⋊Q8D412yes—
43C4⋊Q8D411yescharacteristic
44C4⋊Q8D411yescharacteristic
45#125—21no—
46C22⋊Q8—21no—
47C22⋊Q8—21no—
48C4×Q8—21no—
49C4×Q8—21no—
50C4×D4—21no—
51C4×D4—21no—
52C4×D4—21no—
53C4×D4—21no—
54#117—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4D411yescharacteristic
58C2×C22⋊C4—21no—
59C2×C42C2311yescharacteristic
60#107—22no—
61#106—22no—
62#106—24no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8D412yes—
68C4⋊C8D412yes—
69#103—24no—
70#103—24no—
71#102—22no—
72#102—22no—
73C22⋊C8—21no—
74C22⋊C8D412yes—
75C4×C8—21no—
76C4×C8—21no—
77C24—21no—
78C2×Q8—21no—
79C2×Q8—22no—
80C2×Q8—21no—
81C2×Q8—41no—
82C2×Q8—41no—
83C2×Q8—41no—
84C2×Q8—41no—
85C2×Q8—41no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4SD1612yes—
92C22×C4—21no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—82no—
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4#4111yescharacteristic
100SD16—48no—
101M16—41no—
102M16—41no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—41no—
107C2×C8—41no—
108C2×C8—41no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—22no—
116C4⋊C4—21no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C22⋊C4—41no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C42—41no—
125C42C2×D411yescharacteristic
126C42C2×D411yescharacteristic
127C42C2×D411yescharacteristic
128C42C2×D411yescharacteristic
129C23—22no—
130C23—41no—
131C23—41no—
132C23—81no—
133C23—81no—
134C23#12111yescharacteristic
135Q8—42no—
136Q8—42no—
137Q8—42no—
138Q8—42no—
139Q8—81no—
140Q8—81no—
141D4—22no—
142D4—22no—
143D4—41no—
144D4—41no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—82no—
152C2×C4—81no—
153C2×C4—81no—
154C2×C4—82no—
155C2×C4—81no—
156C2×C4#12011yescharacteristic
157C2×C4#12011yescharacteristic
158C2×C4#12011yescharacteristic
159C2×C4#12111yescharacteristic
160C2×C4#12111yescharacteristic
161C2×C4#12711yescharacteristic
162C2×C4C2×SD1611yescharacteristic
163C2×C4#13511yescharacteristic
164C2×C4AGL(1,ℤ/8)11yescharacteristic
165C2×C4#13711yescharacteristic
166C8—42no—
167C8—42no—
168C8—42no—
169C8—81no—
170C22—21no—
171C22—21no—
172C22—42no—
173C22—82no—
174C22—82no—
175C22—81no—
176C22—81no—
177C22#39211yescharacteristic
178C22#46511yescharacteristic
179C22#46811yescharacteristic
180C4—21no—
181C4—21no—
182C4—21no—
183C4—82no—
184C4—81no—
185C4—81no—
186C4—81no—
187C4#44811yescharacteristic
188C4#44911yescharacteristic
189C4#45011yescharacteristic
190C4#45211yescharacteristic
191C4#47112yes—
192C2—21no—
193C2—21no—
194C2—82no—
195C2#200011yescharacteristic
196C2#200911yescharacteristic
197C2#201911yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3229C2semidirect
4. #1546C2semidirect
5. #1481C2semidirect
7. #1301C2semidirect

Maximal subgroups

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

subgroupindex
#32292
#30772
C2×#4612
#15462
#14812
#14602
#13012