Finite Group Databasebeta

← #10215

Subgroups — #10215

Order 256.

232 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 49 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#3254C211yesmaximal, characteristic
2#3237C211yesmaximal, characteristic, complemented
3#3216C211yesmaximal, characteristic
4#1995C211yesmaximal, characteristic, complemented
5#1506C211yesmaximal, characteristic, complemented
6#1505C211yesmaximal, characteristic
7#1307C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#533C2211yescharacteristic
10#532C2211yescharacteristic
11#501—21no—
12#497—21no—
13#491—21no—
14#489—21no—
15#485—22no—
16#477—21no—
17#477—21no—
18#475—21no—
19#474—21no—
20#468—22no—
21#462—21no—
22#461—21no—
23#460—21no—
24#460—21no—
25#452—22no—
26#432C2211yescharacteristic
27#432C2211yescharacteristic
28#423C2211yescharacteristic
29#376C2211yescharacteristic
30#339—22no—
31#328—22no—
32#327—22no—
33C2×#41—21no—
34C2×Q16—22no—
35C2×SD16—22no—
36C2×SD16—22no—
37C2×M16D412yes—
38C4⋊Q8D411yescharacteristic
39C4⋊Q8D411yescharacteristic
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43#127D411yescharacteristic
44#125—21no—
45#125—21no—
46#124—21no—
47C22⋊Q8—21no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50C22⋊Q8—21no—
51#121—21no—
52#121—21no—
53C4×Q8—21no—
54C4×Q8—21no—
55C4×Q8—21no—
56C4×Q8—21no—
57C4×D4—21no—
58C4×D4—21no—
59#117—21no—
60#117—21no—
61C2×C42—21no—
62C2×C42C2311yescharacteristic
63#107—22no—
64#107—22no—
65#106—22no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68C4⋊C8—21no—
69C4⋊C8—21no—
70#103—22no—
71#103—22no—
72#103—22no—
73#103—22no—
74#102—22no—
75#102—22no—
76C22⋊C8D412yes—
77C22⋊C8D412yes—
78#97—21no—
79#97—21no—
80#95—21no—
81#95—21no—
82#41—82no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—41no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97Q16—82no—
98SD16—82no—
99SD16—82no—
100M16—42no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C42—22no—
123C42—41no—
124C42—41no—
125C42—41no—
126C42C2×D411yescharacteristic
127C42C2×D411yescharacteristic
128C42C2×D411yescharacteristic
129C42#4111yescharacteristic
130C23—41no—
131C23#12711yescharacteristic
132Q8—41no—
133Q8—41no—
134Q8—42no—
135Q8—42no—
136Q8—41no—
137Q8—41no—
138Q8—81no—
139D4—42no—
140D4—41no—
141D4—41no—
142D4—81no—
143D4—82no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—82no—
155C2×C4—81no—
156C2×C4—81no—
157C2×C4#12011yescharacteristic
158C2×C4#12011yescharacteristic
159C2×C4#12011yescharacteristic
160C2×C4#12111yescharacteristic
161C2×C4#12111yescharacteristic
162C2×C4#12111yescharacteristic
163C8—42no—
164C8—42no—
165C8—42no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22#39211yescharacteristic
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—82no—
179C4—81no—
180C4—81no—
181C4—81no—
182C4—81no—
183C4—81no—
184C4#45511yescharacteristic
185C4#45611yescharacteristic
186C2—41no—
187C2—161no—
188C2#200711yescharacteristic
189C2#200811yescharacteristic
190C2#200911yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
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
2. #3237C2semidirect
4. #1995C2semidirect
5. #1506C2semidirect
7. #1307C2semidirect

Maximal subgroups

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

subgroupindex
#32542
#32372
#32162
#19952
#15062
#15052
#13072