Finite Group Databasebeta

← #10214

Subgroups — #10214

Order 256.

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

normal subgroups only

This group has 251 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, complemented
2#3235C211yesmaximal, characteristic, complemented
3#3214C211yesmaximal, characteristic
4#1963C211yesmaximal, characteristic, complemented
5#1506C211yesmaximal, characteristic, complemented
6#1505C211yesmaximal, characteristic, complemented
7#1307C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10#532C2211yescharacteristic
11#501—21no—
12#496—21no—
13#490—21no—
14#489—21no—
15#482—22no—
16#477—21no—
17#476—21no—
18#474—21no—
19#474—21no—
20#468—22no—
21#461—21no—
22#461—21no—
23#460—21no—
24#459—21no—
25#449—22no—
26#432C2211yescharacteristic
27#432C2211yescharacteristic
28#423C2211yescharacteristic
29C2×#104—22no—
30#376C2211yescharacteristic
31#328—22no—
32#327—22no—
33C2×#41—21no—
34C2×#41—21no—
35C2×SD16—22no—
36C2×SD16—22no—
37C2×D8—22no—
38C2×M16D412yes—
39C4⋊Q8D411yescharacteristic
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43#127D411yescharacteristic
44#127D411yescharacteristic
45#125—21no—
46#124—21no—
47#124—21no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50#121—21no—
51#121—21no—
52#121—21no—
53#121—21no—
54C4×Q8—21no—
55C4×Q8—21no—
56C4×D4—21no—
57C4×D4—21no—
58C4×D4—21no—
59C4×D4—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#104—44no—
71#103—22no—
72#103—22no—
73#102—22no—
74#102—22no—
75#102—22no—
76#102—22no—
77C22⋊C8D412yes—
78C22⋊C8D412yes—
79#97—21no—
80#97—21no—
81#95—21no—
82#95—21no—
83#41—42no—
84#41—42no—
85#41—82no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—21no—
96C2×D4—41no—
97C2×D4—41no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104SD16—82no—
105SD16—82no—
106D8—82no—
107M16—42no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C22⋊C4—41no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C42—22no—
127C42—41no—
128C42—41no—
129C42C2×D411yescharacteristic
130C42C2×D411yescharacteristic
131C42C2×D411yescharacteristic
132C42#4111yescharacteristic
133C23—41no—
134C23—41no—
135C23#12711yescharacteristic
136Q8—42no—
137Q8—42no—
138Q8—41no—
139Q8—41no—
140Q8—81no—
141D4—41no—
142D4—42no—
143D4—41no—
144D4—42no—
145D4—42no—
146D4—41no—
147D4—41no—
148D4—81no—
149D4—82no—
150D4—82no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—82no—
161C2×C4—81no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4—81no—
165C2×C4#12011yescharacteristic
166C2×C4#12011yescharacteristic
167C2×C4#12011yescharacteristic
168C2×C4#12111yescharacteristic
169C2×C4#12111yescharacteristic
170C2×C4#12111yescharacteristic
171C8—42no—
172C8—42no—
173C8—42no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22#39211yescharacteristic
184C4—21no—
185C4—21no—
186C4—21no—
187C4—21no—
188C4—21no—
189C4—82no—
190C4—81no—
191C4—81no—
192C4—81no—
193C4—81no—
194C4#45311yescharacteristic
195C4#45411yescharacteristic
196C2—41no—
197C2—161no—
198C2—161no—
199C2#200611yescharacteristic
200C2#200811yescharacteristic
201C2#200911yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3214C2
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. #3254C2semidirect
2. #3235C2semidirect
4. #1963C2semidirect
5. #1506C2semidirect
6. #1505C2semidirect
7. #1307C2semidirect

Maximal subgroups

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

subgroupindex
#32542
#32352
#32142
#19632
#15062
#15052
#13072