Finite Group Databasebeta

← #9999

Subgroups — #9999

Order 256.

277 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 59 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#3103C211yesmaximal, characteristic
2#3037C211yesmaximal, characteristic
3#2581C211yesmaximal, characteristic, complemented
4#1517C211yesmaximal, characteristic
5#1487C211yesmaximal, characteristic, complemented
6#1479C211yesmaximal, characteristic, complemented
7#1286C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9#532C2211yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11C2×C4×Q8C2211yescharacteristic
12#481—21no—
13#475—21no—
14#464—22no—
15#462—21no—
16#460—21no—
17#460—21no—
18#448—21no—
19C4×SD16—22no—
20#431C2211yescharacteristic
21#431C2211yescharacteristic
22#423C2211yescharacteristic
23#417—21no—
24#400—21no—
25#395—21no—
26#391—21no—
27#389—21no—
28#387—22no—
29#382C2211yescharacteristic
30#327—22no—
31Q8⋊C8—22no—
32Q8⋊C8—22no—
33C2×#41—21no—
34C22×Q8D411yescharacteristic
35C22×Q8D411yescharacteristic
36#137—42no—
37#137—42no—
38C2×Q16—22no—
39C2×SD16—22no—
40C2×SD16—22no—
41C2×M16—21no—
42C4⋊Q8D411yescharacteristic
43#127D411yescharacteristic
44#124—21no—
45#121—21no—
46#121—21no—
47C4×Q8—21no—
48C4×Q8—21no—
49C4×Q8—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×Q8—21no—
54C4×Q8—21no—
55C4×D4—21no—
56C4×D4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4D411yescharacteristic
63C2×C42—21no—
64C2×C42—21no—
65C2×C42C2311yescharacteristic
66#106—22no—
67C4⋊C8—21no—
68C4⋊C8—21no—
69C4⋊C8—21no—
70C4⋊C8D412yes—
71#103—22no—
72#103—22no—
73#102—22no—
74#102—22no—
75C22⋊C8—21no—
76C22⋊C8—21no—
77C4×C8—21no—
78C4×C8D412yes—
79#95—22no—
80#95—22no—
81#95—22no—
82#95—21no—
83#95D412yes—
84#41—41no—
85#41—41no—
86#41—41no—
87#41—41no—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—42no—
97C2×Q8SD1612yescomplemented
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—41no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—41no—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111Q16—44no—
112SD16—44no—
113SD16—44no—
114M16—41no—
115M16—41no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—41no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—44no—
126C4⋊C4—41no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C22⋊C4—41no—
133C22⋊C4—41no—
134C42—41no—
135C42—41no—
136C42—41no—
137C42—41no—
138C42C2×D411yescharacteristic
139C42C2×D411yescharacteristic
140C42C2×D411yescharacteristic
141C42#4111yescharacteristic
142C23—41no—
143C23#12011yescharacteristic
144Q8—22no—
145Q8—22no—
146Q8—22no—
147Q8—22no—
148Q8—41no—
149Q8—41no—
150Q8—41no—
151Q8—41no—
152Q8—41no—
153Q8—41no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—81no—
158D4—81no—
159D4—81no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4#12011yescharacteristic
179C2×C4#12011yescharacteristic
180C2×C4#12111yescharacteristic
181C2×C4#12111yescharacteristic
182C2×C4#12111yescharacteristic
183C2×C4#12711yescharacteristic
184C2×C4C2×SD1611yescharacteristic
185C2×C4#13511yescharacteristic
186C2×C4AGL(1,ℤ/8)11yescharacteristic
187C2×C4#13711yescharacteristic
188C8—42no—
189C8—42no—
190C8—81no—
191C22—21no—
192C22—21no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22#39211yescharacteristic
197C22#44811yescharacteristic
198C22#44911yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—42no—
204C4—42no—
205C4—82no—
206C4—81no—
207C4—81no—
208C4—81no—
209C4#45011yescharacteristic
210C4#45211yescharacteristic
211C4#46511yescharacteristic
212C4#46811yescharacteristic
213C2—21no—
214C2—21no—
215C2—161no—
216C2#200011yescharacteristic
217C2#200911yescharacteristic
218C2#201011yescharacteristic
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
3. #2581C2semidirect
5. #1487C2semidirect
6. #1479C2semidirect
7. #1286C2semidirect
97. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#31032
#30372
#25812
#15172
#14872
#14792
#12862