Finite Group Databasebeta

← #10006

Subgroups — #10006

Order 256.

286 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 51 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#3213C211yesmaximal, characteristic
2#3112C211yesmaximal, characteristic, complemented
3#2592C211yesmaximal, characteristic, complemented
4#1980C211yesmaximal, characteristic, complemented
5#1527C211yesmaximal, characteristic, complemented
6#1491C211yesmaximal, characteristic, complemented
7#1285C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×#124C2211yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11#496—21no—
12#489—21no—
13#480—21no—
14#477—21no—
15#463—22no—
16#460—21no—
17#448—21no—
18#448—21no—
19#447—21no—
20#440—22no—
21#432C2211yescharacteristic
22#432C2211yescharacteristic
23C2×#104—22no—
24C4×M16C2211yescharacteristic
25#397—21no—
26#393—21no—
27#391—21no—
28#390—21no—
29#388—22no—
30C4×C4⋊C4C2211yescharacteristic
31#329—22no—
32Q8⋊C8—22no—
33C2×#41—21no—
34C22×Q8D411yescharacteristic
35C22×Q8D411yescharacteristic
36C22×D4D411yescharacteristic
37C2×SD16—22no—
38C2×SD16—22no—
39C2×D8—22no—
40C2×M16—21no—
41C4⋊Q8D411yescharacteristic
42#127D411yescharacteristic
43#124—21no—
44#124—21no—
45#124—21no—
46#124—21no—
47#124—21no—
48#121—21no—
49#121—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×Q8—21no—
54C4×D4—21no—
55C4×D4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4D411yescharacteristic
59C2×C22⋊C4—22no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—22no—
63C2×C42—21no—
64C2×C42D412yes—
65C2×C42C2311yescharacteristic
66#107—22no—
67C4⋊C8—21no—
68C4⋊C8D412yes—
69#104—44no—
70#103—22no—
71#103—22no—
72#102—22no—
73#102—22no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76#97—21no—
77#97—21no—
78#97D412yes—
79C4×C8—21no—
80#95—22no—
81#95—22no—
82#95—21no—
83C24—21no—
84#41—42no—
85#41—42no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—42no—
92C2×Q8—42no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—21no—
96C2×D4—21no—
97C2×D4—21no—
98C2×D4—42no—
99C2×D4—41no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—41no—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108SD16—44no—
109SD16—82no—
110D8—82no—
111M16—22no—
112M16—41no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—41no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—42no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C22⋊C4—44no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C22⋊C4—41no—
130C22⋊C4—82no—
131C42—42no—
132C42—41no—
133C42—41no—
134C42C2×D411yescharacteristic
135C42C2×D411yescharacteristic
136C42C2×D411yescharacteristic
137C42#4111yescharacteristic
138C23—42no—
139C23—42no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23#12011yescharacteristic
145Q8—22no—
146Q8—22no—
147Q8—41no—
148Q8—41no—
149Q8—42no—
150Q8—42no—
151Q8—81no—
152D4—42no—
153D4—42no—
154D4—42no—
155D4—42no—
156D4—81no—
157D4—81no—
158D4—81no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—82no—
176C2×C4#12011yescharacteristic
177C2×C4#12011yescharacteristic
178C2×C4#12111yescharacteristic
179C2×C4#12111yescharacteristic
180C2×C4#12111yescharacteristic
181C2×C4#12711yescharacteristic
182C8—42no—
183C8—42no—
184C8—42no—
185C22—21no—
186C22—21no—
187C22—82no—
188C22—82no—
189C22—81no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22#39211yescharacteristic
196C22#45311yescharacteristic
197C22#45411yescharacteristic
198C4—21no—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—42no—
204C4—82no—
205C4—81no—
206C4—81no—
207C4—81no—
208C4#45512yes—
209C2—21no—
210C2—21no—
211C2—161no—
212C2—161no—
213C2#200411yescharacteristic
214C2#200611yescharacteristic
215C2#201011yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3213C2
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. #3112C2semidirect
3. #2592C2semidirect
4. #1980C2semidirect
5. #1527C2semidirect
6. #1491C2semidirect
7. #1285C2semidirect

Maximal subgroups

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

subgroupindex
#32132
#31122
#25922
#19802
#15272
#14912
#12852