Finite Group Databasebeta

← #9265

Subgroups — #9265

Order 256.

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

normal subgroups only

This group has 219 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#3211C211yesmaximal, characteristic
2#2968C211yesmaximal, characteristic, complemented
3#1532C211yesmaximal, characteristic, complemented
4C2×#329C211yesmaximal, characteristic, complemented
5#1469C211yesmaximal, characteristic, complemented
6#1302C211yesmaximal, characteristic, complemented
7#1281C211yesmaximal, characteristic, complemented
8#529C2211yescharacteristic
9C2×#124C2211yescharacteristic
10#495—21no—
11#488—21no—
12#464—21no—
13#463—21no—
14#451C411yescharacteristic
15#450C411yescharacteristic
16#448C411yescharacteristic
17#447C411yescharacteristic
18#446—21no—
19#435—21no—
20#435—21no—
21C8×D4—21no—
22#433—21no—
23#432C2211yescharacteristic
24#431C2211yescharacteristic
25#431C2211yescharacteristic
26#405—21no—
27C2×#97C2211yescharacteristic
28C2×#97C2211yescharacteristic
29#330—21no—
30#329—21no—
31#329—21no—
32#329—21no—
33#323—21no—
34#323—21no—
35C22×Q8C2×C411yescharacteristic
36C22×D4C2×C411yescharacteristic
37C2×Q16—21no—
38C2×SD16—21no—
39C2×SD16—21no—
40C2×D8—21no—
41C2×M16—21no—
42C22×C8—21no—
43C22×C8—21no—
44C22×C8—21no—
45#126—21no—
46#125D411yescharacteristic
47#124—21no—
48#124—21no—
49#124—21no—
50#124—21no—
51#124D411yescharacteristic
52#123—21no—
53C22⋊Q8C2×C411yescharacteristic
54#121C2×C411yescharacteristic
55C4×Q8D411yescharacteristic
56C4×D4—21no—
57C4×D4D411yescharacteristic
58#117—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C42C2311yescharacteristic
62C4⋊C8—21no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67#103—21no—
68#103—21no—
69#102—21no—
70#102—21no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74C22⋊C8C2×C411yescharacteristic
75C22⋊C8C2×C411yescharacteristic
76C22⋊C8D411yescharacteristic
77C22⋊C8D411yescharacteristic
78#97—21no—
79#97—21no—
80#97—21no—
81#97—21no—
82#97—21no—
83#97D411yescharacteristic
84#97D411yescharacteristic
85C4×C8—21no—
86C4×C8—21no—
87C4×C8—21no—
88C24—21no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—81no—
92C2×Q8C22⋊C411yescharacteristic
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—21no—
96C2×D4—81no—
97C2×D4C22⋊C411yescharacteristic
98C22×C4—21no—
99C22×C4—21no—
100C22×C4C22×C411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4#4111yescharacteristic
103Q16—81no—
104SD16—81no—
105SD16—81no—
106D8—81no—
107M16—42no—
108C2×C8—21no—
109C2×C8—22no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—42no—
116C2×C8—42no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8—41no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4C22⋊C411yescharacteristic
125C4⋊C4C22⋊C411yescharacteristic
126C22⋊C4—21no—
127C22⋊C4—21no—
128C22⋊C4—21no—
129C22⋊C4—21no—
130C22⋊C4—21no—
131C22⋊C4—41no—
132C22⋊C4—41no—
133C22⋊C4—41no—
134C22⋊C4—41no—
135C42—21no—
136C42C2×D411yescharacteristic
137C42C2×D411yescharacteristic
138C42C2×D411yescharacteristic
139C42#4111yescharacteristic
140C23—21no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23—81no—
145C23—81no—
146C23C4×D411yescharacteristic
147Q8—41no—
148Q8—41no—
149Q8—41no—
150Q8—81no—
151D4—41no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—81no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—81no—
168C2×C4C2×C22⋊C411yescharacteristic
169C2×C4C4×D411yescharacteristic
170C2×C4#12011yescharacteristic
171C2×C4#12111yescharacteristic
172C2×C4#12111yescharacteristic
173C2×C4#12311yescharacteristic
174C8—22no—
175C8—42no—
176C8—41no—
177C8—41no—
178C8—41no—
179C8—41no—
180C8—41no—
181C8—81no—
182C22—21no—
183C22—21no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22—161no—
191C22—161no—
192C22#38611yescharacteristic
193C22#44311yescharacteristic
194C22#44411yescharacteristic
195C4—21no—
196C4—21no—
197C4—21no—
198C4—21no—
199C4—21no—
200C4—21no—
201C4—41no—
202C4—41no—
203C4—41no—
204C4—81no—
205C2—21no—
206C2—21no—
207C2—81no—
208C2—161no—
209C2#187611yescharacteristic
210C2#190611yescharacteristic
211C2#191411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#447C4
LCSG ▷ C22×C4 ▷ 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. #2968C2semidirect
3. #1532C2semidirect
4. C2×#329C2semidirect
5. #1469C2semidirect
6. #1302C2semidirect
7. #1281C2semidirect

Maximal subgroups

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

subgroupindex
#32112
#29682
#15322
C2×#3292
#14692
#13022
#12812