Finite Group Databasebeta

← #9052

Subgroups — #9052

Order 256.

244 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 61 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#3105C211yesmaximal, characteristic
2#2992C211yesmaximal, characteristic, complemented
3#1955C211yesmaximal, characteristic, complemented
4#1524C211yesmaximal, characteristic, complemented
5#1490C211yesmaximal, characteristic, complemented
6#1302C211yesmaximal, characteristic, complemented
7#1283C211yesmaximal, characteristic, complemented
8#529C2211yescharacteristic
9C2×C4×D4C2211yescharacteristic
10#484C411yescharacteristic
11#481C411yescharacteristic
12#478—21no—
13#463—21no—
14#450C411yescharacteristic
15#447C411yescharacteristic
16#446—21no—
17C4×SD16—21no—
18C4×D8—21no—
19#435—21no—
20#435—21no—
21C8×D4—21no—
22#433—21no—
23#432C2211yescharacteristic
24#431C2211yescharacteristic
25#424—21no—
26C2×C4⋊C8C2211yescharacteristic
27C2×C22⋊C8—21no—
28C2×C22⋊C8—21no—
29C4×M16C2211yescharacteristic
30C2×#97C2211yescharacteristic
31#330—21no—
32#329—21no—
33D4⋊C8—21no—
34D4⋊C8—21no—
35C22×D4C2×C411yescharacteristic
36C23×C4—21no—
37C2×SD16—21no—
38C2×D8—21no—
39C2×M16—21no—
40C2×M16—21no—
41C22×C8—21no—
42C22×C8—21no—
43C22×C8D411yescharacteristic
44C22×C8D411yescharacteristic
45#126—21no—
46#125D411yescharacteristic
47#124D411yescharacteristic
48#123—21no—
49C22⋊Q8C2×C411yescharacteristic
50#121C2×C411yescharacteristic
51C4×Q8D411yescharacteristic
52C4×D4—21no—
53C4×D4—21no—
54C4×D4—21no—
55C4×D4—21no—
56C4×D4—21no—
57C4×D4D411yescharacteristic
58#117—21no—
59C2×C4⋊C4C2×C411yescharacteristic
60C2×C22⋊C4—21no—
61C2×C42C2311yescharacteristic
62#107—21no—
63#106—21no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68C4⋊C8—21no—
69C4⋊C8D411yescharacteristic
70C4⋊C8D411yescharacteristic
71#103—21no—
72#102—21no—
73#102—21no—
74#102—21no—
75C22⋊C8—21no—
76C22⋊C8—21no—
77C22⋊C8—21no—
78C22⋊C8—42no—
79C22⋊C8—42no—
80C22⋊C8C2×C411yescharacteristic
81C22⋊C8C2×C411yescharacteristic
82#97—21no—
83#97—21no—
84#97—21no—
85#97—21no—
86#97—21no—
87C4×C8—21no—
88C4×C8—21no—
89C4×C8—21no—
90C24—21no—
91C2×Q8C22⋊C411yescharacteristic
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—42no—
96C2×D4C22⋊C411yescharacteristic
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4C22×C411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4#4111yescharacteristic
108SD16—42no—
109D8—42no—
110M16—22no—
111M16—41no—
112M16—81no—
113C2×C8—22no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—81no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4C22⋊C411yescharacteristic
133C4⋊C4C22⋊C411yescharacteristic
134C22⋊C4—21no—
135C22⋊C4—21no—
136C22⋊C4—21no—
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—41no—
140C22⋊C4—41no—
141C42—21no—
142C42C2×D411yescharacteristic
143C42C2×D411yescharacteristic
144C42C2×D411yescharacteristic
145C42#4111yescharacteristic
146C23—22no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23C4×D411yescharacteristic
155Q8—41no—
156D4—22no—
157D4—22no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—41no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4C2×C22⋊C411yescharacteristic
181C2×C4C4×D411yescharacteristic
182C2×C4#12011yescharacteristic
183C2×C4#12111yescharacteristic
184C2×C4#12111yescharacteristic
185C2×C4#12311yescharacteristic
186C8—42no—
187C8—41no—
188C8—41no—
189C8—42no—
190C8—41no—
191C8—41no—
192C8—41no—
193C8—41no—
194C22—21no—
195C22—21no—
196C22—42no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22—81no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22#38611yescharacteristic
207C22#44311yescharacteristic
208C22#44411yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—41no—
215C4—41no—
216C4—41no—
217C4—81no—
218C4#45311yescharacteristic
219C4#45411yescharacteristic
220C2—21no—
221C2—21no—
222C2—81no—
223C2—82no—
224C2#187611yescharacteristic
225C2#188311yescharacteristic
226C2#189111yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radical#447C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #2992C2semidirect
3. #1955C2semidirect
4. #1524C2semidirect
5. #1490C2semidirect
6. #1302C2semidirect
7. #1283C2semidirect

Maximal subgroups

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

subgroupindex
#31052
#29922
#19552
#15242
#14902
#13022
#12832