Finite Group Databasebeta

← #8997

Subgroups — #8997

Order 256.

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

normal subgroups only

This group has 226 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#3171C211yesmaximal, characteristic
2#2995C211yesmaximal, characteristic, complemented
3#1963C211yesmaximal, characteristic, complemented
4#1500C211yesmaximal, characteristic, complemented
5#1496C211yesmaximal, characteristic, complemented
6#1300C211yesmaximal, characteristic, complemented
7#1299C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9#529C2211yescharacteristic
10#501C411yescharacteristic, complemented
11#496C411yescharacteristic, complemented
12#490C411yescharacteristic, complemented
13#489C411yescharacteristic, complemented
14#469—21no—
15#468—21no—
16#446—21no—
17C4×SD16—21no—
18C4×D8—21no—
19#436—21no—
20#435—21no—
21#435—21no—
22#433—21no—
23#432C2211yescharacteristic
24#431C2211yescharacteristic
25#424—21no—
26#423C2211yescharacteristic
27C2×#104—21no—
28C2×#104—21no—
29C4×M16C2211yescharacteristic
30#376C2211yescharacteristic
31#328—21no—
32#327—21no—
33D4⋊C8—21no—
34D4⋊C8—21no—
35C2×#41—21no—
36C2×SD16—21no—
37C2×D8—21no—
38C2×M16—21no—
39C2×M16—21no—
40C2×M16D411yescharacteristic
41C2×M16D411yescharacteristic
42C22×C8—21no—
43C4⋊Q8C2×C411yescharacteristic
44#127C2×C411yescharacteristic
45#126—21no—
46#125C2×C411yescharacteristic
47#124—21no—
48#124C2×C411yescharacteristic
49#123—21no—
50C22⋊Q8D411yescharacteristic
51#121—21no—
52#121—21no—
53#121D411yescharacteristic
54C4×Q8D411yescharacteristic
55C4×D4—21no—
56C4×D4—21no—
57C4×D4—21no—
58C4×D4D411yescharacteristic
59#117—21no—
60C2×C42—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#104—42no—
72#104—42no—
73#103—21no—
74#102—21no—
75#102—21no—
76#102—21no—
77C22⋊C8—21no—
78C22⋊C8—21no—
79C22⋊C8—21no—
80C22⋊C8—21no—
81#97—21no—
82#97—21no—
83#97C2×C411yescharacteristic
84#97C2×C411yescharacteristic
85C4×C8—21no—
86C4×C8—21no—
87C4×C8—21no—
88#95—21no—
89#95—21no—
90#41—41no—
91#41—41no—
92#41—41no—
93#41—41no—
94C2×Q8—21no—
95C2×Q8C22⋊C411yescharacteristic
96C2×D4—21no—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—41no—
101C2×D4C22⋊C411yescharacteristic
102C22×C4—21no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4#4111yescharacteristic
108SD16—42no—
109D8—42no—
110M16—42no—
111M16—41no—
112M16—41no—
113M16—81no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—41no—
121C2×C8—81no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4C22⋊C411yescharacteristic
129C4⋊C4C22⋊C411yescharacteristic
130C22⋊C4—21no—
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—41no—
136C22⋊C4—41no—
137C42—21no—
138C42—22no—
139C42—41no—
140C42C22×C411yescharacteristic
141C42C2×D411yescharacteristic
142C42C2×D411yescharacteristic
143C42#4111yescharacteristic
144C23—21no—
145C23—41no—
146C23#12111yescharacteristic
147Q8—41no—
148Q8—41no—
149Q8—41no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—42no—
154D4—41no—
155D4—41no—
156D4—81no—
157D4—81no—
158D4—81no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4C2×C22⋊C411yescharacteristic
175C2×C4C4×D411yescharacteristic
176C2×C4C4×D411yescharacteristic
177C2×C4#12011yescharacteristic
178C2×C4#12111yescharacteristic
179C2×C4#12311yescharacteristic
180C8—22no—
181C8—41no—
182C8—41no—
183C8—41no—
184C8—41no—
185C8—41no—
186C8—81no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22#38611yescharacteristic
197C4—21no—
198C4—21no—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—41no—
203C4—41no—
204C4—41no—
205C4—82no—
206C4—81no—
207C4#45311yescharacteristic
208C4#45411yescharacteristic
209C2—41no—
210C2—81no—
211C2—161no—
212C2#188111yescharacteristic
213C2#188311yescharacteristic
214C2#188411yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radical#3171C2
LCSG ▷ C42 ▷ 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. #2995C2semidirect
3. #1963C2semidirect
4. #1500C2semidirect
5. #1496C2semidirect
6. #1300C2semidirect
7. #1299C2semidirect
10. #501C4semidirect
11. #496C4semidirect
12. #490C4semidirect
13. #489C4semidirect

Maximal subgroups

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

subgroupindex
#31712
#29952
#19632
#15002
#14962
#13002
#12992