Finite Group Databasebeta

← #12712

Subgroups — #12712

Order 256.

476 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 73 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#3486C211yesmaximal, characteristic
2#2135C211yesmaximal, characteristic, complemented
3#2129C211yesmaximal, characteristic, complemented
4#2120C212yesmaximal, complemented
5#1796C212yesmaximal, complemented
6C2×2−1+4C2211yescharacteristic
7C23×Q8C411yescharacteristic, complemented
8D4×Q8—22no—
9D4×Q8—22no—
10#548—22no—
11#548—22no—
12#536—22no—
13#536C411yescharacteristic, complemented
14#536C2211yescharacteristic
15#525—22no—
16#525C411yescharacteristic, complemented
17#525C2211yescharacteristic
18#519C411yescharacteristic, complemented
19#421—22no—
20#419—22no—
21#413C2212yes—
22#410C2212yes—
23#356—22no—
24#354—22no—
25#352—22no—
26AΣL(1,𝔽16)—22no—
272−1+4—44no—
28C2×#41—22no—
29C2×#41—21no—
30C2×#41C2×C411yescharacteristic
31C2×#41D412yes—
32C2×#41C2311yescharacteristic
33C22×Q8—22no—
34C22×Q8—22no—
35C22×Q8—21no—
36C22×Q8—21no—
37C22×Q8—44no—
38C22×Q8C2×C411yescharacteristic
39C22×Q8C2×C411yescharacteristic
40C22×Q8D412yes—
41C23×C4—21no—
42C23×C4C2×C411yescharacteristic
43C2×M16—22no—
44C4⋊Q8—24no—
45C4⋊Q8—21no—
46C4⋊Q8D412yes—
47#124—24no—
48#124—21no—
49#124D412yes—
50#123—22no—
51#123—21no—
52#123—22no—
53#123D412yes—
54C22⋊Q8—24no—
55C22⋊Q8—24no—
56C22⋊Q8—24no—
57C22⋊Q8—21no—
58C22⋊Q8—22no—
59C22⋊Q8—21no—
60C22⋊Q8—22no—
61#121—24no—
62#121—21no—
63#121—21no—
64#120—21no—
65#120D412yes—
66C4×Q8—22no—
67C4×Q8—22no—
68C4×D4—24no—
69C4×D4—24no—
70C4×D4—22no—
71C4×D4—22no—
72#117—22no—
73#117—22no—
74#117C2×C411yescharacteristic
75#117C2×C411yescharacteristic
76#104—44no—
77#103—44no—
78#101—22no—
79#100—22no—
80#99—22no—
81#99—22no—
82C24C22⋊C411yescharacteristic
83#41—24no—
84#41—42no—
85#41—42no—
86#41—42no—
87#41—48no—
88#41—44no—
89C2×Q8—24no—
90C2×Q8—24no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—22no—
94C2×Q8—24no—
95C2×Q8—42no—
96C2×Q8—42no—
97C2×Q8—44no—
98C2×Q8—48no—
99C2×Q8—44no—
100C2×Q8—44no—
101C2×Q8—44no—
102C2×Q8C22⋊C412yes—
103C2×Q8C22⋊C411yescharacteristic
104C2×Q8C22⋊C412yes—
105C2×Q8C2×D411yescharacteristic
106C2×D4—22no—
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4C22⋊C411yescharacteristic
111C2×D4C2×D411yescharacteristic
112C2×D4C2×D412yes—
113C22×C4—21no—
114C22×C4—24no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—44no—
122C22×C4—42no—
123C22×C4C22⋊C411yescharacteristic
124C22×C4C22×C411yescharacteristic
125C22×C4C2×D412yes—
126M16—42no—
127M16—42no—
128C2×C8—42no—
129C4⋊C4—24no—
130C4⋊C4—24no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—42no—
137C22⋊C4—22no—
138C22⋊C4—24no—
139C22⋊C4—22no—
140C22⋊C4—21no—
141C22⋊C4—24no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—42no—
145C22⋊C4C22⋊C412yes—
146C42—24no—
147C42—21no—
148C42—42no—
149C42C22⋊C412yes—
150C23—21no—
151C23—21no—
152C23—41no—
153C23—42no—
154C23C2×C22⋊C411yescharacteristic
155C23#12012yes—
156Q8—24no—
157Q8—22no—
158Q8—22no—
159Q8—41no—
160Q8—44no—
161Q8—42no—
162Q8—48no—
163Q8—42no—
164Q8#9914yes—
165D4—22no—
166D4—42no—
167D4—41no—
168D4—44no—
169D4—44no—
170D4—44no—
171D4—42no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—48no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—82no—
191C2×C4C2×C22⋊C411yescharacteristic
192C2×C4C2×C22⋊C411yescharacteristic
193C2×C4#12012yes—
194C8—82no—
195C22—22no—
196C22—21no—
197C22—22no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22#37911yescharacteristic
207C4—22no—
208C4—24no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—41no—
213C4—42no—
214C4—41no—
215C4—162no—
216C4C2×#9912yes—
217C2—21no—
218C2—42no—
219C2—42no—
220C2—41no—
221C2—81no—
222C2#178111yescharacteristic
subgroupG/H
center, socleC2#1781
commutatorC22×C4C22×C4
Frattini, cosocleC2×#41C23
prime radical#3486C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2135C2semidirect
3. #2129C2semidirect
4. #2120C2semidirect
5. #1796C2semidirect
7. C23×Q8C4semidirect
13. #536C4semidirect
16. #525C4semidirect
18. #519C4semidirect

Maximal subgroups

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

subgroupindex
#34862
#21352
#21292
#21202
#17962