Finite Group Databasebeta

← #18302

Subgroups — #18302

Order 256.

516 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 127 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#3458C211yesmaximal, characteristic, complemented
2#3208C212yesmaximal, complemented
3#2619C211yesmaximal, characteristic
4#1599C214yesmaximal, complemented
5#1596C214yesmaximal, complemented
6#1518C212yesmaximal, complemented
7#1458C211yesmaximal, characteristic, complemented
8#544—22no—
9#544—22no—
10#544—21no—
11#544C2212yes—
12#543—21no—
13#543C2212yes—
14#533—24no—
15#533—21no—
16#533C2212yes—
17C2×#127C2211yescharacteristic
18C2×#121—21no—
19C2×#117—21no—
20C2×#117C2211yescharacteristic
21#492—22no—
22#486—22no—
23#460—24no—
24#450C2218yes—
25#431C2212yes—
26#394—21no—
27#394C2212yes—
28#392—21no—
29#392C2212yes—
30#390—21no—
31#382C2211yescharacteristic
32#328C2218yes—
33#324C2212yes—
34#323C2212yes—
35C22×D4—22no—
36C22×D4—21no—
37C22×D4—21no—
38C22×D4C2312yes—
39C23×C4D411yescharacteristic
40C2×SD16—28no—
41C4⋊Q8—24no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8—24no—
45C4⋊Q8D412yescomplemented
46C4⋊Q8D412yescomplemented
47#127—22no—
48#127—22no—
49#125—24no—
50#125—22no—
51#125—24no—
52#125—22no—
53C22⋊Q8—24no—
54C22⋊Q8—24no—
55C22⋊Q8—24no—
56C22⋊Q8—22no—
57C22⋊Q8—22no—
58C22⋊Q8—22no—
59C22⋊Q8—24no—
60C22⋊Q8C2314yes—
61#121—44no—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8—22no—
65C4×Q8D414yescomplemented
66#117—22no—
67#117—22no—
68#117—22no—
69#117—22no—
70#117—22no—
71#117—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4D411yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4D411yescharacteristic
82C2×C22⋊C4C2311yescharacteristic
83C2×C42—22no—
84C2×C42D411yescharacteristic
85C2×C42C2311yescharacteristic
86C4⋊C8—22no—
87#102—28no—
88C22⋊C8C2314yes—
89C4×C8—22no—
90#95—21no—
91#95C2312yes—
92C24—22no—
93C24C2×D411yescharacteristic
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8C2×D414yes—
98C2×D4—24no—
99C2×D4—44no—
100C2×D4—42no—
101C2×D4—41no—
102C2×D4—41no—
103C2×D4—42no—
104C2×D4—42no—
105C2×D4—82no—
106C2×D4—82no—
107C2×D4—82no—
108C2×D4—82no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4SD1614yescomplemented
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2411yescharacteristic
122SD16—88no—
123C2×C8—24no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—24no—
128C4⋊C4—24no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—24no—
137C4⋊C4—42no—
138C4⋊C4C2×D414yes—
139C22⋊C4—24no—
140C22⋊C4—24no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—44no—
144C22⋊C4—82no—
145C22⋊C4—82no—
146C42—24no—
147C42—24no—
148C42—21no—
149C42—21no—
150C42C2×D412yes—
151C42C2×D412yes—
152C23—21no—
153C23—21no—
154C23—41no—
155C23—41no—
156C23—42no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23—42no—
163C23C2×SD1612yes—
164C23C22×D411yescharacteristic
165Q8—44no—
166Q8—82no—
167Q8—82no—
168Q8—82no—
169D4—48no—
170D4—84no—
171D4—88no—
172D4—82no—
173D4—82no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—24no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4#12012yes—
192C2×C4#12011yescharacteristic
193C2×C4#12011yescharacteristic
194C2×C4C2×SD1612yes—
195C2×C4C2×SD1612yes—
196C2×C4#13712yes—
197C2×C4C22×D412yes—
198C8—44no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22—82no—
207C22—82no—
208C22—82no—
209C22—82no—
210C22—82no—
211C22—82no—
212C22—82no—
213C22C2×#12011yescharacteristic
214C22C22×SD1611yescharacteristic
215C22C2×#13711yescharacteristic
216C4—22no—
217C4—22no—
218C4—22no—
219C4—41no—
220C4—44no—
221C4—41no—
222C4—44no—
223C4—82no—
224C4#44812yes—
225C4#44812yes—
226C2—21no—
227C2—21no—
228C2—42no—
229C2—162no—
230C2C2×#44811yescharacteristic
231C2#302111yescharacteristic
232C2#302711yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#2619C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3458C2semidirect
2. #3208C2semidirect
4. #1599C2semidirect
5. #1596C2semidirect
6. #1518C2semidirect
7. #1458C2semidirect
45. C4⋊Q8D4semidirect
46. C4⋊Q8D4semidirect
65. C4×Q8D4semidirect
115. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
#34582
#32082
#26192
#15992
#15962
#15182
#14582