Finite Group Databasebeta

← #18300

Subgroups — #18300

Order 256.

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

normal subgroups only

This group has 656 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#3449C211yesmaximal, characteristic, complemented
2#3205C212yesmaximal, complemented
3#2619C211yesmaximal, characteristic, complemented
4#1598C214yesmaximal, complemented
5#1595C214yesmaximal, complemented
6#1517C212yesmaximal, complemented
7#1458C211yesmaximal, characteristic, complemented
8C22×#41—21no—
9#537—21no—
10#537C2212yes—
11#536—22no—
12#535—22no—
13#534—21no—
14#534C2212yes—
15#532—21no—
16#532C2212yes—
17C2×#127C2211yescharacteristic, complemented
18#525—24no—
19C2×#121—21no—
20C2×#117C2211yescharacteristic
21#493—22no—
22#486—22no—
23#459—24no—
24#447C2218yescomplemented
25#431C2212yescomplemented
26#394—21no—
27#394C2212yescomplemented
28#392—21no—
29#392C2212yes—
30#390—21no—
31#382C2211yescharacteristic, complemented
32#327C2218yescomplemented
33#324C2212yescomplemented
34#323C2212yes—
35C2×#41—22no—
36C2×#41—22no—
37C2×#41—22no—
38C2×#41—22no—
39C2×#41—44no—
40C2×#41—44no—
41C22×Q8—21no—
42C22×D4—21no—
43C22×D4—22no—
44C22×D4—22no—
45C22×D4—21no—
46C22×D4—21no—
47C22×D4C2312yes—
48C23×C4—22no—
49C23×C4D411yescharacteristic
50C2×D8—28no—
51C4⋊Q8—21no—
52C4⋊Q8D412yescomplemented
53#127—21no—
54#127—22no—
55#127—22no—
56#127D412yescomplemented
57#124—22no—
58#124—22no—
59#123—24no—
60#123—24no—
61C22⋊Q8—24no—
62C22⋊Q8—24no—
63#121—24no—
64#121—24no—
65#121—22no—
66#121—22no—
67#121—22no—
68#121—44no—
69#121C2314yes—
70#120—24no—
71#120—24no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4D414yescomplemented
76#117—22no—
77#117—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4D411yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4D411yescharacteristic
85C2×C22⋊C4C2311yescharacteristic
86C2×C42D411yescharacteristic
87C2×C42C2311yescharacteristic
88C4⋊C8—22no—
89#102—28no—
90C22⋊C8C2314yes—
91C4×C8—22no—
92#95—21no—
93#95C2312yes—
94C24—22no—
95C24—22no—
96C24C2×D411yescharacteristic
97#41—44no—
98#41—48no—
99#41—48no—
100#41—44no—
101#41—44no—
102#41—44no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—44no—
106C2×D4—24no—
107C2×D4—22no—
108C2×D4—24no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—24no—
114C2×D4—44no—
115C2×D4—44no—
116C2×D4—44no—
117C2×D4—44no—
118C2×D4—42no—
119C2×D4—41no—
120C2×D4—41no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—82no—
124C2×D4—82no—
125C2×D4—82no—
126C2×D4—82no—
127C2×D4C2×D414yes—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—24no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—48no—
136C22×C4D814yescomplemented
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2411yescharacteristic
143D8—88no—
144C2×C8—24no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—42no—
151C4⋊C4C2×D414yes—
152C22⋊C4—24no—
153C22⋊C4—24no—
154C22⋊C4—22no—
155C22⋊C4—24no—
156C22⋊C4—22no—
157C22⋊C4—24no—
158C22⋊C4—44no—
159C22⋊C4—82no—
160C22⋊C4—82no—
161C42—21no—
162C42—21no—
163C42C2×D412yes—
164C42C2×D412yes—
165C23—21no—
166C23—21no—
167C23—24no—
168C23—41no—
169C23—41no—
170C23—44no—
171C23—42no—
172C23—42no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—82no—
179C23—82no—
180C23C2×D812yes—
181C23C22×D411yescharacteristic
182Q8—44no—
183Q8—44no—
184D4—44no—
185D4—44no—
186D4—48no—
187D4—48no—
188D4—44no—
189D4—44no—
190D4—44no—
191D4—82no—
192D4—84no—
193D4—82no—
194D4—82no—
195D4—88no—
196D4—82no—
197D4—82no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—24no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—48no—
208C2×C4—44no—
209C2×C4—48no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4#12012yes—
216C2×C4#12011yescharacteristic
217C2×C4#12011yescharacteristic
218C2×C4C2×D812yes—
219C2×C4C2×D812yes—
220C2×C4AGL(1,ℤ/8)12yes—
221C2×C4C22×D412yes—
222C8—44no—
223C22—21no—
224C22—21no—
225C22—22no—
226C22—42no—
227C22—41no—
228C22—41no—
229C22—44no—
230C22—44no—
231C22—44no—
232C22—81no—
233C22—82no—
234C22—82no—
235C22—82no—
236C22—82no—
237C22—88no—
238C22—82no—
239C22—82no—
240C22—82no—
241C22C2×#12011yescharacteristic
242C22C22×D811yescharacteristic
243C22C2×AGL(1,ℤ/8)11yescharacteristic
244C4—22no—
245C4—22no—
246C4—22no—
247C4—44no—
248C4—41no—
249C4—41no—
250C4—82no—
251C4#44712yes—
252C4#44712yes—
253C2—21no—
254C2—21no—
255C2—42no—
256C2—84no—
257C2—162no—
258C2C2×#44711yescharacteristic
259C2#302111yescharacteristic
260C2#302711yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
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. #3449C2semidirect
2. #3205C2semidirect
3. #2619C2semidirect
4. #1598C2semidirect
5. #1595C2semidirect
6. #1517C2semidirect
7. #1458C2semidirect
17. C2×#127C22semidirect
24. #447C22semidirect
25. #431C22semidirect
27. #394C22semidirect
31. #382C22semidirect
32. #327C22semidirect
33. #324C22semidirect
52. C4⋊Q8D4semidirect
56. #127D4semidirect
75. C4×D4D4semidirect
136. C22×C4D8semidirect

Maximal subgroups

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

subgroupindex
#34492
#32052
#26192
#15982
#15952
#15172
#14582