Finite Group Databasebeta

← #21775

Subgroups — #21775

Order 256.

681 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3449C211yesmaximal, characteristic
2#2879C211yesmaximal, characteristic, complemented
3#2297C211yesmaximal, characteristic, complemented
4#1897C212yesmaximal, complemented
5#1896C212yesmaximal, complemented
6#1871C214yesmaximal, complemented
7#1826C212yesmaximal, complemented
8#1789C212yesmaximal, complemented
9C22×#41C2211yescharacteristic
10C22×#41C2211yescharacteristic
11C22×M16C2212yescomplemented
12#537C414yescomplemented
13#536—21no—
14#536C412yescomplemented
15#535—21no—
16#535C412yescomplemented
17#534—21no—
18#534—21no—
19#532—22no—
20#525—22no—
21#525C2212yes—
22#525C2212yes—
23C2×#117C2211yescharacteristic
24C2×#117C2211yescharacteristic, complemented
25C22×C42C2211yescharacteristic, complemented
26C2×#104C2214yes—
27C2×#104C2214yes—
28#408—22no—
29#408—22no—
30#407C2212yes—
31#407C2212yescomplemented
32#391—21no—
33#390—21no—
34#386—21no—
35#386—21no—
36C4×C22⋊C4C2212yes—
37C4×C22⋊C4C2212yescomplemented
38#339C2214yescomplemented
39#339C2214yescomplemented
40C2×#41—21no—
41C2×#41—21no—
42C2×#41—42no—
43C2×#41—42no—
44C2×#41—42no—
45C2×#41—42no—
46C2×#41C2×C412yes—
47C2×#41C2×C412yes—
48C2×#41C2×C412yescomplemented
49C2×#41D412yes—
50C2×#41D412yes—
51C2×#41C2312yes—
52C22×Q8C2×C411yescharacteristic, complemented
53C22×Q8D411yescharacteristic
54C22×D4—21no—
55C22×D4C2×C411yescharacteristic, complemented
56C22×D4D411yescharacteristic
57C22×D4D411yescharacteristic
58C22×D4D411yescharacteristic
59C23×C4—21no—
60C23×C4—21no—
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C2×M16—22no—
65C2×M16—42no—
66C2×M16—42no—
67C2×M16C2314yes—
68C22×C8—22no—
69C4⋊Q8—22no—
70#127—22no—
71#124—22no—
72#124—22no—
73#123—22no—
74#123—22no—
75#123—22no—
76#123—22no—
77C22⋊Q8—24no—
78C22⋊Q8C2×C414yes—
79C22⋊Q8C2×C414yescomplemented
80#121—24no—
81#121—24no—
82#121—24no—
83#121C2×C414yes—
84#121C2×C414yescomplemented
85#120—22no—
86#120—22no—
87#120—22no—
88#120—22no—
89C4×D4—24no—
90C4×D4—24no—
91#117—21no—
92#117—21no—
93#117—42no—
94#117C2×C412yescomplemented
95#117C2312yes—
96C2×C4⋊C4—21no—
97C2×C4⋊C4C2×C412yescomplemented
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—42no—
105C2×C42C2312yes—
106C2×C42C2312yes—
107#104—44no—
108#104—44no—
109C22⋊C8—22no—
110C22⋊C8—22no—
111C22⋊C8—24no—
112#95—22no—
113#95—22no—
114C24—21no—
115C24C2×D411yescharacteristic
116C24C2×D411yescharacteristic
117C24C2×D411yescharacteristic
118#41—24no—
119#41—24no—
120#41—44no—
121#41—44no—
122#41—42no—
123#41—42no—
124#41—42no—
125#41—42no—
126#41—44no—
127#41—42no—
128#41—42no—
129#41—44no—
130C2×Q8—21no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—42no—
134C2×Q8C22⋊C412yes—
135C2×Q8C22⋊C412yes—
136C2×Q8C22×C412yes—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4—42no—
143C2×D4—41no—
144C2×D4—41no—
145C2×D4—41no—
146C2×D4—41no—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4C22⋊C412yes—
152C2×D4C22⋊C412yes—
153C2×D4C22⋊C412yes—
154C2×D4C22⋊C412yes—
155C2×D4C22⋊C412yes—
156C2×D4C22⋊C412yes—
157C2×D4C22×C412yes—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4C22×C411yescharacteristic
174C22×C4C22×C411yescharacteristic
175C22×C4C22×C412yes—
176C22×C4C22×C412yes—
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D412yes—
183C22×C4C2×D412yes—
184C22×C4C2411yescharacteristic
185M16—24no—
186M16—24no—
187M16—44no—
188C2×C8—24no—
189C2×C8—44no—
190C4⋊C4—22no—
191C4⋊C4—24no—
192C4⋊C4—24no—
193C4⋊C4—42no—
194C4⋊C4C22×C414yes—
195C22⋊C4—22no—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—22no—
199C22⋊C4—24no—
200C22⋊C4—24no—
201C22⋊C4—44no—
202C22⋊C4—44no—
203C22⋊C4—42no—
204C42—22no—
205C42—22no—
206C42—24no—
207C42—24no—
208C42—42no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—22no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—82no—
221C23C2×C22⋊C411yescharacteristic
222C23C2×C22⋊C412yes—
223C23C2×C22⋊C412yes—
224C23C2×C22⋊C411yescharacteristic
225C23C22×D411yescharacteristic
226Q8—24no—
227Q8—42no—
228Q8—42no—
229Q8—42no—
230D4—24no—
231D4—42no—
232D4—42no—
233D4—42no—
234D4—42no—
235D4—42no—
236D4—42no—
237D4—42no—
238D4—44no—
239D4—42no—
240D4—42no—
241D4—84no—
242D4—84no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—48no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—82no—
272C2×C4C2×C22⋊C411yescharacteristic
273C2×C4C2×C22⋊C412yes—
274C2×C4C2×C22⋊C412yes—
275C2×C4C2×C22⋊C411yescharacteristic
276C2×C4C23×C411yescharacteristic
277C2×C4C22×D411yescharacteristic
278C2×C4H(5,𝔽2)12yes—
279C2×C4H(5,𝔽2)12yes—
280C8—44no—
281C22—22no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—24no—
288C22—24no—
289C22—22no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—44no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—84no—
299C22C22×C22⋊C411yescharacteristic
300C22#53411yescharacteristic
301C22#53411yescharacteristic
302C4—21no—
303C4—21no—
304C4—22no—
305C4—22no—
306C4—44no—
307C4—42no—
308C4—44no—
309C4—82no—
310C4#51812yes—
311C4#53412yes—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—41no—
316C2—42no—
317C2—41no—
318C2—82no—
319C2#230811yescharacteristic
320C2#290112yes—
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C22×C22⋊C4
prime radical#3449C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2879C2semidirect
3. #2297C2semidirect
4. #1897C2semidirect
5. #1896C2semidirect
6. #1871C2semidirect
7. #1826C2semidirect
8. #1789C2semidirect
11. C22×M16C22semidirect
12. #537C4semidirect
14. #536C4semidirect
16. #535C4semidirect
24. C2×#117C22semidirect
25. C22×C42C22semidirect
31. #407C22semidirect
37. C4×C22⋊C4C22semidirect
38. #339C22semidirect
39. #339C22semidirect
48. C2×#41C2×C4semidirect
52. C22×Q8C2×C4semidirect
55. C22×D4C2×C4semidirect
79. C22⋊Q8C2×C4semidirect
84. #121C2×C4semidirect
94. #117C2×C4semidirect
97. C2×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#34492
#28792
#22972
#18972
#18962
#18712
#18262
#17892