Finite Group Databasebeta

← #18328

Subgroups — #18328

Order 256.

596 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C2×#538C211yesmaximal, characteristic, complemented
2#3193C211yesmaximal, characteristic, complemented
3#3192C211yesmaximal, characteristic, complemented
4#2539C211yesmaximal, characteristic, complemented
5#1618C214yesmaximal, complemented
6#1600C214yesmaximal, complemented
7#1508C211yesmaximal, characteristic, complemented
8#1507C211yesmaximal, characteristic, complemented
9#1462C211yesmaximal, characteristic, complemented
10#538—22no—
11#538—22no—
12#538—22no—
13#538C2214yes—
14C2×#126—21no—
15C2×#124—21no—
16C2×#123—21no—
17C2×#123—21no—
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121—21no—
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#120—21no—
23C2×#120—21no—
24C2×C4×D4—21no—
25C22×C22⋊C4C2211yescharacteristic
26#484C2212yescomplemented
27#481—21no—
28#481C2212yescomplemented
29#480—21no—
30#469—22no—
31#468—22no—
32#450C2212yescomplemented
33#448—21no—
34#447—21no—
35#447C2212yescomplemented
36#407C2211yescharacteristic
37#407C2211yescharacteristic
38#399C2212yescomplemented
39#392C2212yescomplemented
40#390—22no—
41C4×C22⋊C4C2211yescharacteristic
42#328C2214yescomplemented
43#327C2214yescomplemented
44#323C2214yescomplemented
45C25D411yescharacteristic
46C22×Q8D411yescharacteristic, complemented
47C22×D4—21no—
48C22×D4—21no—
49C22×D4—21no—
50C22×D4—21no—
51C22×D4—22no—
52C22×D4D411yescharacteristic, complemented
53C22×D4C2311yescharacteristic
54C23×C4—21no—
55C23×C4D411yescharacteristic
56C23×C4C2311yescharacteristic
57C2×SD16—22no—
58C2×D8—22no—
59C2×M16—21no—
60C2×M16—21no—
61#126—24no—
62#126—24no—
63#124—24no—
64#124—24no—
65#123—24no—
66#123—24no—
67#123—44no—
68C22⋊Q8—21no—
69C22⋊Q8—21no—
70C22⋊Q8D412yescomplemented
71C22⋊Q8C2312yes—
72#121—24no—
73#121—24no—
74#121—21no—
75#121—21no—
76#121—21no—
77#121—21no—
78#121—21no—
79#121—21no—
80#121D412yescomplemented
81#121C2312yes—
82#120—24no—
83#120—24no—
84#120—44no—
85C4×D4—24no—
86C4×D4—24no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4D411yescharacteristic, complemented
91C2×C4⋊C4D411yescharacteristic, complemented
92C2×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4D412yes—
107C2×C22⋊C4C2312yes—
108C2×C42—21no—
109C2×C42—21no—
110#107—22no—
111#106—22no—
112#103—22no—
113#102—22no—
114#102—22no—
115#102—22no—
116C22⋊C8—21no—
117C22⋊C8—21no—
118C22⋊C8C2312yes—
119C22⋊C8C2312yes—
120#95C2312yes—
121C24—21no—
122C24—21no—
123C24—21no—
124C24—21no—
125C24—42no—
126C24—42no—
127C24—81no—
128C24C2×D411yescharacteristic
129C24C2×D412yes—
130C2×Q8—21no—
131C2×Q8—81no—
132C2×Q8C2×D412yes—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—44no—
141C2×D4—44no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—44no—
145C2×D4—41no—
146C2×D4—41no—
147C2×D4—81no—
148C2×D4—81no—
149C2×D4—82no—
150C2×D4—82no—
151C2×D4—82no—
152C2×D4C2×D412yes—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—42no—
162C22×C4—81no—
163C22×C4—82no—
164C22×C4C2×D412yes—
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2411yescharacteristic
171SD16—82no—
172D8—82no—
173M16—81no—
174M16—81no—
175C2×C8—22no—
176C2×C8—22no—
177C4⋊C4—24no—
178C4⋊C4—22no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—44no—
185C4⋊C4C2×D412yes—
186C4⋊C4C2×D412yes—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22⋊C4—24no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—44no—
200C22⋊C4—44no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C42—24no—
204C42—81no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—41no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—41no—
220C23—41no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—41no—
225C23—42no—
226C23—81no—
227C23—81no—
228C23—88no—
229C23#12011yescharacteristic
230C23#12011yescharacteristic
231C23C22×D411yescharacteristic
232Q8—42no—
233Q8—81no—
234D4—42no—
235D4—42no—
236D4—42no—
237D4—44no—
238D4—44no—
239D4—81no—
240D4—84no—
241D4—84no—
242D4—88no—
243D4—82no—
244D4—82no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—81no—
267C2×C4—84no—
268C2×C4#12011yescharacteristic
269C2×C4#12011yescharacteristic
270C2×C4C22×D411yescharacteristic
271C2×C4C22×D411yescharacteristic
272C8—42no—
273C8—42no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—44no—
282C22—44no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—82no—
290C22—84no—
291C22—84no—
292C22—82no—
293C22—82no—
294C22—84no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22#45312yes—
299C22C2×#12011yescharacteristic
300C22#57512yes—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—44no—
306C4—42no—
307C4—42no—
308C4—42no—
309C4—42no—
310C4—81no—
311C2—22no—
312C2—22no—
313C2—41no—
314C2—82no—
315C2—82no—
316C2—161no—
317C2#300511yescharacteristic
318C2C2×#45311yescharacteristic
319C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#538C2semidirect
2. #3193C2semidirect
3. #3192C2semidirect
4. #2539C2semidirect
5. #1618C2semidirect
6. #1600C2semidirect
7. #1508C2semidirect
8. #1507C2semidirect
9. #1462C2semidirect
26. #484C22semidirect
28. #481C22semidirect
32. #450C22semidirect
35. #447C22semidirect
38. #399C22semidirect
39. #392C22semidirect
42. #328C22semidirect
43. #327C22semidirect
44. #323C22semidirect
46. C22×Q8D4semidirect
52. C22×D4D4semidirect
70. C22⋊Q8D4semidirect
80. #121D4semidirect
90. C2×C4⋊C4D4semidirect
91. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5382
#31932
#31922
#25392
#16182
#16002
#15082
#15072
#14622