Finite Group Databasebeta

← #18296

Subgroups — #18296

Order 256.

581 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 581 conjugacy classes of subgroups — too many to draw. Only its 119 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#3187C211yesmaximal, characteristic, complemented
3#3186C211yesmaximal, characteristic, complemented
4#2617C211yesmaximal, characteristic, complemented
5#1612C214yesmaximal, complemented
6#1600C214yesmaximal, complemented
7#1520C211yesmaximal, characteristic, complemented
8#1519C211yesmaximal, characteristic, complemented
9C2×#323C211yesmaximal, characteristic, complemented
10#538—22no—
11#538—22no—
12#538—22no—
13#538C2214yes—
14C2×#126—21no—
15C2×#124—21no—
16C2×#123—21no—
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121—21no—
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#120—21no—
22C2×#120—21no—
23C2×C4×D4—21no—
24C22×C22⋊C4C2211yescharacteristic
25#483C2212yescomplemented
26#481—21no—
27#480—21no—
28#480C2212yescomplemented
29#466—22no—
30#465—22no—
31#450C2212yescomplemented
32#448—21no—
33#447—21no—
34#447C2212yescomplemented
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#397C2212yescomplemented
38#394—22no—
39#392C2212yescomplemented
40#386—21no—
41#380C2211yescharacteristic
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—
59C22×C8—21no—
60C22×C8—21no—
61#126—24no—
62#126—24no—
63#124—24no—
64#124—24no—
65#123—24no—
66#123—24no—
67C22⋊Q8—21no—
68C22⋊Q8—21no—
69C22⋊Q8D412yescomplemented
70C22⋊Q8C2312yes—
71#121—24no—
72#121—24no—
73#121—21no—
74#121—21no—
75#121—21no—
76#121—21no—
77#121—21no—
78#121—21no—
79#121D412yescomplemented
80#121C2312yes—
81#120—24no—
82#120—24no—
83#120—44no—
84C4×D4—24no—
85C4×D4—24no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4D411yescharacteristic, complemented
90C2×C4⋊C4D411yescharacteristic, complemented
91C2×C4⋊C4C2311yescharacteristic
92C2×C22⋊C4—21no—
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—22no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4D412yes—
106C2×C22⋊C4C2312yes—
107C2×C42—21no—
108#107—22no—
109#106—22no—
110#103—22no—
111#102—22no—
112#102—22no—
113#102—22no—
114C22⋊C8—21no—
115C22⋊C8—21no—
116C22⋊C8C2312yes—
117C22⋊C8C2312yes—
118#95—22no—
119#95C2312yes—
120C24—21no—
121C24—21no—
122C24—21no—
123C24—22no—
124C24—22no—
125C24—21no—
126C24—81no—
127C24—81no—
128C24C2×D412yes—
129C24C2×D411yescharacteristic
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—42no—
141C2×D4—42no—
142C2×D4—44no—
143C2×D4—41no—
144C2×D4—41no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—81no—
148C2×D4—81no—
149C2×D4—82no—
150C2×D4—82no—
151C2×D4—82no—
152C2×D4—82no—
153C2×D4C2×D412yes—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—24no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—81no—
163C22×C4—81no—
164C22×C4—82no—
165C22×C4C2×D412yes—
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2411yescharacteristic
172SD16—82no—
173D8—82no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—81no—
177C2×C8—81no—
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—22no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—81no—
186C4⋊C4C2×D412yes—
187C4⋊C4C2×D412yes—
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—42no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—82no—
204C42—24no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—41no—
210C23—41no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—81no—
226C23—81no—
227C23—84no—
228C23—84no—
229C23#12011yescharacteristic
230C23#12011yescharacteristic
231C23#13512yes—
232C23AGL(1,ℤ/8)12yes—
233C23C22×D411yescharacteristic
234Q8—42no—
235Q8—81no—
236D4—42no—
237D4—42no—
238D4—42no—
239D4—42no—
240D4—42no—
241D4—44no—
242D4—81no—
243D4—84no—
244D4—84no—
245D4—88no—
246D4—82no—
247D4—82no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—81no—
270C2×C4—84no—
271C2×C4#12011yescharacteristic
272C2×C4#12011yescharacteristic
273C2×C4C22×D411yescharacteristic
274C2×C4C22×D411yescharacteristic
275C8—42no—
276C8—42no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—82no—
293C22—84no—
294C22—84no—
295C22—84no—
296C22—82no—
297C22—82no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22#44912yes—
302C22#44912yes—
303C22C2×#12011yescharacteristic
304C22C2×#13511yescharacteristic
305C22C2×AGL(1,ℤ/8)11yescharacteristic
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—42no—
311C4—42no—
312C4—42no—
313C4—42no—
314C4—82no—
315C4—81no—
316C2—21no—
317C2—21no—
318C2—22no—
319C2—22no—
320C2—82no—
321C2—82no—
322C2—161no—
323C2C2×#44911yescharacteristic
324C2#301911yescharacteristic
325C2#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. #3187C2semidirect
3. #3186C2semidirect
4. #2617C2semidirect
5. #1612C2semidirect
6. #1600C2semidirect
7. #1520C2semidirect
8. #1519C2semidirect
9. C2×#323C2semidirect
25. #483C22semidirect
28. #480C22semidirect
31. #450C22semidirect
34. #447C22semidirect
37. #397C22semidirect
39. #392C22semidirect
42. #328C22semidirect
43. #327C22semidirect
44. #323C22semidirect
46. C22×Q8D4semidirect
52. C22×D4D4semidirect
69. C22⋊Q8D4semidirect
79. #121D4semidirect
89. C2×C4⋊C4D4semidirect
90. 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
#31872
#31862
#26172
#16122
#16002
#15202
#15192
C2×#3232