Finite Group Databasebeta

← #15683859

Subgroups — #15683859

Order 2944.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10625080C211yesmaximal, characteristic, complemented
2C2×#10625122C211yesmaximal, characteristic, complemented
3C2×#10625103C211yesmaximal, characteristic, complemented
4#13092963C211yesmaximal, characteristic, complemented
5C2×#10625107C211yesmaximal, characteristic, complemented
6#13092808C212yesmaximal, complemented
7#13092805C212yesmaximal, complemented
8#13092800C212yesmaximal, complemented
9#13092798C212yesmaximal, complemented
10M16×Dic23C211yesmaximal, characteristic, complemented
11C2×C23×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
12C2×#41×C23C2211yescharacteristic
13C22×D4×C23C2211yescharacteristic
14C2×#67288C2211yescharacteristic
15C22×Dic46C2211yescharacteristic
16C23×AGL(1,ℤ/8)—24no—
17C2×SD16×C23C2212yes—
18C2×D8×C23C2212yes—
19C2×M16×C23C2211yescharacteristic
20C2×C22⋊Dic23—22no—
21#10625122—24no—
22#10625118C2211yescharacteristic, complemented
23#10625115C2211yescharacteristic, complemented
24C2×#67275C2212yescomplemented
25#10625111C2211yescharacteristic, complemented
26#10625110—22no—
27#10625107—22no—
28#10625107C2212yes—
29#10625107C2212yes—
30#10625107C2211yescharacteristic, complemented
31C2×#67273C2212yes—
32C2×#67273C2212yescomplemented
33#10625103—24no—
34C2×#67272C2212yes—
35C4×C22⋊D23—22no—
36C2×C4×Dic23C2211yescharacteristic
37C2×#67267C2211yescharacteristic
38#10625080—24no—
39C2×Dic92C2212yescomplemented
40C2×#67263C2212yescomplemented
41#10624997C2212yescomplemented
42C8×Dic23C2212yescomplemented
43C24×C23—21no—
44#41×C23—22no—
45#41×C23—22no—
46C2×Q8×C23C2311yescharacteristic
47C2×D4×C23—21no—
48C2×D4×C23—22no—
49C2×D4×C23—22no—
50C2×D4×C23—21no—
51C2×D4×C23C2312yes—
52C2×D4×C23C2311yescharacteristic
53C22×C4×C23—21no—
54C22×C4×C23C2311yescharacteristic
55C2×C22⋊D23—21no—
56C2×C22⋊D23—22no—
57C22×Dic23—21no—
58C22×Dic23D412yes—
59#67288—22no—
60#67288—22no—
61C2×D92C2311yescharacteristic
62C2×C4×D23—21no—
63C2×Dic46—22no—
64C2×Dic46—22no—
65C2×Dic46—21no—
66C2×Dic46C2312yes—
67C2×Dic46C2311yescharacteristic
68SD16×C23—24no—
69D8×C23—24no—
70M16×C23D414yescomplemented
71C2×C8×C23C2312yes—
72C22⋊Dic23—22no—
73C22⋊Dic23—24no—
74C22⋊Dic23—24no—
75#67275—24no—
76#67273—24no—
77#67273—24no—
78#67272—24no—
79(C2×D23)⋊C4—22no—
80Dic23⋊C4—22no—
81C4×Dic23D412yes—
82C4×Dic23C2311yescharacteristic
83C4×Dic23C2311yescharacteristic
84#67267D414yescomplemented
85C2×C23⋊C8C2312yes—
86Dic92—24no—
87#67263—24no—
88C23×C23—22no—
89C23×C23—21no—
90C23×C23—44no—
91C23×C23—42no—
92C23×C23C2×D411yescharacteristic
93C22×D23—21no—
94Q8×C23—21no—
95Q8×C23—21no—
96D4×C23—22no—
97D4×C23—21no—
98D4×C23—21no—
99D4×C23—22no—
100D4×C23—42no—
101D4×C23—42no—
102C2×C4×C23—21no—
103C2×C4×C23—42no—
104C2×C4×C23C2×D411yescharacteristic
105C2×C4×C23C2×D412yes—
106C2×C4×C23C2×D412yes—
107C2×C4×C23C2411yescharacteristic
108C22⋊D23—44no—
109C22⋊D23—42no—
110C2×Dic23—21no—
111C2×Dic23—22no—
112C2×Dic23—42no—
113C2×Dic23—42no—
114C2×Dic23C2×D412yes—
115C2×Dic23C2×D412yes—
116D92—21no—
117D92—21no—
118C4×D23—42no—
119Dic46—22no—
120Dic46—21no—
121Dic46—21no—
122Dic46—22no—
123Dic46—42no—
124C8×C23C2×D414yes—
125C23⋊C8C2×D414yes—
126#3153—231nomaximal, Sylow
127C22×C23—22no—
128C22×C23—41no—
129C22×C23—42no—
130C22×C23—42no—
131C22×C23—44no—
132C22×C23—84no—
133C22×C23#12712yes—
134C22×C23C22×D411yescharacteristic
135C2×D23—41no—
136C2×D23—42no—
137C4×C23—41no—
138C4×C23#12712yes—
139C4×C23C22×D411yescharacteristic
140C4×C23C22×D411yescharacteristic
141Dic23—22no—
142Dic23—22no—
143Dic23—42no—
144Dic23—41no—
145C2×#137—231no—
146C2×#137—231no—
147C2×AGL(1,ℤ/8)—231no—
148C2×AGL(1,ℤ/8)D2311yescharacteristic, complemented
149#532—231no—
150C2×#124—231no—
151#497—232no—
152#496—232no—
153#495—232no—
154#495—232no—
155C4×M16—231no—
156C2×C23—22no—
157C2×C23—81no—
158C2×C23—82no—
159C2×C23C2×#12711yescharacteristic
160C2×C23#57512yes—
161D23—81no—
162C2×#41—231no—
163C2×#41C2×D2311yescharacteristic
164C22×Q8—231no—
165C22×D4C2×D2311yescharacteristic
166#137—464no—
167#137—464no—
168AGL(1,ℤ/8)—24no—
169AGL(1,ℤ/8)—464no—
170C2×Q16—232no—
171C2×Q16—232no—
172C2×SD16—232no—
173C2×SD16—232no—
174C2×SD16—232no—
175C2×SD16C2×D2312yes—
176C2×D8—232no—
177C2×D8C2×D2312yes—
178C2×M16—231no—
179C2×M16C2×D2311yescharacteristic
180C4⋊Q8—231no—
181#127—231no—
182#124—231no—
183#124—232no—
184#124—232no—
185#124—231no—
186#124—462no—
187#121—462no—
188C4×D4—462no—
189C2×C22⋊C4—462no—
190C2×C42—231no—
191#97—232no—
192C4×C8—232no—
193C23#315311yescharacteristic, complemented, Sylow
194C24—21no—
195#41—22no—
196#41—22no—
197#41—462no—
198#41—462no—
199C2×Q8—232no—
200C2×Q8—231no—
201C2×Q8—461no—
202C2×Q8—462no—
203C2×Q8—462no—
204C2×Q8C22×D2311yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—231no—
210C2×D4—462no—
211C2×D4—461no—
212C2×D4C22×D2312yes—
213C2×D4C22×D2311yescharacteristic
214C22×C4—21no—
215C22×C4—232no—
216C22×C4—461no—
217C22×C4—461no—
218C22×C4C22×D2311yescharacteristic
219Q16—464no—
220Q16—464no—
221SD16—24no—
222SD16—464no—
223SD16—464no—
224SD16—464no—
225D8—24no—
226D8—464no—
227M16—234no—
228M16C22⋊D2314yescomplemented
229C2×C8—232no—
230C2×C8C22×D2312yes—
231C4⋊C4—462no—
232C22⋊C4—462no—
233C22⋊C4—464no—
234C22⋊C4—464no—
235C22⋊C4—462no—
236C42—232no—
237C42—231no—
238C42—231no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—461no—
244C23D4×D2311yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—461no—
248Q8—462no—
249Q8—462no—
250Q8—461no—
251Q8—922no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—461no—
259D4—461no—
260D4—924no—
261D4—922no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—232no—
265C2×C4—232no—
266C2×C4—462no—
267C2×C4—461no—
268C2×C4—922no—
269C2×C4—922no—
270C2×C4—922no—
271C2×C4D4×D2311yescharacteristic
272C2×C4D4×D2312yes—
273C2×C4C2×C22⋊D2312yes—
274C2×C4C23×D2311yescharacteristic
275C8—234no—
276C8C2×C22⋊D2314yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—921no—
284C22—922no—
285C22#1062511112yes—
286C22C2×D4×D2311yescharacteristic
287C4—41no—
288C4—462no—
289C4—462no—
290C4—921no—
291C4—922no—
292C4#1062511112yes—
293C4C2×D4×D2311yescharacteristic
294C4C22×C22⋊D2311yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—1841no—
299C2C2×#1062511111yescharacteristic
300C2#1309321312yes—
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC2×C23×AGL(1,ℤ/8)C2
socleC22×C23C22×D4
hypercenterC2×AGL(1,ℤ/8)D23
hypocenterC23#3153
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#10625080C2semidirect
2. C2×#10625122C2semidirect
3. C2×#10625103C2semidirect
4. #13092963C2semidirect
5. C2×#10625107C2semidirect
6. #13092808C2semidirect
7. #13092805C2semidirect
8. #13092800C2semidirect
9. #13092798C2semidirect
10. M16×Dic23C2semidirect
11. C2×C23×AGL(1,ℤ/8)C2semidirect
22. #10625118C22semidirect
23. #10625115C22semidirect
24. C2×#67275C22semidirect
25. #10625111C22semidirect
30. #10625107C22semidirect
32. C2×#67273C22semidirect
39. C2×Dic92C22semidirect
40. C2×#67263C22semidirect
41. #10624997C22semidirect
42. C8×Dic23C22semidirect
70. M16×C23D4semidirect
84. #67267D4semidirect
148. C2×AGL(1,ℤ/8)D23semidirect
193. C23#3153semidirect
228. M16C22⋊D23semidirect

Maximal subgroups

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

subgroupindex
C2×#106250802
C2×#106251222
C2×#106251032
#130929632
C2×#106251072
#130928082
#130928052
#130928002
#130927982
M16×Dic232
C2×C23×AGL(1,ℤ/8)2
#315323