Finite Group Databasebeta

← #15683871

Subgroups — #15683871

Order 2944.

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

normal subgroups only

This group has 564 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×#10625121C211yesmaximal, characteristic, complemented
3C22×#67273C211yesmaximal, characteristic, complemented
4#13092988C211yesmaximal, characteristic, complemented
5#13092984C211yesmaximal, characteristic, complemented
6#13092808C212yesmaximal, complemented
7#13092805C212yesmaximal, complemented
8#13092800C212yesmaximal, complemented
9#13092788C212yesmaximal, complemented
10#13092375C211yesmaximal, 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
20#10625124—22no—
21#10625121—24no—
22#10625115C2212yescomplemented
23C2×#67275C2211yescharacteristic, complemented
24C2×#67274C2211yescharacteristic, complemented
25#10625111C2212yes—
26#10625109—22no—
27#10625107C2212yes—
28#10625107C2212yescomplemented
29#10625106—22no—
30C2×#67273—24no—
31C2×#67273C2212yes—
32C2×#67273C2212yes—
33C2×#67273C2211yescharacteristic, complemented
34C2×#67272C2211yescharacteristic, complemented
35#10625097C2211yescharacteristic
36#10625095—22no—
37C22×C23⋊C8C2211yescharacteristic
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×C23C2311yescharacteristic
52C2×D4×C23C2312yes—
53C22×C4×C23—21no—
54C22×C4×C23C2311yescharacteristic
55C2×C22⋊D23—21no—
56C2×C22⋊D23—24no—
57C22×Dic23—21no—
58#67288—22no—
59#67288—22no—
60C2×D92C2311yescharacteristic
61C2×C4×D23—21no—
62C2×Dic46—22no—
63C2×Dic46—22no—
64C2×Dic46—21no—
65C2×Dic46C2312yes—
66C2×Dic46C2311yescharacteristic
67SD16×C23—24no—
68D8×C23—24no—
69M16×C23D414yescomplemented
70C2×C8×C23C2312yes—
71C22⋊Dic23—24no—
72C22⋊Dic23—24no—
73C22⋊Dic23D412yes—
74#67275—22no—
75#67274—22no—
76#67273—24no—
77#67273—24no—
78#67273—22no—
79#67272—22no—
80C4⋊Dic23D412yes—
81Dic23⋊C4—24no—
82C4×Dic23C2312yes—
83C2×C23⋊C8D414yescomplemented
84C2×C23⋊C8C2311yescharacteristic
85C2×C23⋊C8C2311yescharacteristic
86Dic92—24no—
87#67263—24no—
88C23×C23—21no—
89C23×C23—22no—
90C23×C23—42no—
91C23×C23—82no—
92C23×C23C2×D411yescharacteristic
93C22×D23—21no—
94Q8×C23—21no—
95Q8×C23—21no—
96D4×C23—21no—
97D4×C23—22no—
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—42no—
109C22⋊D23—84no—
110C2×Dic23—22no—
111C2×Dic23—21no—
112C2×Dic23—42no—
113C2×Dic23C2×D414yes—
114D92—21no—
115D92—21no—
116C4×D23—42no—
117Dic46—22no—
118Dic46—21no—
119Dic46—22no—
120Dic46—21no—
121Dic46—42no—
122C8×C23C2×D414yes—
123C23⋊C8C2×D412yes—
124C23⋊C8C2×D412yes—
125#3154—231nomaximal, Sylow
126C22×C23—22no—
127C22×C23—41no—
128C22×C23—42no—
129C22×C23—44no—
130C22×C23—42no—
131C22×C23—84no—
132C22×C23#12712yes—
133C22×C23C22×D411yescharacteristic
134C2×D23—41no—
135C2×D23—42no—
136C4×C23—41no—
137C4×C23#12712yes—
138C4×C23C22×D411yescharacteristic
139C4×C23C22×D411yescharacteristic
140Dic23—24no—
141Dic23—42no—
142Dic23—41no—
143C2×#137—231no—
144C2×AGL(1,ℤ/8)D2311yescharacteristic, complemented
145C2×#135—231no—
146C22×SD16—231no—
147#536—231no—
148#535—231no—
149#497—232no—
150#496—232no—
151#495—232no—
152#492—232no—
153#405—231no—
154C2×C23—22no—
155C2×C23—81no—
156C2×C23—82no—
157C2×C23C2×#12711yescharacteristic
158C2×C23#57712yes—
159D23—81no—
160C2×#41—231no—
161C2×#41C2×D2311yescharacteristic
162C22×Q8—231no—
163C22×D4C2×D2311yescharacteristic
164#137—464no—
165AGL(1,ℤ/8)—24no—
166#135—464no—
167C2×Q16—231no—
168C2×Q16—232no—
169C2×SD16—231no—
170C2×SD16—232no—
171C2×SD16—232no—
172C2×SD16—231no—
173C2×SD16—232no—
174C2×SD16—464no—
175C2×SD16C2×D2312yes—
176C2×D8—231no—
177C2×D8C2×D2312yes—
178C2×M16C2×D2311yescharacteristic
179C22×C8—231no—
180C4⋊Q8—232no—
181#127—232no—
182#124—232no—
183#124—232no—
184#123—462no—
185C22⋊Q8—462no—
186#121—462no—
187#120—462no—
188#117—231no—
189#97—232no—
190C4×C8—232no—
191C23#315411yescharacteristic, complemented, Sylow
192C24—21no—
193#41—22no—
194#41—22no—
195#41—462no—
196#41—462no—
197C2×Q8—231no—
198C2×Q8—232no—
199C2×Q8—461no—
200C2×Q8—462no—
201C2×Q8—462no—
202C2×Q8C22×D2311yescharacteristic
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—231no—
208C2×D4—464no—
209C2×D4—461no—
210C2×D4C22×D2311yescharacteristic
211C2×D4C22×D2312yes—
212C22×C4—21no—
213C22×C4—461no—
214C22×C4—461no—
215C22×C4C22×D2311yescharacteristic
216Q16—462no—
217Q16—464no—
218SD16—24no—
219SD16—464no—
220SD16—464no—
221SD16—462no—
222SD16—462no—
223SD16—464no—
224D8—24no—
225D8—462no—
226M16C22⋊D2314yescomplemented
227C2×C8—234no—
228C2×C8—231no—
229C2×C8—231no—
230C2×C8C22×D2312yes—
231C4⋊C4—232no—
232C4⋊C4—464no—
233C22⋊C4—232no—
234C22⋊C4—464no—
235C22⋊C4—464no—
236C42—232no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—82no—
241C23—461no—
242C23D4×D2311yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—462no—
246Q8—461no—
247Q8—462no—
248Q8—461no—
249Q8—922no—
250D4—21no—
251D4—22no—
252D4—21no—
253D4—22no—
254D4—42no—
255D4—42no—
256D4—461no—
257D4—461no—
258D4—922no—
259D4—1844no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—234no—
263C2×C4—461no—
264C2×C4—462no—
265C2×C4—922no—
266C2×C4—922no—
267C2×C4D4×D2311yescharacteristic
268C2×C4D4×D2312yes—
269C2×C4C2×C22⋊D2312yes—
270C2×C4C23×D2311yescharacteristic
271C8—232no—
272C8—232no—
273C8C2×C22⋊D2314yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—44no—
278C22—42no—
279C22—84no—
280C22—921no—
281C22—922no—
282C22#1062511112yes—
283C22C2×D4×D2311yescharacteristic
284C4—41no—
285C4—464no—
286C4—921no—
287C4—922no—
288C4#1062511112yes—
289C4C2×D4×D2311yescharacteristic
290C4C22×C22⋊D2311yescharacteristic
291C2—22no—
292C2—82no—
293C2—81no—
294C2—1841no—
295C2C2×#1062511111yescharacteristic
296C2#1309322712yes—
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#3154
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×#10625121C2semidirect
3. C22×#67273C2semidirect
4. #13092988C2semidirect
5. #13092984C2semidirect
6. #13092808C2semidirect
7. #13092805C2semidirect
8. #13092800C2semidirect
9. #13092788C2semidirect
10. #13092375C2semidirect
11. C2×C23×AGL(1,ℤ/8)C2semidirect
22. #10625115C22semidirect
23. C2×#67275C22semidirect
24. C2×#67274C22semidirect
28. #10625107C22semidirect
33. C2×#67273C22semidirect
34. C2×#67272C22semidirect
39. C2×Dic92C22semidirect
40. C2×#67263C22semidirect
41. #10624997C22semidirect
42. C8×Dic23C22semidirect
69. M16×C23D4semidirect
83. C2×C23⋊C8D4semidirect
144. C2×AGL(1,ℤ/8)D23semidirect
191. C23#3154semidirect
226. 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×#106251212
C22×#672732
#130929882
#130929842
#130928082
#130928052
#130928002
#130927882
#130923752
C2×C23×AGL(1,ℤ/8)2
#315423