Finite Group Databasebeta

← #15682700

Subgroups — #15682700

Order 2944.

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

normal subgroups only

This group has 764 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×#10625155C211yesmaximal, characteristic, complemented
2C2×#10625121C211yesmaximal, characteristic, complemented
3#13092995C211yesmaximal, characteristic, complemented
4#13092576C212yesmaximal, complemented
5#13092566C212yesmaximal, complemented
6#13092559C212yesmaximal, complemented
7#13092549C212yesmaximal, complemented
8#13092419C211yesmaximal, characteristic, complemented
9C2×#10625014C211yesmaximal, characteristic, complemented
10#13092385C211yesmaximal, characteristic, complemented
11C2×C23×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
12C2×#41×C23C2211yescharacteristic, complemented
13C22×D4×C23C2211yescharacteristic
14C22×C22⋊D23—22no—
15#10625155—24no—
16#10625155—24no—
17C2×#67290—21no—
18C2×#67290C2212yes—
19C2×D4×D23—21no—
20C2×D4×D23C2212yes—
21C2×#67288C2211yescharacteristic
22C2×#67288C2211yescharacteristic
23C23×AGL(1,ℤ/8)—24no—
24C2×SD16×C23C2212yescomplemented
25C2×D8×C23C2212yescomplemented
26C2×M16×C23C2211yescharacteristic, complemented
27#10625121—24no—
28#10625117C2212yescomplemented
29C2×#67275C2211yescharacteristic, complemented
30C2×#67274C2211yescharacteristic, complemented
31#10625110—22no—
32#10625109C2212yescomplemented
33C2×#67273C2211yescharacteristic, complemented
34C2×#67272C2211yescharacteristic, complemented
35#10625101—21no—
36C2×C4⋊Dic23C2211yescharacteristic, complemented
37#10625095—21no—
38C22×C23⋊C8C2211yescharacteristic, complemented
39#10625014C2212yescomplemented
40#10625014C2212yescomplemented
41#10625003C2212yescomplemented
42(C2×D23)⋊C8C2212yescomplemented
43(C2×D23)⋊C8C2212yescomplemented
44#10624998C2212yescomplemented
45C24×C23—21no—
46C23×D23—22no—
47#41×C23—22no—
48#41×C23—22no—
49C2×Q8×C23C2311yescharacteristic
50C2×D4×C23—21no—
51C2×D4×C23—21no—
52C2×D4×C23—22no—
53C2×D4×C23D414yescomplemented
54C2×D4×C23C2311yescharacteristic
55C2×D4×C23C2312yes—
56C22×C4×C23—21no—
57C22×C4×C23C2311yescharacteristic
58C2×C22⋊D23—216no—
59C2×C22⋊D23—21no—
60C2×C22⋊D23—22no—
61C2×C22⋊D23—22no—
62C2×C22⋊D23—22no—
63C2×C22⋊D23D412yes—
64C22×Dic23—22no—
65C22×Dic23—21no—
66#67290—28no—
67#67290—24no—
68#67290—24no—
69D4×D23—28no—
70D4×D23—24no—
71D4×D23—24no—
72#67288—22no—
73#67288—22no—
74#67288—22no—
75#67288D414yescomplemented
76C2×D92D411yescharacteristic
77C2×D92C2311yescharacteristic
78C2×C4×D23—21no—
79C2×C4×D23C2312yes—
80C2×Dic46D411yescharacteristic
81C2×Dic46C2311yescharacteristic
82SD16×C23—24no—
83D8×C23—24no—
84M16×C23—22no—
85C2×C8×C23C2312yes—
86C22⋊Dic23—22no—
87#67275—22no—
88#67274—22no—
89#67273—22no—
90#67272—22no—
91(C2×D23)⋊C4—22no—
92C4⋊Dic23—21no—
93C4⋊Dic23C2312yes—
94Dic23⋊C4—22no—
95C2×C23⋊C8—22no—
96C2×C23⋊C8C2311yescharacteristic
97C2×C23⋊C8C2311yescharacteristic
98C23×C23—22no—
99C23×C23—24no—
100C23×C23—21no—
101C23×C23—81no—
102C23×C23—81no—
103C23×C23C2×D411yescharacteristic
104C22×D23—21no—
105C22×D23—28no—
106C22×D23—82no—
107C22×D23—82no—
108C22×D23C2×D412yes—
109Q8×C23—21no—
110Q8×C23—21no—
111D4×C23—24no—
112D4×C23—22no—
113D4×C23—21no—
114D4×C23—21no—
115D4×C23—42no—
116D4×C23C2×D414yes—
117C2×C4×C23—21no—
118C2×C4×C23—42no—
119C2×C4×C23C2×D411yescharacteristic
120C2×C4×C23C2×D412yes—
121C2×C4×C23C2×D412yes—
122C2×C4×C23C2411yescharacteristic
123C22⋊D23—28no—
124C22⋊D23—28no—
125C22⋊D23—24no—
126C22⋊D23—24no—
127C22⋊D23—82no—
128C2×Dic23—28no—
129C2×Dic23—21no—
130C2×Dic23—22no—
131C2×Dic23—81no—
132C2×Dic23C2×D412yes—
133D92—22no—
134D92—21no—
135D92C2×D412yes—
136C4×D23—24no—
137C4×D23—22no—
138C4×D23—22no—
139Dic46—22no—
140Dic46—21no—
141Dic46C2×D412yes—
142C8×C23—22no—
143C23⋊C8—21no—
144C23⋊C8—21no—
145#3012—231nomaximal, Sylow
146C22×C23—21no—
147C22×C23—21no—
148C22×C23—24no—
149C22×C23—42no—
150C22×C23—42no—
151C22×C23—41no—
152C22×C23—41no—
153C22×C23—41no—
154C22×C23—84no—
155C22×C23#12012yes—
156C22×C23C22×D411yescharacteristic
157C2×D23—22no—
158C2×D23—22no—
159C2×D23—22no—
160C2×D23—22no—
161C2×D23—41no—
162C2×D23—41no—
163C2×D23—88no—
164C4×C23—41no—
165C4×C23#12012yes—
166C4×C23C22×D411yescharacteristic
167C4×C23C22×D411yescharacteristic
168Dic23—22no—
169Dic23—22no—
170Dic23—42no—
171C2×H(5,𝔽2)—231no—
172C2×AGL(1,ℤ/8)D2311yescharacteristic, complemented
173C2×#135—231no—
174#537—231no—
175#452—232no—
176#450—232no—
177#449—232no—
178#447—232no—
179#417—231no—
180C2×#102—231no—
181#408—231no—
182C2×C23—22no—
183C2×C23—44no—
184C2×C23—81no—
185C2×C23C2×#12011yescharacteristic
186C2×C23#57611yescharacteristic
187C2×C23#57711yescharacteristic
188D23—42no—
189D23—42no—
190H(5,𝔽2)—464no—
191H(5,𝔽2)—464no—
192C2×#41—231no—
193C2×#41—232no—
194C2×#41—231no—
195C2×#41—461no—
196C2×#41C2×D2311yescharacteristic, complemented
197C22×D4—232no—
198C22×D4—461no—
199C22×D4—462no—
200C22×D4C2×D2311yescharacteristic
201AGL(1,ℤ/8)—24no—
202#135—464no—
203C2×Q16—231no—
204C2×SD16—231no—
205C2×SD16—231no—
206C2×SD16C2×D2312yescomplemented
207C2×D8—231no—
208C2×D8C2×D2312yescomplemented
209C2×M16C2×D2311yescharacteristic, complemented
210C22×C8—231no—
211C22⋊Q8—232no—
212C22⋊Q8—461no—
213#121—232no—
214#121—461no—
215#121—462no—
216C2×C4⋊C4—231no—
217#103—232no—
218#102—232no—
219#102—232no—
220#102—232no—
221C22⋊C8—232no—
222C22⋊C8—232no—
223C23#301211yescharacteristic, complemented, Sylow
224C24—21no—
225C24—462no—
226#41—22no—
227#41—22no—
228#41—234no—
229#41—464no—
230#41—468no—
231#41—464no—
232#41—462no—
233#41—462no—
234#41—462no—
235C2×Q8—231no—
236C2×Q8—231no—
237C2×Q8C22×D2311yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—232no—
242C2×D4—231no—
243C2×D4—231no—
244C2×D4—462no—
245C2×D4—464no—
246C2×D4—468no—
247C2×D4—462no—
248C2×D4—4616no—
249C2×D4—461no—
250C2×D4—464no—
251C2×D4—462no—
252C2×D4C22⋊D2314yescomplemented
253C2×D4C22×D2311yescharacteristic
254C2×D4C22×D2312yes—
255C22×C4—21no—
256C22×C4—232no—
257C22×C4—462no—
258C22×C4—461no—
259C22×C4—461no—
260C22×C4C22×D2311yescharacteristic
261Q16—462no—
262SD16—24no—
263SD16—462no—
264SD16—462no—
265D8—24no—
266D8—462no—
267M16—22no—
268C2×C8—231no—
269C2×C8—231no—
270C2×C8—462no—
271C2×C8C22×D2312yes—
272C4⋊C4—232no—
273C4⋊C4—462no—
274C4⋊C4—461no—
275C22⋊C4—462no—
276C22⋊C4—462no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—232no—
283C23—468no—
284C23—461no—
285C23—1842no—
286C23—1842no—
287C23D4×D2311yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—232no—
291Q8—462no—
292Q8—461no—
293D4—24no—
294D4—22no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—232no—
299D4—468no—
300D4—468no—
301D4—464no—
302D4—464no—
303D4—462no—
304D4—461no—
305D4—1842no—
306D4C2×C22⋊D2314yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—232no—
310C2×C4—468no—
311C2×C4—461no—
312C2×C4—464no—
313C2×C4—462no—
314C2×C4—462no—
315C2×C4—462no—
316C2×C4—1841no—
317C2×C4D4×D2311yescharacteristic
318C2×C4D4×D2312yes—
319C2×C4C2×C22⋊D2312yes—
320C2×C4C23×D2311yescharacteristic
321C8—22no—
322C8—461no—
323C8—461no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—462no—
334C22—462no—
335C22—462no—
336C22—462no—
337C22—921no—
338C22—921no—
339C22—1848no—
340C22#1062510812yes—
341C22C2×D4×D2311yescharacteristic
342C4—41no—
343C4—462no—
344C4—462no—
345C4—922no—
346C4#1062510812yes—
347C4C2×D4×D2311yescharacteristic
348C4C22×C22⋊D2311yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—922no—
353C2—922no—
354C2C2×#1062510811yescharacteristic
355C2#1309322111yescharacteristic
356C2#1309322711yescharacteristic
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#3012
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×#10625155C2semidirect
2. C2×#10625121C2semidirect
3. #13092995C2semidirect
4. #13092576C2semidirect
5. #13092566C2semidirect
6. #13092559C2semidirect
7. #13092549C2semidirect
8. #13092419C2semidirect
9. C2×#10625014C2semidirect
10. #13092385C2semidirect
11. C2×C23×AGL(1,ℤ/8)C2semidirect
12. C2×#41×C23C22semidirect
24. C2×SD16×C23C22semidirect
25. C2×D8×C23C22semidirect
26. C2×M16×C23C22semidirect
28. #10625117C22semidirect
29. C2×#67275C22semidirect
30. C2×#67274C22semidirect
32. #10625109C22semidirect
33. C2×#67273C22semidirect
34. C2×#67272C22semidirect
36. C2×C4⋊Dic23C22semidirect
38. C22×C23⋊C8C22semidirect
39. #10625014C22semidirect
40. #10625014C22semidirect
41. #10625003C22semidirect
42. (C2×D23)⋊C8C22semidirect
43. (C2×D23)⋊C8C22semidirect
44. #10624998C22semidirect
53. C2×D4×C23D4semidirect
75. #67288D4semidirect
172. C2×AGL(1,ℤ/8)D23semidirect
196. C2×#41C2×D23semidirect
206. C2×SD16C2×D23semidirect
208. C2×D8C2×D23semidirect
209. C2×M16C2×D23semidirect
223. C23#3012semidirect
252. C2×D4C22⋊D23semidirect

Maximal subgroups

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

subgroupindex
C2×#106251552
C2×#106251212
#130929952
#130925762
#130925662
#130925592
#130925492
#130924192
C2×#106250142
#130923852
C2×C23×AGL(1,ℤ/8)2
#301223