Finite Group Databasebeta

← D23×#450 #15682576

Subgroups — D23×#450 #15682576

Order 2944.

996 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C22×D4×D23C211yesmaximal, characteristic
2C2×SD16×D23C212yesmaximal, complemented
3C22⋊Q8×D23C211yesmaximal, characteristic, complemented
4#13092570C211yesmaximal, characteristic, complemented
5#13092569C212yesmaximal, complemented
6#13092568C211yesmaximal, characteristic, complemented
7#13092567C211yesmaximal, characteristic, complemented
8#13092566C212yesmaximal, complemented
9#102×D23C212yesmaximal, complemented
10C22⋊C8×D23C211yesmaximal, characteristic, complemented
11C23×#450C211yesmaximal, characteristic, complemented
12C24×D23—21no—
13C22×D4×C23C2211yescharacteristic
14C22×C22⋊D23—21no—
15C2×Q8×D23C2211yescharacteristic, complemented
16C2×D4×D23—28no—
17C2×D4×D23—21no—
18C2×D4×D23—21no—
19C2×D4×D23C2212yes—
20C2×D4×D23C2212yes—
21C22×D92C2211yescharacteristic
22C22×C4×D23C2211yescharacteristic
23C2×SD16×C23C2212yescomplemented
24C23×C22⋊Q8C2211yescharacteristic, complemented
25#10625117C2211yescharacteristic, complemented
26C2×#67274C2212yescomplemented
27C2×#67273C2212yescomplemented
28#10625095C2211yescharacteristic, complemented
29SD16×D23—28no—
30C2×#67263C2212yescomplemented
31C2×C8×D23C2212yescomplemented
32#10625068C2211yescharacteristic, complemented
33#10625067—21no—
34C4⋊C4×D23—21no—
35C4⋊C4×D23C2211yescharacteristic, complemented
36C22⋊C4×D23—21no—
37#10625049—21no—
38C23×#102C2212yescomplemented
39C23×C22⋊C8C2211yescharacteristic, complemented
40#10625014C2212yescomplemented
41C22⋊(C23⋊C8)C2211yescharacteristic, complemented
42#10625003C2212yescomplemented
43(C2×D23)⋊C8C2212yescomplemented
44#10624991C2212yescomplemented
45C24×C23—21no—
46C23×D23—28no—
47C23×D23—21no—
48C23×D23—22no—
49C23×D23—22no—
50C23×D23—44no—
51C23×D23—44no—
52C23×D23D411yescharacteristic
53C2×Q8×C23C2311yescharacteristic, complemented
54C2×D4×C23—24no—
55C2×D4×C23—21no—
56C2×D4×C23C2312yes—
57C22×C4×C23C2311yescharacteristic
58C2×C22⋊D23—28no—
59C2×C22⋊D23—22no—
60C2×C22⋊D23—22no—
61C22×Dic23D411yescharacteristic
62Q8×D23—24no—
63D4×D23—28no—
64D4×D23—28no—
65D4×D23—24no—
66D4×D23—24no—
67D4×D23—24no—
68D4×D23D418yes—
69C2×D92—21no—
70C2×D92—24no—
71C2×D92C2312yes—
72C2×C4×D23—24no—
73C2×C4×D23—21no—
74C2×C4×D23—21no—
75C2×C4×D23D411yescharacteristic
76C2×C4×D23D411yescharacteristic
77C2×C4×D23C2311yescharacteristic
78C2×C4×D23C2311yescharacteristic
79C2×Dic46—21no—
80C2×Dic46C2311yescharacteristic, complemented
81SD16×C23—24no—
82C2×C8×C23C2312yescomplemented
83C4⋊C4×C23—21no—
84C4⋊C4×C23C2311yescharacteristic, complemented
85C22⋊C4×C23—21no—
86C22⋊Dic23—21no—
87#67274—24no—
88#67273—24no—
89(C2×D23)⋊C4—21no—
90(C2×D23)⋊C4—21no—
91C4⋊Dic23—21no—
92C4⋊Dic23C2311yescharacteristic, complemented
93Dic23⋊C4—21no—
94Dic23⋊C4—21no—
95Dic23⋊C4—21no—
96C2×C23⋊C8C2312yescomplemented
97#67263—24no—
98C8×D23—24no—
99C23×C23—24no—
100C23×C23—22no—
101C23×C23—42no—
102C23×C23—42no—
103C23×C23C2×D411yescharacteristic
104C22×D23—22no—
105C22×D23—22no—
106C22×D23—24no—
107C22×D23—28no—
108C22×D23—22no—
109C22×D23—28no—
110C22×D23—44no—
111C22×D23—432no—
112C22×D23—42no—
113C22×D23—42no—
114C22×D23—44no—
115C22×D23—44no—
116C22×D23—44no—
117C22×D23SD1614yescomplemented
118C22×D23C2×D411yescharacteristic
119C22×D23C2×D411yescharacteristic
120Q8×C23—22no—
121D4×C23—24no—
122D4×C23—22no—
123D4×C23C2×D414yes—
124C2×C4×C23—22no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23C2×D411yescharacteristic
128C2×C4×C23C2411yescharacteristic
129C22⋊D23—28no—
130C22⋊D23—28no—
131C2×Dic23—22no—
132C2×Dic23—21no—
133C2×Dic23—21no—
134C2×Dic23C2×D411yescharacteristic
135C2×Dic23C2×D411yescharacteristic
136D92—24no—
137D92—22no—
138D92C2×D414yes—
139C4×D23—22no—
140C4×D23—22no—
141C4×D23—22no—
142C4×D23—42no—
143C4×D23—42no—
144C4×D23C2×D414yes—
145Dic46—22no—
146Dic46—42no—
147C8×C23—22no—
148C23⋊C8—22no—
149C2×#450—231nomaximal, Sylow
150C22×C23—21no—
151C22×C23—21no—
152C22×C23—24no—
153C22×C23—48no—
154C22×C23—42no—
155C22×C23—42no—
156C22×C23C2×SD1612yescomplemented
157C22×C23C22×D411yescharacteristic
158C2×D23—21no—
159C2×D23—21no—
160C2×D23—28no—
161C2×D23—24no—
162C2×D23—416no—
163C2×D23—48no—
164C2×D23—42no—
165C2×D23—42no—
166C2×D23—416no—
167C2×D23#12012yes—
168C2×D23C2×SD1612yes—
169C2×D23C2×SD1612yescomplemented
170C2×D23AGL(1,ℤ/8)12yes—
171C4×C23—21no—
172C4×C23—41no—
173C4×C23—41no—
174C4×C23C22×D412yes—
175Dic23—21no—
176Dic23—41no—
177Dic23—41no—
178Dic23#12012yes—
179C23×D4—231no—
180C22×SD16—232no—
181C2×C22⋊Q8—231no—
182#450—231no—
183#450—232no—
184#450—231no—
185#450—232no—
186#450—231no—
187#450D2311yescharacteristic, direct product factor
188C2×#102—232no—
189C2×C22⋊C8—231no—
190C2×C23—22no—
191C2×C23—44no—
192C2×C23C2×#12011yescharacteristic
193C2×C23C22×SD1611yescharacteristic
194C2×C23C2×AGL(1,ℤ/8)11yescharacteristic
195D23—22no—
196D23—44no—
197D23#45014yesdirect product factor
198C25—461no—
199C22×Q8—231no—
200C22×D4—232no—
201C22×D4—232no—
202C22×D4—231no—
203C22×D4—461no—
204C22×D4—461no—
205C22×D4—461no—
206C22×D4—468no—
207C22×D4C2×D2311yescharacteristic
208C23×C4—231no—
209C2×SD16—232no—
210C2×SD16—232no—
211C2×SD16—232no—
212C2×SD16—468no—
213C2×SD16C2×D2312yescomplemented
214C22×C8—232no—
215C22⋊Q8—231no—
216C22⋊Q8—231no—
217C22⋊Q8—231no—
218C22⋊Q8—461no—
219C22⋊Q8—461no—
220C22⋊Q8C2×D2311yescharacteristic, complemented
221C2×C4⋊C4—231no—
222C2×C4⋊C4—461no—
223C2×C22⋊C4—461no—
224#102—232no—
225#102—232no—
226#102—232no—
227#102C2×D2312yescomplemented
228C22⋊C8—231no—
229C22⋊C8—232no—
230C22⋊C8C2×D2311yescharacteristic, complemented
231C23C2×#45011yescharacteristic, complemented, Sylow
232C24—21no—
233C24—231no—
234C24—462no—
235C24—462no—
236C24—468no—
237C24—461no—
238C24—924no—
239C24—924no—
240C2×Q8—231no—
241C2×Q8—461no—
242C2×Q8—464no—
243C2×Q8C22×D2311yescharacteristic, complemented
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—238no—
247C2×D4—232no—
248C2×D4—464no—
249C2×D4—464no—
250C2×D4—462no—
251C2×D4—462no—
252C2×D4—464no—
253C2×D4—468no—
254C2×D4—468no—
255C2×D4—468no—
256C2×D4—464no—
257C2×D4—461no—
258C2×D4C22×D2312yes—
259C22×C4—231no—
260C22×C4—231no—
261C22×C4—231no—
262C22×C4—231no—
263C22×C4—231no—
264C22×C4—464no—
265C22×C4—461no—
266C22×C4—461no—
267C22×C4C22×D2311yescharacteristic
268SD16—24no—
269SD16—464no—
270SD16—464no—
271SD16—464no—
272C2×C8—232no—
273C2×C8—464no—
274C2×C8C22×D2312yescomplemented
275C4⋊C4—21no—
276C4⋊C4—231no—
277C4⋊C4—461no—
278C4⋊C4—461no—
279C4⋊C4—461no—
280C4⋊C4—461no—
281C4⋊C4C22×D2311yescharacteristic, complemented
282C22⋊C4—21no—
283C22⋊C4—461no—
284C22⋊C4—461no—
285C22⋊C4—461no—
286C23—24no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—231no—
291C23—234no—
292C23—231no—
293C23—462no—
294C23—462no—
295C23—468no—
296C23—464no—
297C23—468no—
298C23—462no—
299C23—922no—
300C23—924no—
301C23—924no—
302C23—924no—
303C23—9232no—
304C23—924no—
305C23—922no—
306C23D4×D2311yescharacteristic
307Q8—22no—
308Q8—462no—
309Q8—922no—
310D4—24no—
311D4—22no—
312D4—234no—
313D4—464no—
314D4—468no—
315D4—468no—
316D4—462no—
317D4D4×D2314yes—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—231no—
322C2×C4—234no—
323C2×C4—231no—
324C2×C4—462no—
325C2×C4—462no—
326C2×C4—462no—
327C2×C4—462no—
328C2×C4—461no—
329C2×C4—461no—
330C2×C4—922no—
331C2×C4—922no—
332C2×C4D4×D2311yescharacteristic
333C2×C4C23×D2311yescharacteristic
334C8—22no—
335C8—462no—
336C22—21no—
337C22—21no—
338C22—24no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22—232no—
343C22—232no—
344C22—232no—
345C22—232no—
346C22—468no—
347C22—461no—
348C22—461no—
349C22—464no—
350C22—928no—
351C22—9216no—
352C22—9216no—
353C22—922no—
354C22—922no—
355C22SD16×D2312yescomplemented
356C22C2×D4×D2311yescharacteristic
357C4—21no—
358C4—41no—
359C4—41no—
360C4—232no—
361C4—461no—
362C4—921no—
363C4—921no—
364C4C2×D4×D2312yes—
365C2—22no—
366C2—44no—
367C2—234no—
368C2—462no—
369C2—924no—
370C2#120×D2311yescharacteristic
371C2C2×SD16×D2311yescharacteristic
372C2AGL(1,ℤ/8)×D2311yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#450C2
socleC22×C23C22×D4
prime radicalC22×D4×D23C2
hypercenter#450D23
hypocenterC23C2×#450
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×SD16×D23C2semidirect
3. C22⋊Q8×D23C2semidirect
4. #13092570C2semidirect
5. #13092569C2semidirect
6. #13092568C2semidirect
7. #13092567C2semidirect
8. #13092566C2semidirect
9. #102×D23C2semidirect
10. C22⋊C8×D23C2semidirect
11. C23×#450C2semidirect
15. C2×Q8×D23C22semidirect
23. C2×SD16×C23C22semidirect
24. C23×C22⋊Q8C22semidirect
25. #10625117C22semidirect
26. C2×#67274C22semidirect
27. C2×#67273C22semidirect
28. #10625095C22semidirect
30. C2×#67263C22semidirect
31. C2×C8×D23C22semidirect
32. #10625068C22semidirect
35. C4⋊C4×D23C22semidirect
38. C23×#102C22semidirect
39. C23×C22⋊C8C22semidirect
40. #10625014C22semidirect
41. C22⋊(C23⋊C8)C22semidirect
42. #10625003C22semidirect
43. (C2×D23)⋊C8C22semidirect
44. #10624991C22semidirect
53. C2×Q8×C23C23semidirect
80. C2×Dic46C23semidirect
82. C2×C8×C23C23semidirect
84. C4⋊C4×C23C23semidirect
92. C4⋊Dic23C23semidirect
96. C2×C23⋊C8C23semidirect
117. C22×D23SD16semidirect
156. C22×C23C2×SD16semidirect
169. C2×D23C2×SD16semidirect
187. #450D23direct (N × H)
197. D23#450direct (N × H)
213. C2×SD16C2×D23semidirect
220. C22⋊Q8C2×D23semidirect
227. #102C2×D23semidirect
230. C22⋊C8C2×D23semidirect
231. C23C2×#450semidirect
243. C2×Q8C22×D23semidirect
274. C2×C8C22×D23semidirect
281. C4⋊C4C22×D23semidirect
355. C22SD16×D23semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D232
C2×SD16×D232
C22⋊Q8×D232
#130925702
#130925692
#130925682
#130925672
#130925662
#102×D232
C22⋊C8×D232
C23×#4502
C2×#45023