Finite Group Databasebeta

← #15684886

Subgroups — #15684886

Order 2944.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D23C211yesmaximal, characteristic, complemented
2#13093074C211yesmaximal, characteristic, complemented
3D4×C22⋊D23C211yesmaximal, characteristic, complemented
4#13092805C211yesmaximal, characteristic, complemented
5#13092663C211yesmaximal, characteristic, complemented
6#13092650C211yesmaximal, characteristic, complemented
7#13092630C211yesmaximal, characteristic, complemented
8#13092615C211yesmaximal, characteristic, complemented
9#13092568C211yesmaximal, characteristic, complemented
10#13092562C211yesmaximal, characteristic, complemented
11#13092554C211yesmaximal, characteristic, complemented
12#13092549C211yesmaximal, characteristic, complemented
13#13092518C211yesmaximal, characteristic, complemented
14#13092486C211yesmaximal, characteristic, complemented
15C2×C23×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic, complemented
17C22×D4×C23C2211yescharacteristic
18C22×C22⋊D23—21no—
19C2×Q8×D23C2211yescharacteristic, complemented
20C2×D4×D23C2211yescharacteristic
21C23×AGL(1,ℤ/8)—24no—
22C2×SD16×C23C2211yescharacteristic
23C2×SD16×C23C2211yescharacteristic, complemented
24C2×D8×C23C2211yescharacteristic
25C2×D8×C23C2211yescharacteristic, complemented
26C2×M16×C23C2211yescharacteristic, complemented
27#10625124—21no—
28Q8×Dic23C2211yescharacteristic, complemented
29C2×#67274C2211yescharacteristic, complemented
30#10625111C2211yescharacteristic
31#10625110—21no—
32#10625109—21no—
33#10625109C2211yescharacteristic
34#10625109C2211yescharacteristic, complemented
35#10625108—21no—
36#10625107—21no—
37#10625107C2211yescharacteristic, complemented
38D4×Dic23C2211yescharacteristic
39C2×#67273C2211yescharacteristic
40C2×#67273C2211yescharacteristic, complemented
41C2×#67272C2211yescharacteristic, complemented
42#10625101C2211yescharacteristic
43C4×C22⋊D23—21no—
44C4×C22⋊D23C2211yescharacteristic
45#10625095—21no—
46#10625095C2211yescharacteristic, complemented
47SD16×D23—24no—
48C2×#67263C2211yescharacteristic, complemented
49C2×C8×D23C2211yescharacteristic, complemented
50#10625017C2211yescharacteristic, complemented
51#10625014C2211yescharacteristic
52#10625014C2211yescharacteristic, complemented
53#10625014C2211yescharacteristic, complemented
54C22⋊(C23⋊C8)C2211yescharacteristic, complemented
55#10625003C2211yescharacteristic, complemented
56(C2×D23)⋊C8C2211yescharacteristic, complemented
57#10625000C2211yescharacteristic, complemented
58#10624998C2211yescharacteristic, complemented
59#10624997C2211yescharacteristic, complemented
60Dic23⋊C8C2211yescharacteristic, complemented
61C24×C23—21no—
62C23×D23—21no—
63#41×C23—22no—
64#41×C23—41no—
65C2×Q8×C23C2311yescharacteristic
66C2×D4×C23—22no—
67C2×D4×C23—21no—
68C2×D4×C23—21no—
69C2×D4×C23—41no—
70C2×D4×C23C2311yescharacteristic
71C2×D4×C23C2311yescharacteristic
72C2×D4×C23C2311yescharacteristic
73C22×C4×C23—21no—
74C22×C4×C23C2311yescharacteristic
75C2×C22⋊D23—21no—
76C2×C22⋊D23—21no—
77C2×C22⋊D23—21no—
78C2×C22⋊D23—21no—
79C2×C22⋊D23—21no—
80C2×C22⋊D23—21no—
81C2×C22⋊D23—44no—
82C2×C22⋊D23D411yescharacteristic
83C22×Dic23—21no—
84Q8×D23—22no—
85Q8×D23—41no—
86D4×D23—22no—
87D4×D23—41no—
88C2×D92C2311yescharacteristic
89C2×C4×D23—21no—
90C2×C4×D23C2311yescharacteristic
91C2×Dic46—21no—
92C2×Dic46C2311yescharacteristic
93SD16×C23—22no—
94SD16×C23D414yescomplemented
95D8×C23—22no—
96D8×C23—22no—
97M16×C23—22no—
98C2×C8×C23C2311yescharacteristic
99C2×C8×C23C2311yescharacteristic
100C22⋊Dic23—21no—
101C22⋊Dic23—21no—
102C22⋊Dic23—21no—
103C22⋊Dic23—21no—
104C22⋊Dic23—21no—
105C22⋊Dic23—21no—
106C22⋊Dic23D411yescharacteristic
107#67274—22no—
108#67273—22no—
109#67273—41no—
110#67272—41no—
111(C2×D23)⋊C4—21no—
112(C2×D23)⋊C4D411yescharacteristic
113C4⋊Dic23—21no—
114C4⋊Dic23C2311yescharacteristic
115C4⋊Dic23C2311yescharacteristic
116Dic23⋊C4—21no—
117Dic23⋊C4D411yescharacteristic
118C4×Dic23—21no—
119C4×Dic23C2311yescharacteristic
120C2×C23⋊C8C2311yescharacteristic
121C2×C23⋊C8C2311yescharacteristic
122#67263—22no—
123C8×D23—22no—
124C23×C23—21no—
125C23×C23—21no—
126C23×C23—21no—
127C23×C23—42no—
128C23×C23—42no—
129C23×C23—81no—
130C23×C23C2×D411yescharacteristic
131C22×D23—21no—
132C22×D23—42no—
133C22×D23—42no—
134C22×D23—81no—
135C22×D23C2×D411yescharacteristic
136Q8×C23—21no—
137Q8×C23C2×D412yescomplemented
138D4×C23—21no—
139D4×C23—21no—
140D4×C23—21no—
141D4×C23—21no—
142D4×C23—21no—
143D4×C23—42no—
144D4×C23—42no—
145D4×C23C2×D412yes—
146C2×C4×C23—21no—
147C2×C4×C23—21no—
148C2×C4×C23—21no—
149C2×C4×C23—42no—
150C2×C4×C23C2×D411yescharacteristic
151C2×C4×C23C2411yescharacteristic
152C22⋊D23—22no—
153C22⋊D23—42no—
154C22⋊D23—42no—
155C22⋊D23—42no—
156C22⋊D23—42no—
157C22⋊D23—42no—
158C22⋊D23—81no—
159C2×Dic23—21no—
160C2×Dic23—21no—
161C2×Dic23—21no—
162C2×Dic23—42no—
163C2×Dic23C2×D411yescharacteristic
164C2×Dic23C2×D411yescharacteristic
165C2×Dic23C2×D411yescharacteristic
166D92—21no—
167D92—21no—
168C4×D23—21no—
169C4×D23—21no—
170C4×D23—42no—
171Dic46—21no—
172Dic46—21no—
173Dic46—42no—
174C8×C23—21no—
175C8×C23C2×D412yescomplemented
176C23⋊C8—21no—
177C23⋊C8—21no—
178#3265—231nomaximal, Sylow
179C22×C23—22no—
180C22×C23—21no—
181C22×C23—21no—
182C22×C23—21no—
183C22×C23—41no—
184C22×C23—41no—
185C22×C23—41no—
186C22×C23—41no—
187C22×C23—41no—
188C22×C23—41no—
189C22×C23—41no—
190C22×C23—41no—
191C22×C23—84no—
192C22×C23C22×D411yescharacteristic
193C2×D23—21no—
194C2×D23—21no—
195C2×D23—41no—
196C2×D23—41no—
197C2×D23—41no—
198C2×D23—84no—
199C2×D23AGL(1,ℤ/8)12yescomplemented
200C4×C23—22no—
201C4×C23—21no—
202C4×C23C22×D411yescharacteristic
203C4×C23H(5,𝔽2)11yescharacteristic
204Dic23—21no—
205Dic23—21no—
206Dic23—21no—
207Dic23—41no—
208Dic23—41no—
209Dic23—41no—
210C2×AGL(1,ℤ/8)D2311yescharacteristic, complemented
211C22×SD16—231no—
212#548—231no—
213D42—231no—
214#496—231no—
215#468—231no—
216#466—231no—
217#463—231no—
218#460—231no—
219#450—231no—
220#449—231no—
221#448—231no—
222#447—231no—
223#440—231no—
224#435—231no—
225C2×C23—42no—
226C2×C23—41no—
227C2×C23—81no—
228C2×C23—81no—
229C2×C23D4211yescharacteristic
230C2×C23C2×AGL(1,ℤ/8)11yescharacteristic
231C2×C23#57711yescharacteristic
232D23—22no—
233D23—81no—
234C2×#41C2×D2311yescharacteristic, complemented
235C22×Q8—231no—
236C22×D4—231no—
237C22×D4—461no—
238C22×D4C2×D2311yescharacteristic
239AGL(1,ℤ/8)—24no—
240C2×SD16—231no—
241C2×SD16—231no—
242C2×SD16—231no—
243C2×SD16—231no—
244C2×SD16—464no—
245C2×SD16C2×D2311yescharacteristic
246C2×SD16C2×D2311yescharacteristic, complemented
247C2×D8—231no—
248C2×D8C2×D2311yescharacteristic
249C2×D8C2×D2311yescharacteristic, complemented
250C2×M16C2×D2311yescharacteristic, complemented
251C22×C8—231no—
252#127—231no—
253#124—231no—
254#124—461no—
255C22⋊Q8—231no—
256C22⋊Q8—461no—
257#121—231no—
258#121—231no—
259#121—231no—
260#121—461no—
261#121—461no—
262#120—461no—
263#120—461no—
264C4×Q8—231no—
265C4×D4—231no—
266C4×D4—231no—
267C4×D4—461no—
268#107—231no—
269C4⋊C8—231no—
270#103—231no—
271#103—231no—
272#102—231no—
273#102—231no—
274#102—231no—
275#102—231no—
276C22⋊C8—231no—
277C22⋊C8—231no—
278#97—231no—
279C23#326511yescharacteristic, complemented, Sylow
280C24—21no—
281C24—461no—
282#41—22no—
283#41—41no—
284C2×Q8—231no—
285C2×Q8—461no—
286C2×Q8—462no—
287C2×Q8—921no—
288C2×Q8C22×D2311yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—231no—
294C2×D4—231no—
295C2×D4—461no—
296C2×D4—461no—
297C2×D4—461no—
298C2×D4—462no—
299C2×D4—461no—
300C2×D4—461no—
301C2×D4—461no—
302C2×D4—921no—
303C2×D4—924no—
304C2×D4C22×D2311yescharacteristic
305C2×D4C22×D2311yescharacteristic
306C2×D4C22×D2311yescharacteristic
307C22×C4—21no—
308C22×C4—231no—
309C22×C4—461no—
310C22×C4—461no—
311C22×C4C22×D2311yescharacteristic
312SD16—22no—
313SD16—462no—
314SD16—462no—
315SD16—462no—
316SD16—921no—
317SD16C22⋊D2314yescomplemented
318D8—22no—
319D8—22no—
320D8—921no—
321M16—22no—
322C2×C8—231no—
323C2×C8—231no—
324C2×C8—462no—
325C2×C8C22×D2311yescharacteristic
326C2×C8C22×D2311yescharacteristic
327C4⋊C4—231no—
328C4⋊C4—231no—
329C4⋊C4—231no—
330C4⋊C4—461no—
331C4⋊C4—461no—
332C22⋊C4—231no—
333C22⋊C4—231no—
334C22⋊C4—461no—
335C22⋊C4—461no—
336C22⋊C4—461no—
337C22⋊C4—461no—
338C22⋊C4—461no—
339C22⋊C4—461no—
340C22⋊C4—461no—
341C42—231no—
342C42—461no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—231no—
350C23—461no—
351C23—922no—
352C23—922no—
353C23—1841no—
354C23D4×D2311yescharacteristic
355Q8—21no—
356Q8—461no—
357Q8—461no—
358Q8—922no—
359Q8C2×C22⋊D2312yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—42no—
366D4—42no—
367D4—462no—
368D4—461no—
369D4—461no—
370D4—922no—
371D4—922no—
372D4—922no—
373D4—922no—
374D4—922no—
375D4—1841no—
376D4C2×C22⋊D2312yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—231no—
382C2×C4—231no—
383C2×C4—231no—
384C2×C4—461no—
385C2×C4—461no—
386C2×C4—461no—
387C2×C4—461no—
388C2×C4—461no—
389C2×C4—922no—
390C2×C4—922no—
391C2×C4D4×D2311yescharacteristic
392C2×C4C23×D2311yescharacteristic
393C8—21no—
394C8—461no—
395C8—461no—
396C8C2×C22⋊D2312yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—232no—
411C22—461no—
412C22—461no—
413C22—921no—
414C22—921no—
415C22—921no—
416C22—1844no—
417C22C2×D4×D2311yescharacteristic
418C4—21no—
419C4—22no—
420C4—461no—
421C4—461no—
422C4—461no—
423C4—921no—
424C4—921no—
425C4—921no—
426C4#1062515511yescharacteristic
427C4C22×C22⋊D2311yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—81no—
432C2—462no—
433C2—1841no—
434C2D4×C22⋊D2311yescharacteristic
435C2AGL(1,ℤ/8)×D2311yescharacteristic
436C2#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#3265
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×SD16×D23C2semidirect
2. #13093074C2semidirect
3. D4×C22⋊D23C2semidirect
4. #13092805C2semidirect
5. #13092663C2semidirect
6. #13092650C2semidirect
7. #13092630C2semidirect
8. #13092615C2semidirect
9. #13092568C2semidirect
10. #13092562C2semidirect
11. #13092554C2semidirect
12. #13092549C2semidirect
13. #13092518C2semidirect
14. #13092486C2semidirect
15. C2×C23×AGL(1,ℤ/8)C2semidirect
16. C2×#41×C23C22semidirect
19. C2×Q8×D23C22semidirect
23. C2×SD16×C23C22semidirect
25. C2×D8×C23C22semidirect
26. C2×M16×C23C22semidirect
28. Q8×Dic23C22semidirect
29. C2×#67274C22semidirect
34. #10625109C22semidirect
37. #10625107C22semidirect
40. C2×#67273C22semidirect
41. C2×#67272C22semidirect
46. #10625095C22semidirect
48. C2×#67263C22semidirect
49. C2×C8×D23C22semidirect
50. #10625017C22semidirect
52. #10625014C22semidirect
53. #10625014C22semidirect
54. C22⋊(C23⋊C8)C22semidirect
55. #10625003C22semidirect
56. (C2×D23)⋊C8C22semidirect
57. #10625000C22semidirect
58. #10624998C22semidirect
59. #10624997C22semidirect
60. Dic23⋊C8C22semidirect
94. SD16×C23D4semidirect
137. Q8×C23C2×D4semidirect
175. C8×C23C2×D4semidirect
199. C2×D23AGL(1,ℤ/8)semidirect
210. C2×AGL(1,ℤ/8)D23semidirect
234. C2×#41C2×D23semidirect
246. C2×SD16C2×D23semidirect
249. C2×D8C2×D23semidirect
250. C2×M16C2×D23semidirect
279. C23#3265semidirect
317. SD16C22⋊D23semidirect
359. Q8C2×C22⋊D23semidirect
396. C8C2×C22⋊D23semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D232
#130930742
D4×C22⋊D232
#130928052
#130926632
#130926502
#130926302
#130926152
#130925682
#130925622
#130925542
#130925492
#130925182
#130924862
C2×C23×AGL(1,ℤ/8)2
#326523