Finite Group Databasebeta

← #15684877

Subgroups — #15684877

Order 2944.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10625082C211yesmaximal, characteristic, complemented
2#13093061C211yesmaximal, characteristic, complemented
3D4×C22⋊D23C211yesmaximal, characteristic, complemented
4#13092805C211yesmaximal, characteristic, complemented
5#13092667C211yesmaximal, characteristic, complemented
6#13092646C211yesmaximal, characteristic, complemented
7#13092630C211yesmaximal, characteristic, complemented
8#13092610C211yesmaximal, characteristic, complemented
9#13092568C211yesmaximal, characteristic, complemented
10#13092559C211yesmaximal, characteristic, complemented
11#13092551C211yesmaximal, characteristic, complemented
12#13092549C211yesmaximal, characteristic, complemented
13#13092529C211yesmaximal, characteristic, complemented
14#13092488C211yesmaximal, characteristic, complemented
15C22×D8×C23C211yesmaximal, characteristic, complemented
16C22×D4×C23C2211yescharacteristic
17C22×D4×C23C2211yescharacteristic, complemented
18C22×C22⋊D23—21no—
19C2×#67290C2211yescharacteristic, complemented
20C2×D4×D23C2211yescharacteristic, complemented
21C2×D8×C23—24no—
22C2×D8×C23C2211yescharacteristic
23C2×D8×C23C2211yescharacteristic
24C2×D8×C23C2211yescharacteristic, complemented
25C2×D8×C23C2211yescharacteristic, complemented
26C22×C8×C23C2211yescharacteristic, complemented
27#10625124—21no—
28#10625124—21no—
29C2×C22⋊Dic23—21no—
30#10625111C2211yescharacteristic, complemented
31#10625110—21no—
32#10625110—21no—
33#10625109C2211yescharacteristic, complemented
34#10625109C2211yescharacteristic, complemented
35#10625108—21no—
36#10625107C2211yescharacteristic, complemented
37#10625106—21no—
38D4×Dic23C2211yescharacteristic, complemented
39D4×Dic23C2211yescharacteristic, complemented
40C2×#67273C2211yescharacteristic, complemented
41C2×#67273C2211yescharacteristic, complemented
42C2×#67272C2211yescharacteristic, complemented
43C2×#67272C2211yescharacteristic, complemented
44#10625101C2211yescharacteristic
45C4×C22⋊D23C2211yescharacteristic
46#10625095C2211yescharacteristic, complemented
47#10625082—24no—
48C2×#67263C2211yescharacteristic, complemented
49C2×#67262C2211yescharacteristic, complemented
50#10625014C2211yescharacteristic, complemented
51#10625014C2211yescharacteristic, complemented
52#10625014C2211yescharacteristic, complemented
53#10625014C2211yescharacteristic, complemented
54C22⋊(C23⋊C8)C2211yescharacteristic, complemented
55#10625003C2211yescharacteristic, complemented
56(C2×D23)⋊C8C2211yescharacteristic, complemented
57#10624999C2211yescharacteristic, complemented
58#10624998C2211yescharacteristic, complemented
59#10624997C2211yescharacteristic, complemented
60Dic23⋊C8C2211yescharacteristic, complemented
61C24×C23—21no—
62C24×C23—21no—
63C23×D23—21no—
64C2×D4×C23—22no—
65C2×D4×C23—22no—
66C2×D4×C23—21no—
67C2×D4×C23—21no—
68C2×D4×C23—41no—
69C2×D4×C23—41no—
70C2×D4×C23C2311yescharacteristic
71C2×D4×C23C2311yescharacteristic
72C2×D4×C23C2311yescharacteristic
73C2×D4×C23C2311yescharacteristic
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—21no—
82C2×C22⋊D23—44no—
83C2×C22⋊D23D411yescharacteristic
84C22×Dic23—21no—
85C22×Dic23—21no—
86C22×Dic23—21no—
87#67290—22no—
88#67290—41no—
89D4×D23—22no—
90D4×D23—41no—
91C2×D92C2311yescharacteristic
92C2×C4×D23C2311yescharacteristic
93C2×Dic46C2311yescharacteristic
94D8×C23—22no—
95D8×C23—22no—
96D8×C23—22no—
97D8×C23D414yescomplemented
98C2×C8×C23—22no—
99C2×C8×C23C2311yescharacteristic
100C2×C8×C23C2311yescharacteristic
101C22⋊Dic23—21no—
102C22⋊Dic23—21no—
103C22⋊Dic23—21no—
104C22⋊Dic23—21no—
105C22⋊Dic23—21no—
106C22⋊Dic23—21no—
107C22⋊Dic23—21no—
108C22⋊Dic23D411yescharacteristic
109#67273—22no—
110#67273—41no—
111#67272—22no—
112#67272—41no—
113(C2×D23)⋊C4—21no—
114(C2×D23)⋊C4D411yescharacteristic
115C4⋊Dic23C2311yescharacteristic
116C4⋊Dic23C2311yescharacteristic
117Dic23⋊C4—21no—
118Dic23⋊C4D411yescharacteristic
119C4×Dic23C2311yescharacteristic
120C2×C23⋊C8C2311yescharacteristic
121C2×C23⋊C8C2311yescharacteristic
122#67263—22no—
123#67262—22no—
124C23×C23—21no—
125C23×C23—21no—
126C23×C23—21no—
127C23×C23—21no—
128C23×C23—42no—
129C23×C23—42no—
130C23×C23—42no—
131C23×C23—42no—
132C23×C23—81no—
133C23×C23—81no—
134C23×C23C2×D411yescharacteristic
135C22×D23—21no—
136C22×D23—42no—
137C22×D23—42no—
138C22×D23—81no—
139C22×D23C2×D411yescharacteristic
140D4×C23—21no—
141D4×C23—21no—
142D4×C23—21no—
143D4×C23—21no—
144D4×C23—21no—
145D4×C23—21no—
146D4×C23—42no—
147D4×C23—42no—
148D4×C23C2×D412yes—
149D4×C23C2×D412yescomplemented
150C2×C4×C23—21no—
151C2×C4×C23—21no—
152C2×C4×C23C2×D411yescharacteristic
153C2×C4×C23C2411yescharacteristic
154C22⋊D23—22no—
155C22⋊D23—42no—
156C22⋊D23—42no—
157C22⋊D23—42no—
158C22⋊D23—42no—
159C22⋊D23—42no—
160C22⋊D23—81no—
161C22⋊D23—81no—
162C2×Dic23—21no—
163C2×Dic23—21no—
164C2×Dic23—21no—
165C2×Dic23—42no—
166C2×Dic23—42no—
167C2×Dic23—81no—
168C2×Dic23C2×D411yescharacteristic
169C2×Dic23C2×D411yescharacteristic
170C2×Dic23C2×D411yescharacteristic
171D92—21no—
172D92—21no—
173C4×D23—21no—
174C4×D23—21no—
175Dic46—21no—
176Dic46—21no—
177C8×C23—21no—
178C8×C23C2×D412yescomplemented
179C23⋊C8—21no—
180C23⋊C8—21no—
181#3264—231nomaximal, Sylow
182C22×C23—21no—
183C22×C23—21no—
184C22×C23—22no—
185C22×C23—22no—
186C22×C23—41no—
187C22×C23—41no—
188C22×C23—41no—
189C22×C23—41no—
190C22×C23—41no—
191C22×C23—41no—
192C22×C23—41no—
193C22×C23—41no—
194C22×C23—41no—
195C22×C23—41no—
196C22×C23—84no—
197C22×C23—84no—
198C22×C23AGL(1,ℤ/8)12yescomplemented
199C22×C23C22×D411yescharacteristic
200C2×D23—21no—
201C2×D23—21no—
202C2×D23—21no—
203C2×D23—41no—
204C2×D23—41no—
205C2×D23—41no—
206C2×D23—84no—
207C4×C23—21no—
208C4×C23C22×D411yescharacteristic
209C4×C23H(5,𝔽2)11yescharacteristic
210Dic23—21no—
211Dic23—21no—
212Dic23—21no—
213Dic23—41no—
214Dic23—41no—
215Dic23—41no—
216C2×AGL(1,ℤ/8)—231no—
217C22×D8D2311yescharacteristic, complemented
218#546—231no—
219D42—231no—
220#496—231no—
221#469—231no—
222#465—231no—
223#463—231no—
224#459—231no—
225#450—231no—
226#449—231no—
227#447—231no—
228#447—231no—
229#442—231no—
230#435—231no—
231C2×C23—22no—
232C2×C23—42no—
233C2×C23—42no—
234C2×C23—81no—
235C2×C23—81no—
236C2×C23D4211yescharacteristic
237C2×C23C2×AGL(1,ℤ/8)11yescharacteristic
238C2×C23#57611yescharacteristic
239D23—41no—
240D23—81no—
241C2×#41—231no—
242C22×D4—231no—
243C22×D4—461no—
244C22×D4C2×D2311yescharacteristic
245C22×D4C2×D2311yescharacteristic, complemented
246AGL(1,ℤ/8)—464no—
247C2×SD16—231no—
248C2×SD16—231no—
249C2×SD16—231no—
250C2×D8—24no—
251C2×D8—231no—
252C2×D8—231no—
253C2×D8C2×D2311yescharacteristic
254C2×D8C2×D2311yescharacteristic
255C2×D8C2×D2311yescharacteristic, complemented
256C2×D8C2×D2311yescharacteristic, complemented
257C2×M16—231no—
258C22×C8C2×D2311yescharacteristic, complemented
259#127—231no—
260#124—231no—
261#123—461no—
262C22⋊Q8—231no—
263#121—231no—
264#121—231no—
265#121—231no—
266#121—461no—
267#121—461no—
268#120—461no—
269#120—461no—
270#120—461no—
271C4×D4—231no—
272C4×D4—231no—
273C4×D4—231no—
274C2×C22⋊C4—461no—
275#106—231no—
276C4⋊C8—231no—
277#103—231no—
278#102—231no—
279#102—231no—
280#102—231no—
281#102—231no—
282#102—231no—
283C22⋊C8—231no—
284C22⋊C8—231no—
285#97—231no—
286C23#326411yescharacteristic, complemented, Sylow
287C24—21no—
288C24—21no—
289C24—461no—
290#41—462no—
291#41—921no—
292C2×Q8—231no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—231no—
300C2×D4—231no—
301C2×D4—461no—
302C2×D4—461no—
303C2×D4—461no—
304C2×D4—462no—
305C2×D4—461no—
306C2×D4—461no—
307C2×D4—461no—
308C2×D4—461no—
309C2×D4—921no—
310C2×D4—924no—
311C2×D4C22×D2311yescharacteristic
312C2×D4C22×D2311yescharacteristic
313C2×D4C22×D2311yescharacteristic
314C2×D4C22×D2311yescharacteristic
315C22×C4—231no—
316C22×C4—461no—
317C22×C4—461no—
318C22×C4—461no—
319C22×C4C22×D2311yescharacteristic
320SD16—462no—
321SD16—462no—
322SD16—921no—
323D8—22no—
324D8—22no—
325D8—22no—
326D8—462no—
327D8—921no—
328D8C22⋊D2314yescomplemented
329M16—462no—
330C2×C8—22no—
331C2×C8—231no—
332C2×C8—231no—
333C2×C8C22×D2311yescharacteristic
334C2×C8C22×D2311yescharacteristic
335C4⋊C4—231no—
336C4⋊C4—231no—
337C4⋊C4—231no—
338C4⋊C4—461no—
339C22⋊C4—231no—
340C22⋊C4—231no—
341C22⋊C4—461no—
342C22⋊C4—461no—
343C22⋊C4—461no—
344C22⋊C4—461no—
345C22⋊C4—461no—
346C22⋊C4—461no—
347C22⋊C4—461no—
348C22⋊C4—461no—
349C42—231no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23—231no—
361C23—461no—
362C23—922no—
363C23—922no—
364C23—1841no—
365C23D4×D2311yescharacteristic
366Q8—461no—
367Q8—461no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—462no—
377D4—461no—
378D4—461no—
379D4—922no—
380D4—922no—
381D4—922no—
382D4—922no—
383D4—922no—
384D4—1841no—
385D4—1841no—
386D4C2×C22⋊D2312yes—
387D4C2×C22⋊D2312yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—231no—
391C2×C4—231no—
392C2×C4—231no—
393C2×C4—461no—
394C2×C4—461no—
395C2×C4—461no—
396C2×C4—461no—
397C2×C4—461no—
398C2×C4—922no—
399C2×C4—922no—
400C2×C4—1841no—
401C2×C4D4×D2311yescharacteristic
402C2×C4C23×D2311yescharacteristic
403C8—21no—
404C8—461no—
405C8—461no—
406C8C2×C22⋊D2312yescomplemented
407C22—21no—
408C22—21no—
409C22—22no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—461no—
424C22—461no—
425C22—461no—
426C22—921no—
427C22—921no—
428C22—921no—
429C22—1844no—
430C22#1062508212yescomplemented
431C22C2×D4×D2311yescharacteristic
432C4—21no—
433C4—461no—
434C4—461no—
435C4—461no—
436C4—921no—
437C4—921no—
438C4—921no—
439C4#1062515511yescharacteristic
440C4C22×C22⋊D2311yescharacteristic
441C2—22no—
442C2—42no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—921no—
447C2—1841no—
448C2D4×C22⋊D2311yescharacteristic
449C2C2×#1062508211yescharacteristic
450C2#1309321811yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC22×D8×C23C2
socleC22×C23C22×D4
hypercenterC22×D8D23
hypocenterC23#3264
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#10625082C2semidirect
2. #13093061C2semidirect
3. D4×C22⋊D23C2semidirect
4. #13092805C2semidirect
5. #13092667C2semidirect
6. #13092646C2semidirect
7. #13092630C2semidirect
8. #13092610C2semidirect
9. #13092568C2semidirect
10. #13092559C2semidirect
11. #13092551C2semidirect
12. #13092549C2semidirect
13. #13092529C2semidirect
14. #13092488C2semidirect
15. C22×D8×C23C2semidirect
17. C22×D4×C23C22semidirect
19. C2×#67290C22semidirect
20. C2×D4×D23C22semidirect
24. C2×D8×C23C22semidirect
25. C2×D8×C23C22semidirect
26. C22×C8×C23C22semidirect
30. #10625111C22semidirect
33. #10625109C22semidirect
34. #10625109C22semidirect
36. #10625107C22semidirect
38. D4×Dic23C22semidirect
39. D4×Dic23C22semidirect
40. C2×#67273C22semidirect
41. C2×#67273C22semidirect
42. C2×#67272C22semidirect
43. C2×#67272C22semidirect
46. #10625095C22semidirect
48. C2×#67263C22semidirect
49. C2×#67262C22semidirect
50. #10625014C22semidirect
51. #10625014C22semidirect
52. #10625014C22semidirect
53. #10625014C22semidirect
54. C22⋊(C23⋊C8)C22semidirect
55. #10625003C22semidirect
56. (C2×D23)⋊C8C22semidirect
57. #10624999C22semidirect
58. #10624998C22semidirect
59. #10624997C22semidirect
60. Dic23⋊C8C22semidirect
97. D8×C23D4semidirect
149. D4×C23C2×D4semidirect
178. C8×C23C2×D4semidirect
198. C22×C23AGL(1,ℤ/8)semidirect
217. C22×D8D23semidirect
245. C22×D4C2×D23semidirect
255. C2×D8C2×D23semidirect
256. C2×D8C2×D23semidirect
258. C22×C8C2×D23semidirect
286. C23#3264semidirect
328. D8C22⋊D23semidirect
387. D4C2×C22⋊D23semidirect
406. C8C2×C22⋊D23semidirect
430. C22#10625082semidirect

Maximal subgroups

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

subgroupindex
C2×#106250822
#130930612
D4×C22⋊D232
#130928052
#130926672
#130926462
#130926302
#130926102
#130925682
#130925592
#130925512
#130925492
#130925292
#130924882
C22×D8×C232
#326423