Finite Group Databasebeta

← #15684916

Subgroups — #15684916

Order 2944.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10625083C211yesmaximal, characteristic, complemented
2#13093066C211yesmaximal, characteristic, complemented
3#13093061C211yesmaximal, characteristic, complemented
4#13092808C211yesmaximal, characteristic, complemented
5#13092675C211yesmaximal, characteristic, complemented
6#13092643C211yesmaximal, characteristic, complemented
7#13092630C211yesmaximal, characteristic, complemented
8#13092620C211yesmaximal, characteristic, complemented
9#13092579C211yesmaximal, characteristic, complemented
10#13092576C211yesmaximal, characteristic, complemented
11#13092568C211yesmaximal, characteristic, complemented
12#13092559C211yesmaximal, characteristic, complemented
13#13092529C211yesmaximal, characteristic, complemented
14#13092486C211yesmaximal, characteristic, complemented
15C2×C23×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17C22×D4×C23C2211yescharacteristic
18C2×#67290C2211yescharacteristic
19C2×#67290C2211yescharacteristic, complemented
20C23×AGL(1,ℤ/8)—24no—
21C2×SD16×C23C2211yescharacteristic
22C2×SD16×C23C2211yescharacteristic
23C2×D8×C23C2211yescharacteristic
24C2×D8×C23C2211yescharacteristic
25C2×M16×C23C2211yescharacteristic
26#10625124—21no—
27C2×C22⋊Dic23—21no—
28#10625117C2211yescharacteristic, complemented
29#10625115C2211yescharacteristic, complemented
30C2×#67275C2211yescharacteristic, complemented
31#10625110—21no—
32#10625110—21no—
33#10625109C2211yescharacteristic
34#10625107C2211yescharacteristic
35#10625106—21no—
36#10625106—21no—
37D4×Dic23C2211yescharacteristic
38D4×Dic23C2211yescharacteristic, complemented
39C2×#67273C2211yescharacteristic
40C2×#67273C2211yescharacteristic
41C2×#67273C2211yescharacteristic, complemented
42#10625100—21no—
43C4×C22⋊D23C2211yescharacteristic
44#10625095C2211yescharacteristic
45#10625095C2211yescharacteristic
46C2×Dic23⋊C4—21no—
47#10625083—24no—
48C2×Dic92C2211yescharacteristic, complemented
49C2×C8×D23C2211yescharacteristic, complemented
50#10625017C2211yescharacteristic, complemented
51#10625014C2211yescharacteristic
52#10625014C2211yescharacteristic
53#10625014C2211yescharacteristic, complemented
54C22⋊(C23⋊C8)C2211yescharacteristic
55(C2×D23)⋊C8C2211yescharacteristic, complemented
56#10624999C2211yescharacteristic, complemented
57#10624998C2211yescharacteristic, complemented
58#10624998C2211yescharacteristic, complemented
59#10624997C2211yescharacteristic, complemented
60Dic23⋊C8C2211yescharacteristic, complemented
61C24×C23—21no—
62#41×C23—22no—
63#41×C23—41no—
64C2×Q8×C23C2311yescharacteristic
65C2×D4×C23—22no—
66C2×D4×C23—21no—
67C2×D4×C23—21no—
68C2×D4×C23—41no—
69C2×D4×C23C2311yescharacteristic
70C2×D4×C23C2311yescharacteristic
71C2×D4×C23C2311yescharacteristic
72C22×C4×C23—21no—
73C22×C4×C23C2311yescharacteristic
74C2×C22⋊D23—21no—
75C2×C22⋊D23—21no—
76C2×C22⋊D23—21no—
77C2×C22⋊D23D411yescharacteristic
78C22×Dic23—21no—
79C22×Dic23—21no—
80C22×Dic23—21no—
81C22×Dic23—21no—
82#67290—22no—
83#67290—22no—
84#67290—41no—
85#67290—41no—
86C2×C4×D23C2311yescharacteristic
87C2×Dic46C2311yescharacteristic
88C2×Dic46C2311yescharacteristic
89SD16×C23—22no—
90SD16×C23—22no—
91D8×C23—22no—
92D8×C23D414yescomplemented
93M16×C23—22no—
94C2×C8×C23C2311yescharacteristic
95C2×C8×C23C2311yescharacteristic
96C22⋊Dic23—21no—
97C22⋊Dic23—21no—
98C22⋊Dic23—21no—
99C22⋊Dic23—21no—
100C22⋊Dic23—21no—
101C22⋊Dic23—21no—
102C22⋊Dic23—21no—
103C22⋊Dic23D411yescharacteristic
104#67275—41no—
105#67273—22no—
106#67273—22no—
107#67273—41no—
108(C2×D23)⋊C4—21no—
109(C2×D23)⋊C4D411yescharacteristic
110C4⋊Dic23C2311yescharacteristic
111C4⋊Dic23C2311yescharacteristic
112Dic23⋊C4—21no—
113Dic23⋊C4—21no—
114Dic23⋊C4—21no—
115Dic23⋊C4—21no—
116Dic23⋊C4—21no—
117Dic23⋊C4D411yescharacteristic
118C4×Dic23C2311yescharacteristic
119C2×C23⋊C8C2311yescharacteristic
120C2×C23⋊C8C2311yescharacteristic
121Dic92—22no—
122C8×D23—22no—
123C23×C23—21no—
124C23×C23—21no—
125C23×C23—21no—
126C23×C23—42no—
127C23×C23—42no—
128C23×C23—81no—
129C23×C23C2×D411yescharacteristic
130C22×D23C2×D411yescharacteristic
131Q8×C23—21no—
132Q8×C23—21no—
133D4×C23—21no—
134D4×C23—21no—
135D4×C23—21no—
136D4×C23—21no—
137D4×C23—42no—
138D4×C23—42no—
139D4×C23C2×D412yes—
140D4×C23C2×D412yes—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—42no—
145C2×C4×C23C2×D411yescharacteristic
146C2×C4×C23C2411yescharacteristic
147C22⋊D23—41no—
148C22⋊D23—42no—
149C22⋊D23—42no—
150C22⋊D23—81no—
151C2×Dic23—21no—
152C2×Dic23—21no—
153C2×Dic23—21no—
154C2×Dic23—21no—
155C2×Dic23—42no—
156C2×Dic23—42no—
157C2×Dic23—42no—
158C2×Dic23—81no—
159C2×Dic23C2×D411yescharacteristic
160C2×Dic23C2×D411yescharacteristic
161C2×Dic23C2×D411yescharacteristic
162C4×D23—21no—
163C4×D23—21no—
164Dic46—21no—
165Dic46—21no—
166Dic46—21no—
167Dic46—21no—
168C8×C23—21no—
169C8×C23C2×D412yes—
170C23⋊C8—21no—
171C23⋊C8—21no—
172#3267—231nomaximal, Sylow
173C22×C23—22no—
174C22×C23—22no—
175C22×C23—21no—
176C22×C23—21no—
177C22×C23—21no—
178C22×C23—41no—
179C22×C23—41no—
180C22×C23—41no—
181C22×C23—41no—
182C22×C23—41no—
183C22×C23—41no—
184C22×C23—41no—
185C22×C23—84no—
186C22×C23C22×D411yescharacteristic
187C2×D23—21no—
188C2×D23—21no—
189C2×D23—21no—
190C4×C23—21no—
191C4×C23—41no—
192C4×C23C22×D411yescharacteristic
193C4×C23H(5,𝔽2)11yescharacteristic
194Dic23—21no—
195Dic23—21no—
196Dic23—21no—
197Dic23—41no—
198Dic23—41no—
199Dic23—41no—
200Dic23—41no—
201C2×AGL(1,ℤ/8)D2311yescharacteristic, complemented
202C2×#135—231no—
203#547—231no—
204#546—231no—
205#497—231no—
206#470—231no—
207#465—231no—
208#463—231no—
209#461—231no—
210#452—231no—
211#452—231no—
212#450—231no—
213#449—231no—
214#442—231no—
215#435—231no—
216C2×C23—41no—
217C2×C23—42no—
218C2×C23—42no—
219C2×C23—81no—
220C2×C23D4211yescharacteristic
221C2×C23#57511yescharacteristic
222C2×C23#57711yescharacteristic
223D23—41no—
224C2×#41—231no—
225C2×#41—231no—
226C2×#41C2×D2311yescharacteristic
227C22×D4C2×D2311yescharacteristic
228AGL(1,ℤ/8)—24no—
229#135—464no—
230C2×Q16—231no—
231C2×Q16—231no—
232C2×SD16—231no—
233C2×SD16—231no—
234C2×SD16—231no—
235C2×SD16C2×D2311yescharacteristic
236C2×SD16C2×D2311yescharacteristic
237C2×D8C2×D2311yescharacteristic
238C2×D8C2×D2311yescharacteristic
239C2×M16C2×D2311yescharacteristic
240C22×C8—231no—
241C4⋊Q8—231no—
242#124—231no—
243#123—461no—
244#123—461no—
245#123—461no—
246C22⋊Q8—231no—
247C22⋊Q8—231no—
248C22⋊Q8—231no—
249#121—231no—
250#121—461no—
251#121—461no—
252#120—461no—
253C4×D4—231no—
254C4×D4—231no—
255C4×D4—231no—
256C2×C4⋊C4—461no—
257C2×C22⋊C4—461no—
258#106—231no—
259C4⋊C8—231no—
260#103—231no—
261#103—231no—
262#103—231no—
263#102—231no—
264#102—231no—
265#102—231no—
266C22⋊C8—231no—
267C22⋊C8—231no—
268#97—231no—
269C23#326711yescharacteristic, complemented, Sylow
270C24—21no—
271#41—22no—
272#41—41no—
273#41—462no—
274#41—462no—
275#41—921no—
276#41—921no—
277C2×Q8—231no—
278C2×Q8—231no—
279C2×Q8C22×D2311yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—231no—
285C2×D4—461no—
286C2×D4—461no—
287C2×D4—461no—
288C2×D4C22×D2311yescharacteristic
289C2×D4C22×D2311yescharacteristic
290C2×D4C22×D2311yescharacteristic
291C22×C4—21no—
292C22×C4—231no—
293C22×C4—461no—
294C22×C4—461no—
295C22×C4—461no—
296C22×C4—461no—
297C22×C4C22×D2311yescharacteristic
298Q16—462no—
299Q16—921no—
300SD16—22no—
301SD16—22no—
302SD16—462no—
303SD16—462no—
304SD16—921no—
305D8—22no—
306D8C22⋊D2314yescomplemented
307M16—22no—
308C2×C8—231no—
309C2×C8—231no—
310C2×C8—462no—
311C2×C8C22×D2311yescharacteristic
312C2×C8C22×D2311yescharacteristic
313C4⋊C4—231no—
314C4⋊C4—231no—
315C4⋊C4—231no—
316C4⋊C4—461no—
317C4⋊C4—461no—
318C4⋊C4—461no—
319C4⋊C4—461no—
320C4⋊C4—461no—
321C22⋊C4—231no—
322C22⋊C4—231no—
323C22⋊C4—461no—
324C22⋊C4—461no—
325C22⋊C4—461no—
326C22⋊C4—461no—
327C22⋊C4—461no—
328C22⋊C4—461no—
329C22⋊C4—461no—
330C22⋊C4—461no—
331C42—231no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—231no—
339C23D4×D2311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—461no—
343Q8—461no—
344Q8—461no—
345Q8—461no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—921no—
353D4—922no—
354D4—922no—
355D4—1841no—
356D4C2×C22⋊D2312yes—
357D4C2×C22⋊D2312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—231no—
363C2×C4—231no—
364C2×C4—231no—
365C2×C4—461no—
366C2×C4—461no—
367C2×C4—461no—
368C2×C4—461no—
369C2×C4—461no—
370C2×C4—461no—
371C2×C4—922no—
372C2×C4—922no—
373C2×C4—922no—
374C2×C4—1841no—
375C2×C4D4×D2311yescharacteristic
376C2×C4C23×D2311yescharacteristic
377C8—21no—
378C8—461no—
379C8—461no—
380C8C2×C22⋊D2312yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—461no—
395C22—461no—
396C22—461no—
397C22C2×D4×D2311yescharacteristic
398C4—21no—
399C4—41no—
400C4—461no—
401C4—461no—
402C4—461no—
403C4—921no—
404C4—921no—
405C4—921no—
406C4—921no—
407C4#1062515511yescharacteristic
408C4C22×C22⋊D2311yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—81no—
413C2—921no—
414C2D4×C22⋊D2311yescharacteristic
415C2#1309321311yescharacteristic
416C2#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#3267
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×#10625083C2semidirect
2. #13093066C2semidirect
3. #13093061C2semidirect
4. #13092808C2semidirect
5. #13092675C2semidirect
6. #13092643C2semidirect
7. #13092630C2semidirect
8. #13092620C2semidirect
9. #13092579C2semidirect
10. #13092576C2semidirect
11. #13092568C2semidirect
12. #13092559C2semidirect
13. #13092529C2semidirect
14. #13092486C2semidirect
15. C2×C23×AGL(1,ℤ/8)C2semidirect
19. C2×#67290C22semidirect
28. #10625117C22semidirect
29. #10625115C22semidirect
30. C2×#67275C22semidirect
38. D4×Dic23C22semidirect
41. C2×#67273C22semidirect
48. C2×Dic92C22semidirect
49. C2×C8×D23C22semidirect
50. #10625017C22semidirect
53. #10625014C22semidirect
55. (C2×D23)⋊C8C22semidirect
56. #10624999C22semidirect
57. #10624998C22semidirect
58. #10624998C22semidirect
59. #10624997C22semidirect
60. Dic23⋊C8C22semidirect
92. D8×C23D4semidirect
201. C2×AGL(1,ℤ/8)D23semidirect
269. C23#3267semidirect
306. D8C22⋊D23semidirect

Maximal subgroups

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

subgroupindex
C2×#106250832
#130930662
#130930612
#130928082
#130926752
#130926432
#130926302
#130926202
#130925792
#130925762
#130925682
#130925592
#130925292
#130924862
C2×C23×AGL(1,ℤ/8)2
#326723