Finite Group Databasebeta

← #15684931

Subgroups — #15684931

Order 2944.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10625087C211yesmaximal, characteristic, complemented
2#13093086C211yesmaximal, characteristic, complemented
3#13093061C211yesmaximal, characteristic, complemented
4#13092805C211yesmaximal, characteristic, complemented
5#13092663C211yesmaximal, characteristic, complemented
6#13092650C211yesmaximal, characteristic, complemented
7#13092637C211yesmaximal, characteristic, complemented
8#13092610C211yesmaximal, characteristic, complemented
9#13092579C211yesmaximal, characteristic, complemented
10#13092561C211yesmaximal, characteristic, complemented
11#13092559C211yesmaximal, characteristic, complemented
12#13092551C211yesmaximal, characteristic, complemented
13#13092518C211yesmaximal, characteristic, complemented
14#13092486C211yesmaximal, characteristic, complemented
15C2×C23×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17C22×D4×C23C2211yescharacteristic
18C2×#67292C2211yescharacteristic, complemented
19C2×#67290C2211yescharacteristic
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—
28Q8×Dic23C2211yescharacteristic, complemented
29C2×#67275C2211yescharacteristic, complemented
30#10625111—21no—
31#10625111C2211yescharacteristic, complemented
32#10625110—21no—
33#10625109—21no—
34#10625109C2211yescharacteristic
35#10625109C2211yescharacteristic, complemented
36#10625107C2211yescharacteristic
37#10625106—21no—
38D4×Dic23C2211yescharacteristic
39C2×#67273C2211yescharacteristic, complemented
40C2×#67272C2211yescharacteristic
41C2×#67272C2211yescharacteristic
42#10625101—21no—
43#10625101C2211yescharacteristic
44C4×C22⋊D23—21no—
45C4×C22⋊D23C2211yescharacteristic
46#10625095C2211yescharacteristic
47#10625087—24no—
48C2×#67263C2211yescharacteristic, complemented
49C2×C8×D23C2211yescharacteristic, complemented
50#10625017C2211yescharacteristic, complemented
51#10625014C2211yescharacteristic
52#10625014C2211yescharacteristic
53#10625014C2211yescharacteristic, complemented
54C22⋊(C23⋊C8)C2211yescharacteristic
55#10625003C2211yescharacteristic, complemented
56(C2×D23)⋊C8C2211yescharacteristic, complemented
57#10625000C2211yescharacteristic, 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⋊D23—21no—
78C2×C22⋊D23—21no—
79C2×C22⋊D23—21no—
80C2×C22⋊D23D411yescharacteristic
81C22×Dic23—21no—
82C22×Dic23—21no—
83#67292—22no—
84#67292—41no—
85#67290—22no—
86#67290—41no—
87C2×D92—21no—
88C2×D92C2311yescharacteristic
89C2×C4×D23—21no—
90C2×C4×D23C2311yescharacteristic
91C2×Dic46C2311yescharacteristic
92SD16×C23—22no—
93SD16×C23D414yescomplemented
94D8×C23—22no—
95D8×C23—22no—
96M16×C23—22no—
97C2×C8×C23C2311yescharacteristic
98C2×C8×C23C2311yescharacteristic
99C22⋊Dic23—21no—
100C22⋊Dic23—21no—
101C22⋊Dic23—21no—
102C22⋊Dic23—21no—
103C22⋊Dic23—21no—
104C22⋊Dic23—21no—
105C22⋊Dic23D411yescharacteristic
106#67275—22no—
107#67273—41no—
108#67272—22no—
109#67272—41no—
110(C2×D23)⋊C4—21no—
111(C2×D23)⋊C4D411yescharacteristic
112C4⋊Dic23—21no—
113C4⋊Dic23C2311yescharacteristic
114C4⋊Dic23C2311yescharacteristic
115Dic23⋊C4—21no—
116Dic23⋊C4D411yescharacteristic
117C4×Dic23—21no—
118C4×Dic23C2311yescharacteristic
119C2×C23⋊C8C2311yescharacteristic
120C2×C23⋊C8C2311yescharacteristic
121#67263—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×D23—21no—
131C22×D23C2×D411yescharacteristic
132Q8×C23—21no—
133Q8×C23C2×D412yes—
134D4×C23—21no—
135D4×C23—21no—
136D4×C23—21no—
137D4×C23—21no—
138D4×C23—21no—
139D4×C23—42no—
140D4×C23—42no—
141D4×C23C2×D412yes—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—42no—
146C2×C4×C23C2×D411yescharacteristic
147C2×C4×C23C2411yescharacteristic
148C22⋊D23—41no—
149C22⋊D23—42no—
150C22⋊D23—42no—
151C22⋊D23—81no—
152C22⋊D23—81no—
153C22⋊D23—81no—
154C22⋊D23—81no—
155C2×Dic23—21no—
156C2×Dic23—21no—
157C2×Dic23—21no—
158C2×Dic23—42no—
159C2×Dic23—81no—
160C2×Dic23C2×D411yescharacteristic
161C2×Dic23C2×D411yescharacteristic
162C2×Dic23C2×D411yescharacteristic
163D92—21no—
164D92—21no—
165D92—42no—
166C4×D23—21no—
167C4×D23—21no—
168C4×D23—42no—
169Dic46—21no—
170Dic46—21no—
171C8×C23—21no—
172C8×C23C2×D412yes—
173C23⋊C8—21no—
174C23⋊C8—21no—
175#3268—231nomaximal, Sylow
176C22×C23—22no—
177C22×C23—21no—
178C22×C23—21no—
179C22×C23—21no—
180C22×C23—41no—
181C22×C23—41no—
182C22×C23—41no—
183C22×C23—41no—
184C22×C23—41no—
185C22×C23—41no—
186C22×C23—41no—
187C22×C23—41no—
188C22×C23—84no—
189C22×C23C22×D411yescharacteristic
190C2×D23—21no—
191C2×D23—21no—
192C2×D23—21no—
193C2×D23—41no—
194C2×D23—41no—
195C2×D23—41no—
196C4×C23—22no—
197C4×C23—21no—
198C4×C23C22×D411yescharacteristic
199C4×C23H(5,𝔽2)11yescharacteristic
200Dic23—21no—
201Dic23—21no—
202Dic23—21no—
203Dic23—41no—
204Dic23—41no—
205Dic23—41no—
206C2×AGL(1,ℤ/8)D2311yescharacteristic, complemented
207C2×#135—231no—
208#550—231no—
209#546—231no—
210#496—231no—
211#468—231no—
212#466—231no—
213#464—231no—
214#459—231no—
215#452—231no—
216#449—231no—
217#449—231no—
218#447—231no—
219#440—231no—
220#435—231no—
221C2×C23—41no—
222C2×C23—42no—
223C2×C23—81no—
224C2×C23—81no—
225C2×C23D4211yescharacteristic
226C2×C23#57511yescharacteristic
227C2×C23#57611yescharacteristic
228D23—41no—
229D23—81no—
230C2×#41—231no—
231C2×#41—231no—
232C2×#41C2×D2311yescharacteristic
233C22×D4C2×D2311yescharacteristic
234AGL(1,ℤ/8)—24no—
235#135—464no—
236C2×Q16—231no—
237C2×SD16—231no—
238C2×SD16—231no—
239C2×SD16C2×D2311yescharacteristic
240C2×SD16C2×D2311yescharacteristic
241C2×D8—231no—
242C2×D8—231no—
243C2×D8C2×D2311yescharacteristic
244C2×D8C2×D2311yescharacteristic
245C2×M16C2×D2311yescharacteristic
246C22×C8—231no—
247#127—231no—
248#127—461no—
249#124—231no—
250#123—461no—
251C22⋊Q8—231no—
252#121—231no—
253#121—231no—
254#121—231no—
255#121—461no—
256#121—461no—
257#121—461no—
258#120—461no—
259C4×Q8—231no—
260C4×D4—231no—
261C4×D4—231no—
262C4×D4—461no—
263C2×C22⋊C4—461no—
264#107—231no—
265C4⋊C8—231no—
266#103—231no—
267#103—231no—
268#102—231no—
269#102—231no—
270#102—231no—
271#102—231no—
272C22⋊C8—231no—
273C22⋊C8—231no—
274#97—231no—
275C23#326811yescharacteristic, complemented, Sylow
276C24—21no—
277#41—22no—
278#41—41no—
279#41—462no—
280#41—462no—
281#41—921no—
282#41—921no—
283C2×Q8—231no—
284C2×Q8C22×D2311yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—231no—
290C2×D4—231no—
291C2×D4—461no—
292C2×D4—461no—
293C2×D4—461no—
294C2×D4—461no—
295C2×D4—461no—
296C2×D4—461no—
297C2×D4—461no—
298C2×D4C22×D2311yescharacteristic
299C2×D4C22×D2311yescharacteristic
300C2×D4C22×D2311yescharacteristic
301C22×C4—21no—
302C22×C4—231no—
303C22×C4—461no—
304C22×C4—461no—
305C22×C4—461no—
306C22×C4C22×D2311yescharacteristic
307Q16—462no—
308SD16—22no—
309SD16—462no—
310SD16—921no—
311SD16C22⋊D2314yescomplemented
312D8—22no—
313D8—22no—
314D8—462no—
315D8—921no—
316M16—22no—
317C2×C8—231no—
318C2×C8—231no—
319C2×C8—462no—
320C2×C8C22×D2311yescharacteristic
321C2×C8C22×D2311yescharacteristic
322C4⋊C4—231no—
323C4⋊C4—231no—
324C4⋊C4—231no—
325C4⋊C4—461no—
326C4⋊C4—461no—
327C22⋊C4—231no—
328C22⋊C4—231no—
329C22⋊C4—461no—
330C22⋊C4—461no—
331C22⋊C4—461no—
332C22⋊C4—461no—
333C22⋊C4—461no—
334C22⋊C4—461no—
335C22⋊C4—461no—
336C42—231no—
337C42—461no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—231no—
345C23—461no—
346C23D4×D2311yescharacteristic
347Q8—21no—
348Q8—461no—
349Q8—461no—
350Q8C2×C22⋊D2312yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—42no—
358D4—461no—
359D4—461no—
360D4—922no—
361D4—921no—
362D4—922no—
363D4—922no—
364D4—1841no—
365D4—1841no—
366D4—1841no—
367D4—1841no—
368D4C2×C22⋊D2312yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—231no—
374C2×C4—231no—
375C2×C4—231no—
376C2×C4—461no—
377C2×C4—461no—
378C2×C4—461no—
379C2×C4—461no—
380C2×C4—461no—
381C2×C4—922no—
382C2×C4—922no—
383C2×C4—1841no—
384C2×C4D4×D2311yescharacteristic
385C2×C4C23×D2311yescharacteristic
386C8—21no—
387C8—461no—
388C8—461no—
389C8C2×C22⋊D2312yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—461no—
404C22—461no—
405C22—461no—
406C22—921no—
407C22—921no—
408C22—921no—
409C22C2×D4×D2311yescharacteristic
410C4—21no—
411C4—22no—
412C4—461no—
413C4—461no—
414C4—461no—
415C4—921no—
416C4—921no—
417C4—921no—
418C4#1062515511yescharacteristic
419C4C22×C22⋊D2311yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—81no—
424C2—921no—
425C2—1841no—
426C2D4×C22⋊D2311yescharacteristic
427C2#1309321311yescharacteristic
428C2#1309322111yescharacteristic
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#3268
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×#10625087C2semidirect
2. #13093086C2semidirect
3. #13093061C2semidirect
4. #13092805C2semidirect
5. #13092663C2semidirect
6. #13092650C2semidirect
7. #13092637C2semidirect
8. #13092610C2semidirect
9. #13092579C2semidirect
10. #13092561C2semidirect
11. #13092559C2semidirect
12. #13092551C2semidirect
13. #13092518C2semidirect
14. #13092486C2semidirect
15. C2×C23×AGL(1,ℤ/8)C2semidirect
18. C2×#67292C22semidirect
28. Q8×Dic23C22semidirect
29. C2×#67275C22semidirect
31. #10625111C22semidirect
35. #10625109C22semidirect
39. C2×#67273C22semidirect
48. C2×#67263C22semidirect
49. C2×C8×D23C22semidirect
50. #10625017C22semidirect
53. #10625014C22semidirect
55. #10625003C22semidirect
56. (C2×D23)⋊C8C22semidirect
57. #10625000C22semidirect
58. #10624998C22semidirect
59. #10624997C22semidirect
60. Dic23⋊C8C22semidirect
93. SD16×C23D4semidirect
206. C2×AGL(1,ℤ/8)D23semidirect
275. C23#3268semidirect
311. SD16C22⋊D23semidirect

Maximal subgroups

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

subgroupindex
C2×#106250872
#130930862
#130930612
#130928052
#130926632
#130926502
#130926372
#130926102
#130925792
#130925612
#130925592
#130925512
#130925182
#130924862
C2×C23×AGL(1,ℤ/8)2
#326823