Finite Group Databasebeta

← #19143964

Subgroups — #19143964

Order 5504.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#14583735C211yesmaximal, characteristic, complemented
2C2×#13063351C211yesmaximal, characteristic, complemented
3#14583466C211yesmaximal, characteristic, complemented
4#14583070C214yesmaximal, complemented
5#14583065C214yesmaximal, complemented
6#14582916C211yesmaximal, characteristic, complemented
7#14582915C211yesmaximal, characteristic, complemented
8#14582881C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C43C211yesmaximal, characteristic, complemented
10#13063373—21no—
11#13063372—21no—
12#13063372C2212yescomplemented
13#13063371—21no—
14#13063371C2212yescomplemented
15#13063370—21no—
16#13063370—22no—
17C2×#10621898—22no—
18C2×#10621898C2211yescharacteristic
19C2×#10621898C2211yescharacteristic
20#41×D43—22no—
21#41×D43—22no—
22#13063351—22no—
23#13063351C2214yescomplemented
24C2×#10621883—21no—
25C2×#10621882—21no—
26C2×#10621877C2211yescharacteristic, complemented
27#13063321C2212yescomplemented
28#13063312C2212yescomplemented
29#13063296—22no—
30#13063294—22no—
31#13063284C2211yescharacteristic, complemented
32#13063252C2214yescomplemented
33#13063251C2214yescomplemented
34#13063243C2212yescomplemented
35#13063241C2212yescomplemented
36C2×#41×C43C2211yescharacteristic
37C22×D4×C43C2211yescharacteristic, complemented
38AGL(1,ℤ/8)×C43—22no—
39AGL(1,ℤ/8)×C43C2214yescomplemented
40C2×SD16×C43—21no—
41C2×D8×C43—21no—
42C2×M16×C43C2211yescharacteristic, complemented
43C24×C43—21no—
44#41×C43—21no—
45#41×C43—21no—
46#41×C43D412yescomplemented
47#41×C43C2312yes—
48C2×Q8×C43D411yescharacteristic, complemented
49C2×D4×C43—21no—
50C2×D4×C43—21no—
51C2×D4×C43—21no—
52C2×D4×C43—21no—
53C2×D4×C43—21no—
54C2×D4×C43—21no—
55C2×D4×C43D411yescharacteristic, complemented
56C2×D4×C43C2312yes—
57C22×C4×C43—21no—
58C22×C4×C43C2311yescharacteristic
59C2×C22⋊D43—24no—
60C2×C22⋊D43—21no—
61C2×C22⋊D43—24no—
62C2×C22⋊D43D412yes—
63#10621902—22no—
64#10621902—22no—
65Q8×D43—22no—
66Q8×D43—22no—
67#10621900—24no—
68#10621900—22no—
69#10621900—22no—
70#10621900—24no—
71D4×D43—22no—
72D4×D43—24no—
73D4×D43—22no—
74D4×D43—24no—
75#10621898—24no—
76#10621898—24no—
77#10621898—21no—
78#10621898—21no—
79#10621898—21no—
80#10621898—21no—
81#10621898—21no—
82#10621898—21no—
83#10621898—22no—
84#10621898—22no—
85#10621898D412yescomplemented
86#10621898C2312yes—
87C2×D172—22no—
88C2×D172D411yescharacteristic, complemented
89C2×D172C2311yescharacteristic
90C2×C4×D43—24no—
91C2×C4×D43—21no—
92C2×C4×D43D412yes—
93C2×Dic86—22no—
94C2×Dic86D411yescharacteristic, complemented
95C2×Dic86C2311yescharacteristic
96SD16×C43—24no—
97D8×C43—24no—
98M16×C43—21no—
99M16×C43C2312yes—
100C2×C8×C43—21no—
101C22⋊Dic43—24no—
102C22⋊Dic43—21no—
103#10621883—24no—
104#10621882—24no—
105C4⋊Dic43—21no—
106C4×Dic43C2312yes—
107#10621877—21no—
108#10621877C2312yes—
109C2×C43⋊C8—21no—
110C23×C43—21no—
111C23×C43—22no—
112C23×C43—41no—
113C23×C43—41no—
114C23×C43—82no—
115C23×C43C2×D411yescharacteristic
116C22×D43—24no—
117C22×D43—21no—
118C22×D43C2×D412yes—
119Q8×C43—21no—
120Q8×C43C2×D412yes—
121D4×C43—22no—
122D4×C43—22no—
123D4×C43—21no—
124D4×C43—22no—
125D4×C43—22no—
126D4×C43—41no—
127D4×C43—41no—
128D4×C43C2×D412yes—
129C2×C4×C43—21no—
130C2×C4×C43—22no—
131C2×C4×C43—22no—
132C2×C4×C43C2×D411yescharacteristic
133C2×C4×C43C2×D411yescharacteristic
134C2×C4×C43C2×D411yescharacteristic
135C2×C4×C43C2×D411yescharacteristic
136C2×C4×C43C2×D411yescharacteristic
137C2×C4×C43C2411yescharacteristic
138C22⋊D43—24no—
139C22⋊D43—24no—
140C22⋊D43—22no—
141C22⋊D43—22no—
142C22⋊D43—22no—
143C22⋊D43—22no—
144C22⋊D43—84no—
145C2×Dic43—24no—
146C2×Dic43—21no—
147C2×Dic43—22no—
148C2×Dic43C2×D412yes—
149D172—24no—
150D172—21no—
151D172—21no—
152D172—21no—
153D172C2×D412yes—
154C4×D43—24no—
155C4×D43—24no—
156C4×D43—22no—
157C4×D43—22no—
158C4×D43—22no—
159C4×D43—22no—
160Dic86—24no—
161Dic86—21no—
162Dic86—21no—
163Dic86—21no—
164Dic86C2×D412yes—
165C8×C43—22no—
166C43⋊C8—22no—
167C22×C43—22no—
168C22×C43—41no—
169C22×C43—44no—
170C22×C43—42no—
171C22×C43—81no—
172C22×C43—82no—
173C22×C43—82no—
174C22×C43#12011yescharacteristic
175C22×C43#12011yescharacteristic
176C22×C43C22×D411yescharacteristic
177C2×D43—22no—
178C2×D43—22no—
179C2×D43—42no—
180C2×D43—81no—
181C2×D43—84no—
182C4×C43—22no—
183C4×C43#12011yescharacteristic
184C4×C43#12011yescharacteristic
185C4×C43C22×D411yescharacteristic
186C4×C43C22×D411yescharacteristic
187Dic43—22no—
188Dic43—22no—
189Dic43—42no—
190#3021—431nomaximal, Sylow
191C2×C43—21no—
192C2×C43—21no—
193C2×C43—21no—
194C2×C43—42no—
195C2×C43—82no—
196C2×C43C2×#12011yescharacteristic
197D43—42no—
198D43—42no—
199#585—431no—
200C2×AGL(1,ℤ/8)—431no—
201C2×AGL(1,ℤ/8)D4311yescharacteristic, complemented
202#535—431no—
203#454—434no—
204#453—434no—
205#421—431no—
206#421—431no—
207#413—431no—
208C43#302111yescharacteristic, complemented, Sylow
2092−1+4—432no—
2102−1+4—861no—
2112−1+4—861no—
212H(5,𝔽2)—432no—
213H(5,𝔽2)—862no—
214H(5,𝔽2)—861no—
215H(5,𝔽2)—861no—
216C2×#41—431no—
217C2×#41—431no—
218C2×#41—862no—
219C2×#41—862no—
220C2×#41—862no—
221C2×#41C2×D4311yescharacteristic
222C22×D4C2×D4311yescharacteristic, complemented
223AGL(1,ℤ/8)—22no—
224AGL(1,ℤ/8)—434no—
225AGL(1,ℤ/8)—862no—
226AGL(1,ℤ/8)C2×D4314yescomplemented
227C2×SD16—21no—
228C2×SD16—861no—
229C2×D8—21no—
230C2×D8—861no—
231C2×M16—431no—
232C2×M16C2×D4311yescharacteristic, complemented
233#127—432no—
234#124—432no—
235#121—862no—
236#120—862no—
237#117—431no—
238#104—434no—
239#104—434no—
240#101—432no—
241#100—432no—
242C24—21no—
243#41—21no—
244#41—21no—
245#41—432no—
246#41—432no—
247#41—862no—
248#41—862no—
249#41—864no—
250#41—864no—
251#41—864no—
252#41—864no—
253#41—862no—
254#41—862no—
255#41—862no—
256#41—862no—
257#41—861no—
258#41—861no—
259#41—861no—
260#41—861no—
261#41—861no—
262#41—861no—
263#41C22⋊D4312yescomplemented
264#41C22×D4312yes—
265C2×Q8—431no—
266C2×Q8—431no—
267C2×Q8—862no—
268C2×Q8—862no—
269C2×Q8—862no—
270C2×Q8C22⋊D4311yescharacteristic, complemented
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—432no—
278C2×D4—431no—
279C2×D4—431no—
280C2×D4—862no—
281C2×D4—864no—
282C2×D4—864no—
283C2×D4—862no—
284C2×D4—864no—
285C2×D4—861no—
286C2×D4—862no—
287C2×D4—864no—
288C2×D4C22⋊D4311yescharacteristic, complemented
289C2×D4C22×D4312yes—
290C22×C4—21no—
291C22×C4—432no—
292C22×C4—864no—
293C22×C4—861no—
294C22×C4C22×D4311yescharacteristic
295SD16—24no—
296SD16—864no—
297D8—24no—
298D8—864no—
299M16—21no—
300M16—432no—
301M16—861no—
302M16C22×D4312yes—
303C2×C8—21no—
304C2×C8—861no—
305C4⋊C4—861no—
306C22⋊C4—864no—
307C22⋊C4—861no—
308C42—432no—
309C23—21no—
310C23—22no—
311C23—41no—
312C23—41no—
313C23—82no—
314C23—432no—
315C23—864no—
316C23—861no—
317C23D4×D4311yescharacteristic
318Q8—21no—
319Q8—432no—
320Q8—864no—
321Q8—861no—
322Q8—861no—
323Q8—861no—
324Q8C2×C22⋊D4312yes—
325D4—22no—
326D4—22no—
327D4—21no—
328D4—22no—
329D4—22no—
330D4—41no—
331D4—41no—
332D4—432no—
333D4—864no—
334D4—864no—
335D4—864no—
336D4—862no—
337D4—862no—
338D4—862no—
339D4—862no—
340D4—861no—
341D4—861no—
342D4—861no—
343D4—3444no—
344D4C2×C22⋊D4312yes—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—432no—
349C2×C4—864no—
350C2×C4—864no—
351C2×C4—864no—
352C2×C4—861no—
353C2×C4—862no—
354C2×C4—862no—
355C2×C4—862no—
356C2×C4—862no—
357C2×C4—862no—
358C2×C4D4×D4311yescharacteristic
359C2×C4D4×D4311yescharacteristic
360C2×C4D4×D4311yescharacteristic
361C2×C4C2×C22⋊D4311yescharacteristic
362C2×C4C2×C22⋊D4311yescharacteristic
363C2×C4C23×D4311yescharacteristic
364C8—22no—
365C8—862no—
366C22—22no—
367C22—41no—
368C22—44no—
369C22—42no—
370C22—81no—
371C22—82no—
372C22—82no—
373C22—862no—
374C22—862no—
375C22—3441no—
376C22—3444no—
377C22—1722no—
378C22#1306329311yescharacteristic
379C22#1306329311yescharacteristic
380C22C2×D4×D4311yescharacteristic
381C4—22no—
382C4—862no—
383C4—862no—
384C4—1722no—
385C4#1306329311yescharacteristic
386C4#1306329311yescharacteristic
387C4C2×D4×D4311yescharacteristic
388C4C22×C22⋊D4311yescharacteristic
389C2—21no—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—82no—
394C2—1722no—
395C2—1722no—
396C2C2×#1306329311yescharacteristic
subgroupG/H
centerC2C2×#13063293
commutatorC2×C4×C43C24
FrattiniC2×C4C23×D43
FittingC2×AGL(1,ℤ/8)×C43C2
socleC2×C43C2×#120
hypercenterC2×AGL(1,ℤ/8)D43
hypocenterC43#3021
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #14583735C2semidirect
2. C2×#13063351C2semidirect
3. #14583466C2semidirect
4. #14583070C2semidirect
5. #14583065C2semidirect
6. #14582916C2semidirect
7. #14582915C2semidirect
8. #14582881C2semidirect
9. C2×AGL(1,ℤ/8)×C43C2semidirect
12. #13063372C22semidirect
14. #13063371C22semidirect
23. #13063351C22semidirect
26. C2×#10621877C22semidirect
27. #13063321C22semidirect
28. #13063312C22semidirect
31. #13063284C22semidirect
32. #13063252C22semidirect
33. #13063251C22semidirect
34. #13063243C22semidirect
35. #13063241C22semidirect
37. C22×D4×C43C22semidirect
39. AGL(1,ℤ/8)×C43C22semidirect
42. C2×M16×C43C22semidirect
46. #41×C43D4semidirect
48. C2×Q8×C43D4semidirect
55. C2×D4×C43D4semidirect
85. #10621898D4semidirect
88. C2×D172D4semidirect
94. C2×Dic86D4semidirect
201. C2×AGL(1,ℤ/8)D43semidirect
208. C43#3021semidirect
222. C22×D4C2×D43semidirect
226. AGL(1,ℤ/8)C2×D43semidirect
232. C2×M16C2×D43semidirect
263. #41C22⋊D43semidirect
270. C2×Q8C22⋊D43semidirect
288. C2×D4C22⋊D43semidirect