Finite Group Databasebeta

← #46820193

Subgroups — #46820193

Order 34432.

764 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575499C211yesmaximal, characteristic, complemented
2C2×#22575480C211yesmaximal, characteristic, complemented
3C2×#22575476C211yesmaximal, characteristic, complemented
4#31771185C211yesmaximal, characteristic, complemented
5#31770766C211yesmaximal, characteristic, complemented
6#31770755C211yesmaximal, characteristic, complemented
7#31770754C211yesmaximal, characteristic, complemented
8#31770753C211yesmaximal, characteristic, complemented
9#31770750C211yesmaximal, characteristic, complemented
10#31770741C211yesmaximal, characteristic, complemented
11#31770740C211yesmaximal, characteristic, complemented
12#31770605C211yesmaximal, characteristic, complemented
13#102×D269C211yesmaximal, characteristic, complemented
14#31770570C211yesmaximal, characteristic, complemented
15#452×C269C211yesmaximal, characteristic, complemented
16#22575499—24no—
17#22575499—24no—
18C2×#16864403C2211yescharacteristic
19C2×#16864403C2211yescharacteristic
20C2×#16864403C2211yescharacteristic, complemented
21C2×#16864402—21no—
22C2×#16864402C2211yescharacteristic
23C22×D1076—21no—
24C22×D1076C2211yescharacteristic
25C2×D4×D269—21no—
26C2×D4×D269—21no—
27C2×D4×D269C2211yescharacteristic
28C2×D4×D269C2211yescharacteristic
29#22575480—24no—
30#22575476—24no—
31C2×#16864389C2211yescharacteristic, complemented
32C2×#16864389C2211yescharacteristic, complemented
33C2×#16864389C2211yescharacteristic, complemented
34C2×D2152C2211yescharacteristic, complemented
35C2×D2152C2211yescharacteristic, complemented
36C2×#16864387C2211yescharacteristic, complemented
37C2×#16864385C2211yescharacteristic, complemented
38C2×C8×D269C2211yescharacteristic, complemented
39#22575431—21no—
40#22575429C2211yescharacteristic, complemented
41#22575428—21no—
42#22575428C2211yescharacteristic, complemented
43#22575427C2211yescharacteristic, complemented
44#22575426—21no—
45#22575426—21no—
46C4⋊C4×D269C2211yescharacteristic, complemented
47#22575377C2211yescharacteristic, complemented
48#22575375C2211yescharacteristic, complemented
49#22575375C2211yescharacteristic, complemented
50#22575374C2211yescharacteristic, complemented
51#22575373C2211yescharacteristic, complemented
52#22575373C2211yescharacteristic, complemented
53C22⋊#14524130C2211yescharacteristic, complemented
54#22575365C2211yescharacteristic, complemented
55#22575365C2211yescharacteristic, complemented
56C2×#41×C269C2211yescharacteristic
57C2×Q16×C269C2211yescharacteristic, complemented
58C2×SD16×C269C2211yescharacteristic, complemented
59C22⋊Q8×C269C2211yescharacteristic, complemented
60#103×C269C2211yescharacteristic, complemented
61#102×C269C2211yescharacteristic, complemented
62C22⋊C8×C269C2211yescharacteristic, complemented
63C23×D269—21no—
64C23×D269—21no—
65C23×D269—21no—
66#16864403—22no—
67#16864403—22no—
68#16864403—22no—
69#16864403—22no—
70#16864403—22no—
71#16864403D414yescomplemented
72#16864402—28no—
73#16864402—22no—
74#16864402—22no—
75C2×Dic538D411yescharacteristic
76C2×D1076—28no—
77C2×D1076—21no—
78C2×D1076—21no—
79C2×D1076—22no—
80C2×D1076—22no—
81C2×D1076—21no—
82C2×D1076—21no—
83C2×D1076—21no—
84C2×D1076D411yescharacteristic
85C2×D1076C2311yescharacteristic
86C2×D1076C2311yescharacteristic
87C2×D1076C2311yescharacteristic
88C2×C22⋊D269—21no—
89C2×C22⋊D269—21no—
90C2×C22⋊D269—21no—
91C2×C22⋊D269D411yescharacteristic
92C2×C22⋊D269D411yescharacteristic
93D4×D269—28no—
94D4×D269—28no—
95D4×D269—22no—
96D4×D269—22no—
97D4×D269—22no—
98D4×D269D414yescomplemented
99C2×C4×D269—21no—
100C2×C4×D269—21no—
101C2×C4×D269—21no—
102C2×C4×D269—21no—
103C2×C4×D269C2311yescharacteristic
104C2×C4×D269C2311yescharacteristic
105#16864389—22no—
106#16864389—22no—
107#16864389—22no—
108D2152—22no—
109D2152—22no—
110#16864387—22no—
111#16864385—22no—
112C2×#14524130C2311yescharacteristic
113C2×#14524130C2311yescharacteristic
114C8×D269—22no—
115C4⋊Dic269C2311yescharacteristic
116#16864381—21no—
117#16864381—21no—
118#16864379—21no—
119#16864379—21no—
120#16864379—21no—
121#41×C269—22no—
122#41×C269—22no—
123C2×Q8×C269C2311yescharacteristic
124C2×Q8×C269C2311yescharacteristic
125C2×D4×C269—21no—
126C2×D4×C269C2311yescharacteristic
127C22×C4×C269—21no—
128C22×C4×C269C2311yescharacteristic
129Q16×C269—22no—
130SD16×C269—22no—
131C2×C8×C269C2311yescharacteristic
132C2×C8×C269C2311yescharacteristic
133C4⋊C4×C269—21no—
134C4⋊C4×C269C2311yescharacteristic
135C22⋊C4×C269—21no—
136C22×D269—21no—
137C22×D269—21no—
138C22×D269—24no—
139C22×D269—24no—
140C22×D269—24no—
141C22×D269—21no—
142C22×D269—81no—
143C22×D269—81no—
144C22×D269—81no—
145C22×D269—81no—
146C22×D269—81no—
147C22×D269—81no—
148C22×D269C2×D411yescharacteristic
149C22×D269C2×D411yescharacteristic
150Dic538—22no—
151D1076—24no—
152D1076—24no—
153D1076—24no—
154D1076—22no—
155D1076—21no—
156D1076—21no—
157D1076—21no—
158D1076—21no—
159D1076—42no—
160D1076—81no—
161D1076C2×D412yes—
162D1076C2×D412yes—
163C22⋊D269—24no—
164C22⋊D269—22no—
165C22⋊D269—22no—
166C22⋊D269—24no—
167C22⋊D269—81no—
168C2×Dic269—21no—
169C2×Dic269C2×D411yescharacteristic
170C2×Dic269C2×D411yescharacteristic
171C4×D269—24no—
172C4×D269—21no—
173C4×D269—21no—
174C4×D269—24no—
175C4×D269—42no—
176C4×D269—81no—
177C4×D269C2×D412yes—
178C4×D269C2×D412yes—
179#14524130—21no—
180#14524130—21no—
181C23×C269—21no—
182C23×C269C2×D411yescharacteristic
183Q8×C269—21no—
184Q8×C269—21no—
185Q8×C269—21no—
186Q8×C269C2×D412yes—
187D4×C269—24no—
188D4×C269—21no—
189D4×C269C2×D412yes—
190C2×C4×C269—24no—
191C2×C4×C269—21no—
192C2×C4×C269—21no—
193C2×C4×C269—21no—
194C2×C4×C269—21no—
195C2×C4×C269—21no—
196C2×C4×C269C2×D411yescharacteristic
197C2×C4×C269C2411yescharacteristic
198C8×C269—21no—
199C8×C269—21no—
200C22×C269—21no—
201C22×C269—22no—
202C22×C269—21no—
203C22×C269—21no—
204C22×C269—41no—
205C22×C269—41no—
206C22×C269C22×D411yescharacteristic
207C2×D269—21no—
208C2×D269—21no—
209C2×D269—21no—
210C2×D269—22no—
211C2×D269—22no—
212C2×D269—21no—
213C2×D269—21no—
214C2×D269—41no—
215C2×D269—41no—
216C2×D269—41no—
217C2×D269—41no—
218C2×D269—41no—
219C2×D269—41no—
220C2×D269—41no—
221C2×D269—84no—
222C2×D269—84no—
223C2×D269—84no—
224C2×D269#12012yes—
225C4×C269—21no—
226C4×C269—22no—
227C4×C269—41no—
228C4×C269—41no—
229C4×C269C22×D411yescharacteristic
230C4×C269C22×D411yescharacteristic
231Dic269—21no—
232Dic269—41no—
233Dic269#12012yes—
234C2×C269—41no—
235C2×C269—42no—
236C2×C269C2×#12011yescharacteristic
237C2×C269#57611yescharacteristic
238C2×C269#57711yescharacteristic
239D269—22no—
240D269—41no—
241D269—42no—
242D269—42no—
243D269—81no—
244C269#301211yescharacteristic, complemented, Sylow
245#3012—2691nomaximal, Sylow
246C2×H(5,𝔽2)—2691no—
247C2×AGL(1,ℤ/8)—2691no—
248C2×#135—2691no—
249#537—2691no—
250#452—2691no—
251#452D26911yescharacteristic, complemented
252#450—2691no—
253#450—2691no—
254#449—2691no—
255#449—2691no—
256#447—2691no—
257#447—2691no—
258#417—2691no—
259C2×#102—2691no—
260#408—2691no—
261H(5,𝔽2)—5384no—
262H(5,𝔽2)—5384no—
263C2×#41—2691no—
264C2×#41—2691no—
265C2×#41—2691no—
266C2×#41—2691no—
267C2×#41—5381no—
268C2×#41C2×D26911yescharacteristic
269C22×D4—2691no—
270C22×D4—2691no—
271C22×D4—2691no—
272C22×D4—5381no—
273C22×D4—5381no—
274C22×D4—5381no—
275AGL(1,ℤ/8)—5384no—
276#135—5384no—
277C2×Q16C2×D26911yescharacteristic, complemented
278C2×SD16—2691no—
279C2×SD16—2691no—
280C2×SD16—2691no—
281C2×SD16C2×D26911yescharacteristic, complemented
282C2×D8—2691no—
283C2×D8—2691no—
284C2×D8—2691no—
285C2×M16—2691no—
286C22×C8—2691no—
287C22⋊Q8—2691no—
288C22⋊Q8—5381no—
289C22⋊Q8C2×D26911yescharacteristic, complemented
290#121—2691no—
291#121—2691no—
292#121—5381no—
293#121—5381no—
294#121—5381no—
295C2×C4⋊C4—2691no—
296#103—2691no—
297#103C2×D26911yescharacteristic, complemented
298#102—2691no—
299#102—2691no—
300#102—2691no—
301#102—2691no—
302#102—2691no—
303#102C2×D26911yescharacteristic, complemented
304C22⋊C8—2691no—
305C22⋊C8—2691no—
306C22⋊C8—2691no—
307C22⋊C8C2×D26911yescharacteristic, complemented
308C24—5381no—
309C24—5381no—
310C24—5381no—
311#41—22no—
312#41—22no—
313#41—2694no—
314#41—5382no—
315#41—5382no—
316#41—5388no—
317#41—5382no—
318#41—5382no—
319#41—5382no—
320#41—5382no—
321#41—5382no—
322C2×Q8—2691no—
323C2×Q8C22×D26911yescharacteristic
324C2×Q8C22×D26911yescharacteristic
325C2×D4—21no—
326C2×D4—2694no—
327C2×D4—2691no—
328C2×D4—2691no—
329C2×D4—2691no—
330C2×D4—2691no—
331C2×D4—2691no—
332C2×D4—2691no—
333C2×D4—5382no—
334C2×D4—5382no—
335C2×D4—5381no—
336C2×D4—5381no—
337C2×D4—5388no—
338C2×D4—5381no—
339C2×D4—5381no—
340C2×D4—5382no—
341C2×D4—5381no—
342C2×D4—5381no—
343C2×D4—5382no—
344C2×D4—5388no—
345C2×D4—5388no—
346C2×D4—5382no—
347C2×D4—5381no—
348C2×D4—5381no—
349C2×D4C22×D26911yescharacteristic
350C22×C4—21no—
351C22×C4—2691no—
352C22×C4—2691no—
353C22×C4—5381no—
354C22×C4—5381no—
355C22×C4—5381no—
356C22×C4—5381no—
357C22×C4C22×D26911yescharacteristic
358Q16—22no—
359SD16—22no—
360SD16—5382no—
361SD16—5382no—
362SD16—5382no—
363D8—5382no—
364D8—5382no—
365D8—5382no—
366M16—5382no—
367C2×C8—2691no—
368C2×C8—2691no—
369C2×C8—5382no—
370C2×C8C22×D26911yescharacteristic
371C2×C8C22×D26911yescharacteristic
372C4⋊C4—21no—
373C4⋊C4—2691no—
374C4⋊C4—5381no—
375C4⋊C4—5381no—
376C4⋊C4C22×D26911yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—5381no—
379C22⋊C4—5381no—
380C22⋊C4—5381no—
381C23—21no—
382C23—2691no—
383C23—2691no—
384C23—5384no—
385C23—5384no—
386C23—5384no—
387C23—5381no—
388C23—5381no—
389C23—5381no—
390C23—21521no—
391C23—21521no—
392C23—21521no—
393C23—21521no—
394C23—21521no—
395C23—21521no—
396C23D4×D26911yescharacteristic
397Q8—21no—
398Q8—21no—
399Q8—21no—
400Q8—5382no—
401Q8D4×D26912yes—
402D4—24no—
403D4—21no—
404D4—2692no—
405D4—2692no—
406D4—5382no—
407D4—5384no—
408D4—5384no—
409D4—5384no—
410D4—5382no—
411D4—5384no—
412D4—5382no—
413D4—5384no—
414D4—5381no—
415D4—5381no—
416D4—5381no—
417D4—5381no—
418D4—10762no—
419D4—21521no—
420D4—21521no—
421D4D4×D26912yes—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—2691no—
429C2×C4—2692no—
430C2×C4—2692no—
431C2×C4—2691no—
432C2×C4—5384no—
433C2×C4—5384no—
434C2×C4—5381no—
435C2×C4—5381no—
436C2×C4—5381no—
437C2×C4—10762no—
438C2×C4—21521no—
439C2×C4D4×D26911yescharacteristic
440C2×C4C23×D26911yescharacteristic
441C8—21no—
442C8—21no—
443C8—5381no—
444C8—5381no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—2692no—
452C22—5381no—
453C22—5381no—
454C22—5381no—
455C22—5381no—
456C22—5382no—
457C22—5382no—
458C22—5381no—
459C22—10761no—
460C22—10761no—
461C22—10761no—
462C22—10761no—
463C22—10761no—
464C22—10761no—
465C22—10761no—
466C22—21524no—
467C22—21524no—
468C22—21524no—
469C22C2×D4×D26911yescharacteristic
470C4—21no—
471C4—22no—
472C4—41no—
473C4—41no—
474C4—2692no—
475C4—5381no—
476C4—10761no—
477C4C2×D4×D26911yescharacteristic
478C4C2×D4×D26911yescharacteristic
479C2—41no—
480C2—42no—
481C2—5382no—
482C2—10761no—
483C2—10762no—
484C2—10762no—
485C2—21521no—
486C2#120×D26911yescharacteristic
487C2#3177140711yescharacteristic
488C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#452×C269C2
socleC22×C269C22×D4
hypercenter#452D269
hypocenterC269#3012
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#22575499C2semidirect
2. C2×#22575480C2semidirect
3. C2×#22575476C2semidirect
4. #31771185C2semidirect
5. #31770766C2semidirect
6. #31770755C2semidirect
7. #31770754C2semidirect
8. #31770753C2semidirect
9. #31770750C2semidirect
10. #31770741C2semidirect
11. #31770740C2semidirect
12. #31770605C2semidirect
13. #102×D269C2semidirect
14. #31770570C2semidirect
15. #452×C269C2semidirect
20. C2×#16864403C22semidirect
31. C2×#16864389C22semidirect
32. C2×#16864389C22semidirect
33. C2×#16864389C22semidirect
34. C2×D2152C22semidirect
35. C2×D2152C22semidirect
36. C2×#16864387C22semidirect
37. C2×#16864385C22semidirect
38. C2×C8×D269C22semidirect
40. #22575429C22semidirect
42. #22575428C22semidirect
43. #22575427C22semidirect
46. C4⋊C4×D269C22semidirect
47. #22575377C22semidirect
48. #22575375C22semidirect
49. #22575375C22semidirect
50. #22575374C22semidirect
51. #22575373C22semidirect
52. #22575373C22semidirect
53. C22⋊#14524130C22semidirect
54. #22575365C22semidirect
55. #22575365C22semidirect
57. C2×Q16×C269C22semidirect
58. C2×SD16×C269C22semidirect
59. C22⋊Q8×C269C22semidirect
60. #103×C269C22semidirect
61. #102×C269C22semidirect
62. C22⋊C8×C269C22semidirect
71. #16864403D4semidirect
98. D4×D269D4semidirect
244. C269#3012semidirect
251. #452D269semidirect
277. C2×Q16C2×D269semidirect
281. C2×SD16C2×D269semidirect
289. C22⋊Q8C2×D269semidirect
297. #103C2×D269semidirect
303. #102C2×D269semidirect
307. C22⋊C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754992
C2×#225754802
C2×#225754762
#317711852
#317707662
#317707552
#317707542
#317707532
#317707502
#317707412
#317707402
#317706052
#102×D2692
#317705702
#452×C2692
#3012269