Finite Group Databasebeta

← #49168508

Subgroups — #49168508

Order 37504.

466 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036928C211yesmaximal, characteristic, complemented
2C2×#22627651C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×D293C211yesmaximal, characteristic, complemented
4#34036894C211yesmaximal, characteristic, complemented
5#34036400C211yesmaximal, characteristic, complemented
6#34036399C211yesmaximal, characteristic, complemented
7#34036398C211yesmaximal, characteristic, complemented
8#34036397C211yesmaximal, characteristic, complemented
9#34036395C211yesmaximal, characteristic, complemented
10#34036394C211yesmaximal, characteristic, complemented
11#34036389C211yesmaximal, characteristic, complemented
12#34036176C211yesmaximal, characteristic, complemented
13#34036112C211yesmaximal, characteristic, complemented
14#100×D293C211yesmaximal, characteristic, complemented
15#471×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17#41×D293C2211yescharacteristic
18C2×D4×D293C2211yescharacteristic
19#22627656—21no—
20#22627655C2211yescharacteristic, complemented
21#22627652C2211yescharacteristic
22#22627652C2211yescharacteristic, complemented
23#22627651—21no—
24#22627651—21no—
25#22627651—21no—
26#22627651—21no—
27#22627651—21no—
28#22627651—21no—
29#22627650C2211yescharacteristic
30#22627649C2211yescharacteristic
31#22627649C2211yescharacteristic, complemented
32#22627648—21no—
33#22627647—21no—
34#22627645—21no—
35C2×#17989789C2211yescharacteristic, complemented
36SD16×D293—21no—
37C2×D2344C2211yescharacteristic, complemented
38C2×#17989787C2211yescharacteristic, complemented
39D8×D293—21no—
40#22627632—21no—
41#22627632C2211yescharacteristic, complemented
42#22627631C2211yescharacteristic, complemented
43#22627630—21no—
44#22627630C2211yescharacteristic, complemented
45C2×#17989785—21no—
46C2×#17989785C2211yescharacteristic
47M16×D293—21no—
48M16×D293C2211yescharacteristic
49M16×D293C2211yescharacteristic
50M16×D293C2211yescharacteristic
51#22627560C2211yescharacteristic, complemented
52#22627559C2211yescharacteristic, complemented
53#22627559C2211yescharacteristic, complemented
54#22627543C2211yescharacteristic, complemented
55#22627541C2211yescharacteristic, complemented
56#22627540C2211yescharacteristic, complemented
57#22627540C2211yescharacteristic, complemented
58#22627540C2211yescharacteristic, complemented
59#22627540C2211yescharacteristic, complemented
60#137×C293C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C293C2211yescharacteristic
62C2×SD16×C293C2211yescharacteristic, complemented
63#131×C293C2211yescharacteristic, complemented
64#108×C293C2211yescharacteristic, complemented
65#101×C293C2211yescharacteristic, complemented
66#100×C293C2211yescharacteristic, complemented
67C23×D293—21no—
68#17989803—21no—
69#17989803—21no—
70#17989803—21no—
71#17989803—21no—
72#17989803D411yescharacteristic
73#17989802—21no—
74#17989802C2311yescharacteristic
75#17989801—21no—
76#17989801D411yescharacteristic
77Q8×D293D411yescharacteristic
78C2×D1172—21no—
79C2×D1172C2311yescharacteristic
80C2×D1172C2311yescharacteristic
81C2×C22⋊D293—21no—
82D4×D293—21no—
83D4×D293—21no—
84D4×D293—21no—
85D4×D293—21no—
86D4×D293—21no—
87D4×D293D411yescharacteristic
88C2×C4×D293—21no—
89C2×C4×D293—21no—
90C2×C4×D293C2311yescharacteristic
91#17989792—21no—
92#17989791—21no—
93#17989791—21no—
94#17989790—21no—
95#17989789—21no—
96#17989789—21no—
97#17989789—21no—
98#17989789—21no—
99#17989789—21no—
100D2344—21no—
101D2344—21no—
102D2344—21no—
103D2344—21no—
104#17989787—21no—
105#17989787—21no—
106#17989787—21no—
107#17989787—21no—
108#17989787—21no—
109#17989786—21no—
110#17989786C2311yescharacteristic
111#17989786C2311yescharacteristic
112#17989786C2311yescharacteristic
113#17989785—21no—
114#17989785—21no—
115#17989785—21no—
116#17989785—21no—
117#17989785—21no—
118#17989785D411yescharacteristic
119#17989785D411yescharacteristic
120#17989785D411yescharacteristic
121#17989785D411yescharacteristic
122C2×#14556873—21no—
123C2×#14556873C2311yescharacteristic
124C8×D293—21no—
125C8×D293—21no—
126C8×D293—21no—
127C8×D293—21no—
128C8×D293—21no—
129#41×C293C2311yescharacteristic
130C2×Q8×C293C2311yescharacteristic
131C2×D4×C293C2311yescharacteristic
132Q16×C293—21no—
133SD16×C293—21no—
134SD16×C293—21no—
135SD16×C293—21no—
136SD16×C293—21no—
137D8×C293—21no—
138M16×C293—21no—
139M16×C293C2311yescharacteristic
140M16×C293C2311yescharacteristic
141M16×C293C2311yescharacteristic
142C2×C8×C293—21no—
143C2×C8×C293C2311yescharacteristic
144C22×D293—21no—
145C22×D293—21no—
146C22×D293—21no—
147C22×D293—41no—
148C22×D293—41no—
149C22×D293—82no—
150C22×D293#4111yescharacteristic
151Dic586—21no—
152Dic586C2×D411yescharacteristic
153D1172—21no—
154D1172—21no—
155D1172—21no—
156D1172—21no—
157D1172—21no—
158D1172—41no—
159D1172—41no—
160D1172C2×D411yescharacteristic
161C22⋊D293—21no—
162C22⋊D293—21no—
163C22⋊D293—21no—
164C22⋊D293—41no—
165C22⋊D293—41no—
166C2×Dic293—21no—
167C2×Dic293#4111yescharacteristic
168C4×D293—21no—
169C4×D293—21no—
170C4×D293—21no—
171C4×D293—41no—
172C4×D293—41no—
173C4×D293C2×D411yescharacteristic
174C4×D293C2×D411yescharacteristic
175C4×D293C2×D411yescharacteristic
176C4×D293C2×D411yescharacteristic
177#14556873—21no—
178#14556873—21no—
179#14556873—21no—
180#14556873C2×D411yescharacteristic
181#14556873C2×D411yescharacteristic
182C23×C293—21no—
183Q8×C293—21no—
184Q8×C293—21no—
185Q8×C293C2×D411yescharacteristic
186D4×C293—21no—
187D4×C293—21no—
188D4×C293—21no—
189D4×C293C2×D411yescharacteristic
190C2×C4×C293—21no—
191C2×C4×C293—21no—
192C2×C4×C293C2411yescharacteristic
193C8×C293—21no—
194C8×C293—21no—
195C8×C293—21no—
196C8×C293C2×D411yescharacteristic
197C8×C293C2×D411yescharacteristic
198C22×C293—21no—
199C22×C293—41no—
200C22×C293—81no—
201C22×C293C2×#4111yescharacteristic
202C2×D293—21no—
203C2×D293—41no—
204C2×D293—41no—
205C2×D293—41no—
206C2×D293—81no—
207C2×D293—82no—
208C2×D293—82no—
209C2×D293—81no—
210C2×D293—81no—
211C2×D293#12111yescharacteristic
212C2×D293#12111yescharacteristic
213C4×C293—21no—
214C4×C293—41no—
215C4×C293C22×D411yescharacteristic
216C4×C293C22×D411yescharacteristic
217Dic293—21no—
218Dic293#12111yescharacteristic
219Dic293#12111yescharacteristic
220C2×C293—21no—
221C2×C293—41no—
222C2×C293—81no—
223C2×C293C2×#12111yescharacteristic
224D293—21no—
225D293—21no—
226D293—41no—
227D293—81no—
228D293—81no—
229C293#306811yescharacteristic, complemented, Sylow
230#3068—2931nomaximal, Sylow
231#575—2931no—
232C2×AGL(1,ℤ/8)—2931no—
233C2×AGL(1,ℤ/8)—2931no—
234#568—2931no—
235#472—2931no—
236#472—2931no—
237#472—2931no—
238#472—2931no—
239#471—2931no—
240#471—2931no—
241#471—2931no—
242#471D29311yescharacteristic, complemented
243#430—2931no—
244#413—2931no—
245C2×#100—2931no—
246C2×#41—2931no—
247C2×#41—2931no—
248C22×D4—2931no—
249#137—2931no—
250#137—5861no—
251#137C2×D29311yescharacteristic, complemented
252AGL(1,ℤ/8)—2931no—
253AGL(1,ℤ/8)—2931no—
254AGL(1,ℤ/8)—2931no—
255AGL(1,ℤ/8)—2931no—
256AGL(1,ℤ/8)—2931no—
257AGL(1,ℤ/8)—5861no—
258AGL(1,ℤ/8)—5861no—
259AGL(1,ℤ/8)—5861no—
260AGL(1,ℤ/8)—5861no—
261AGL(1,ℤ/8)—5861no—
262AGL(1,ℤ/8)—5861no—
263AGL(1,ℤ/8)—5861no—
264AGL(1,ℤ/8)C2×D29311yescharacteristic
265#135—5861no—
266#135—5861no—
267C2×SD16—2931no—
268C2×SD16—5861no—
269C2×SD16C2×D29311yescharacteristic, complemented
270C2×D8—2931no—
271C2×D8—2931no—
272C2×D8—5861no—
273#131—2931no—
274#131—2931no—
275#131—2931no—
276#131—5861no—
277#131—5861no—
278#131C2×D29311yescharacteristic, complemented
279C2×M16—2931no—
280C2×M16—2931no—
281C2×M16—2931no—
282C2×M16—2931no—
283C2×M16—5861no—
284C2×M16—5861no—
285#108—2931no—
286#108—2931no—
287#108—2931no—
288#108C2×D29311yescharacteristic, complemented
289#101—2931no—
290#101C2×D29311yescharacteristic, complemented
291#100—2931no—
292#100—2931no—
293#100—2931no—
294#100—2931no—
295#100—2931no—
296#100C2×D29311yescharacteristic, complemented
297C24—5861no—
298#41—2931no—
299#41—2931no—
300#41—2931no—
301#41—5861no—
302#41—5861no—
303#41—5861no—
304#41—5861no—
305#41—5861no—
306#41—5861no—
307#41C22×D29311yescharacteristic
308C2×Q8—2931no—
309C2×Q8C22×D29311yescharacteristic
310C2×D4—2931no—
311C2×D4—2931no—
312C2×D4—2931no—
313C2×D4—5861no—
314C2×D4—5861no—
315C2×D4—5861no—
316C2×D4—5861no—
317C2×D4—5861no—
318C2×D4—5861no—
319C2×D4—5861no—
320C2×D4C22×D29311yescharacteristic
321C22×C4—2931no—
322C22×C4—5861no—
323C22×C4—5861no—
324Q16—21no—
325Q16—5861no—
326SD16—21no—
327SD16—21no—
328SD16—21no—
329SD16—21no—
330SD16—5861no—
331SD16—5861no—
332SD16—5861no—
333SD16—5861no—
334SD16—5861no—
335SD16—5861no—
336SD16—5861no—
337SD16—5861no—
338D8—21no—
339D8—5861no—
340D8—5861no—
341D8—5861no—
342D8—5861no—
343D8—5861no—
344D8—5861no—
345D8—5861no—
346D8—5861no—
347D8—5861no—
348M16—21no—
349M16—2931no—
350M16—2931no—
351M16—2931no—
352M16—2931no—
353M16—2931no—
354M16—2931no—
355M16—2931no—
356M16—5861no—
357M16—5861no—
358M16—5861no—
359M16—5861no—
360M16—5861no—
361M16—5861no—
362M16C22×D29311yescharacteristic
363M16C22×D29311yescharacteristic
364M16C22×D29311yescharacteristic
365C2×C8—21no—
366C2×C8—2931no—
367C2×C8—5861no—
368C2×C8—5861no—
369C2×C8—5861no—
370C2×C8—5861no—
371C2×C8—5861no—
372C2×C8—5861no—
373C2×C8C22×D29311yescharacteristic
374C23—21no—
375C23—2931no—
376C23—23442no—
377C23—5861no—
378C23—5861no—
379C23—5861no—
380C23—11721no—
381C23—11721no—
382Q8—21no—
383Q8—21no—
384Q8—2931no—
385Q8—5861no—
386Q8D4×D29311yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—2931no—
391D4—5861no—
392D4—5861no—
393D4—5861no—
394D4—5861no—
395D4—5861no—
396D4—5861no—
397D4—5861no—
398D4—5861no—
399D4—11721no—
400D4—11721no—
401D4—11721no—
402D4—11721no—
403D4D4×D29311yescharacteristic
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—2931no—
407C2×C4—2931no—
408C2×C4—2931no—
409C2×C4—2931no—
410C2×C4—2931no—
411C2×C4—5861no—
412C2×C4—5861no—
413C2×C4—5861no—
414C2×C4—5861no—
415C2×C4—11721no—
416C2×C4—11721no—
417C2×C4C23×D29311yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—2931no—
422C8—2931no—
423C8—5861no—
424C8—5861no—
425C8—5861no—
426C8D4×D29311yescharacteristic
427C8D4×D29311yescharacteristic
428C22—21no—
429C22—41no—
430C22—81no—
431C22—2931no—
432C22—2931no—
433C22—23441no—
434C22—23441no—
435C22—23441no—
436C22—23442no—
437C22—23442no—
438C22—5861no—
439C22—11721no—
440C22—11721no—
441C22—11721no—
442C22#41×D29311yescharacteristic
443C4—21no—
444C4—41no—
445C4—2931no—
446C4—2931no—
447C4—5861no—
448C4C2×D4×D29311yescharacteristic
449C4C2×D4×D29311yescharacteristic
450C2—21no—
451C2—41no—
452C2—81no—
453C2—23441no—
454C2—23441no—
455C2—5861no—
456C2—5861no—
457C2—11721no—
458C2#121×D29311yescharacteristic
subgroupG/H
centerC2#121×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#471×C293C2
socleC2×C293C2×#121
hypercenter#471D293
hypocenterC293#3068
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #34036928C2semidirect
2. C2×#22627651C2semidirect
3. AGL(1,ℤ/8)×D293C2semidirect
4. #34036894C2semidirect
5. #34036400C2semidirect
6. #34036399C2semidirect
7. #34036398C2semidirect
8. #34036397C2semidirect
9. #34036395C2semidirect
10. #34036394C2semidirect
11. #34036389C2semidirect
12. #34036176C2semidirect
13. #34036112C2semidirect
14. #100×D293C2semidirect
15. #471×C293C2semidirect
20. #22627655C22semidirect
22. #22627652C22semidirect
31. #22627649C22semidirect
35. C2×#17989789C22semidirect
37. C2×D2344C22semidirect
38. C2×#17989787C22semidirect
41. #22627632C22semidirect
42. #22627631C22semidirect
44. #22627630C22semidirect
51. #22627560C22semidirect
52. #22627559C22semidirect
53. #22627559C22semidirect
54. #22627543C22semidirect
55. #22627541C22semidirect
56. #22627540C22semidirect
57. #22627540C22semidirect
58. #22627540C22semidirect
59. #22627540C22semidirect
60. #137×C293C22semidirect
62. C2×SD16×C293C22semidirect
63. #131×C293C22semidirect
64. #108×C293C22semidirect
65. #101×C293C22semidirect
66. #100×C293C22semidirect
229. C293#3068semidirect
242. #471D293semidirect
251. #137C2×D293semidirect
269. C2×SD16C2×D293semidirect
278. #131C2×D293semidirect
288. #108C2×D293semidirect
290. #101C2×D293semidirect
296. #100C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340369282
C2×#226276512
AGL(1,ℤ/8)×D2932
#340368942
#340364002
#340363992
#340363982
#340363972
#340363952
#340363942
#340363892
#340361762
#340361122
#100×D2932
#471×C2932
#3068293