Finite Group Databasebeta

← #37543447

Subgroups — #37543447

Order 23168.

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#26042075C211yesmaximal, characteristic, complemented
2C2×#19153840C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×D181C211yesmaximal, characteristic, complemented
4#26042044C211yesmaximal, characteristic, complemented
5#26041548C211yesmaximal, characteristic, complemented
6#26041547C211yesmaximal, characteristic, complemented
7#26041546C211yesmaximal, characteristic, complemented
8#26041543C211yesmaximal, characteristic, complemented
9#26041542C211yesmaximal, characteristic, complemented
10#26041541C211yesmaximal, characteristic, complemented
11#26041540C211yesmaximal, characteristic, complemented
12#26041326C211yesmaximal, characteristic, complemented
13#26041260C211yesmaximal, characteristic, complemented
14#100×D181C211yesmaximal, characteristic, complemented
15#472×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17#41×D181C2211yescharacteristic
18C2×D4×D181C2211yescharacteristic
19#19153849C2211yescharacteristic, complemented
20#19153846C2211yescharacteristic, complemented
21#19153845—21no—
22#19153844C2211yescharacteristic
23#19153843—21no—
24#19153842C2211yescharacteristic
25#19153842C2211yescharacteristic, complemented
26#19153841C2211yescharacteristic
27#19153840—21no—
28#19153840—21no—
29#19153840—21no—
30#19153840—21no—
31#19153840—21no—
32#19153840—21no—
33#19153837—21no—
34#19153835—21no—
35C2×#15668438C2211yescharacteristic, complemented
36C2×#15668437C2211yescharacteristic, complemented
37SD16×D181—21no—
38C2×#15668434C2211yescharacteristic, complemented
39D8×D181—21no—
40#19153824—21no—
41#19153824C2211yescharacteristic, complemented
42#19153823C2211yescharacteristic, complemented
43#19153822—21no—
44#19153822C2211yescharacteristic, complemented
45C2×#15668432—21no—
46C2×#15668432C2211yescharacteristic
47M16×D181—21no—
48M16×D181C2211yescharacteristic
49M16×D181C2211yescharacteristic
50M16×D181C2211yescharacteristic
51#19153752C2211yescharacteristic, complemented
52#19153751C2211yescharacteristic, complemented
53#19153751C2211yescharacteristic, complemented
54#19153734C2211yescharacteristic, complemented
55#19153734C2211yescharacteristic, complemented
56#19153733C2211yescharacteristic, complemented
57#19153733C2211yescharacteristic, complemented
58#19153732C2211yescharacteristic, complemented
59#19153732C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C181C2211yescharacteristic
61AGL(1,ℤ/8)×C181C2211yescharacteristic, complemented
62C2×D8×C181C2211yescharacteristic, complemented
63#131×C181C2211yescharacteristic, complemented
64#108×C181C2211yescharacteristic, complemented
65#100×C181C2211yescharacteristic, complemented
66#100×C181C2211yescharacteristic, complemented
67C23×D181—21no—
68#15668450D411yescharacteristic
69#15668449—21no—
70#15668449C2311yescharacteristic
71#15668448—21no—
72#15668448—21no—
73#15668448—21no—
74#15668448—21no—
75#15668448—21no—
76#15668448D411yescharacteristic
77C2×Dic362C2311yescharacteristic
78Q8×D181D411yescharacteristic
79C2×D724C2311yescharacteristic
80C2×C22⋊D181—21no—
81C2×C22⋊D181—21no—
82D4×D181—21no—
83D4×D181—21no—
84D4×D181—21no—
85D4×D181—21no—
86D4×D181—21no—
87D4×D181D411yescharacteristic
88C22×Dic181—21no—
89C2×C4×D181—21no—
90C2×C4×D181C2311yescharacteristic
91Dic724—21no—
92#15668439—21no—
93#15668438—21no—
94#15668438—21no—
95#15668438—21no—
96#15668438—21no—
97#15668437—21no—
98#15668437—21no—
99#15668437—21no—
100#15668437—21no—
101#15668437—21no—
102#15668436—21no—
103D1448—21no—
104#15668434—21no—
105#15668434—21no—
106#15668434—21no—
107#15668434—21no—
108#15668434—21no—
109#15668433—21no—
110#15668433C2311yescharacteristic
111#15668433C2311yescharacteristic
112#15668433C2311yescharacteristic
113#15668432—21no—
114#15668432—21no—
115#15668432—21no—
116#15668432—21no—
117#15668432—21no—
118#15668432D411yescharacteristic
119#15668432D411yescharacteristic
120#15668432D411yescharacteristic
121#15668432D411yescharacteristic
122C2×#13089678—21no—
123C2×#13089678C2311yescharacteristic
124C8×D181—21no—
125C8×D181—21no—
126C8×D181—21no—
127C8×D181—21no—
128C8×D181—21no—
129#41×C181C2311yescharacteristic
130C2×D4×C181C2311yescharacteristic
131C2×D4×C181C2311yescharacteristic
132SD16×C181—21no—
133SD16×C181—21no—
134D8×C181—21no—
135D8×C181—21no—
136D8×C181—21no—
137D8×C181—21no—
138M16×C181—21no—
139M16×C181C2311yescharacteristic
140M16×C181C2311yescharacteristic
141M16×C181C2311yescharacteristic
142C2×C8×C181—21no—
143C2×C8×C181C2311yescharacteristic
144C22×D181—21no—
145C22×D181—21no—
146C22×D181—41no—
147C22×D181—41no—
148C22×D181—82no—
149C22×D181#4111yescharacteristic
150Dic362—21no—
151Dic362—21no—
152Dic362—21no—
153Dic362C2×D411yescharacteristic
154D724—21no—
155D724—21no—
156D724—21no—
157D724C2×D411yescharacteristic
158C22⋊D181—21no—
159C22⋊D181—21no—
160C22⋊D181—21no—
161C22⋊D181—41no—
162C22⋊D181—41no—
163C22⋊D181—41no—
164C22⋊D181—41no—
165C2×Dic181—21no—
166C2×Dic181—21no—
167C2×Dic181—41no—
168C2×Dic181—41no—
169C2×Dic181#4111yescharacteristic
170C4×D181—21no—
171C4×D181—21no—
172C4×D181—21no—
173C4×D181C2×D411yescharacteristic
174C4×D181C2×D411yescharacteristic
175C4×D181C2×D411yescharacteristic
176C4×D181C2×D411yescharacteristic
177#13089678—21no—
178#13089678—21no—
179#13089678—21no—
180#13089678C2×D411yescharacteristic
181#13089678C2×D411yescharacteristic
182C23×C181—21no—
183C23×C181—21no—
184Q8×C181C2×D411yescharacteristic
185D4×C181—21no—
186D4×C181—21no—
187D4×C181—21no—
188D4×C181—21no—
189D4×C181—21no—
190D4×C181C2×D411yescharacteristic
191C2×C4×C181—21no—
192C2×C4×C181C2411yescharacteristic
193C8×C181—21no—
194C8×C181—21no—
195C8×C181—21no—
196C8×C181C2×D411yescharacteristic
197C8×C181C2×D411yescharacteristic
198C22×C181—21no—
199C22×C181—41no—
200C22×C181—41no—
201C22×C181—81no—
202C22×C181—81no—
203C22×C181C2×#4111yescharacteristic
204C2×D181—21no—
205C2×D181—41no—
206C2×D181—41no—
207C2×D181—81no—
208C2×D181—82no—
209C2×D181—82no—
210C2×D181—81no—
211C2×D181#12111yescharacteristic
212C2×D181#12111yescharacteristic
213C4×C181—21no—
214C4×C181C22×D411yescharacteristic
215C4×C181C22×D411yescharacteristic
216Dic181—21no—
217Dic181—41no—
218Dic181#12111yescharacteristic
219Dic181#12111yescharacteristic
220C2×C181—21no—
221C2×C181—41no—
222C2×C181—81no—
223C2×C181—81no—
224C2×C181C2×#12111yescharacteristic
225D181—21no—
226D181—21no—
227D181—41no—
228D181—81no—
229C181#306811yescharacteristic, complemented, Sylow
230#3068—1811nomaximal, Sylow
231#575—1811no—
232C2×AGL(1,ℤ/8)—1811no—
233C2×AGL(1,ℤ/8)—1811no—
234#568—1811no—
235#472—1811no—
236#472—1811no—
237#472—1811no—
238#472D18111yescharacteristic, complemented
239#471—1811no—
240#471—1811no—
241#471—1811no—
242#471—1811no—
243#430—1811no—
244#413—1811no—
245C2×#100—1811no—
246C2×#41—1811no—
247C2×#41—1811no—
248C22×D4—1811no—
249#137—3621no—
250#137—1811no—
251#137—1811no—
252AGL(1,ℤ/8)—3621no—
253AGL(1,ℤ/8)—3621no—
254AGL(1,ℤ/8)—3621no—
255AGL(1,ℤ/8)—3621no—
256AGL(1,ℤ/8)—3621no—
257AGL(1,ℤ/8)—3621no—
258AGL(1,ℤ/8)—3621no—
259AGL(1,ℤ/8)—1811no—
260AGL(1,ℤ/8)—1811no—
261AGL(1,ℤ/8)—1811no—
262AGL(1,ℤ/8)—1811no—
263AGL(1,ℤ/8)C2×D18111yescharacteristic
264AGL(1,ℤ/8)C2×D18111yescharacteristic, complemented
265#135—3621no—
266#135—3621no—
267C2×SD16—3621no—
268C2×SD16—1811no—
269C2×SD16—1811no—
270C2×D8—3621no—
271C2×D8—1811no—
272C2×D8C2×D18111yescharacteristic, complemented
273#131—3621no—
274#131—3621no—
275#131—1811no—
276#131—1811no—
277#131—1811no—
278#131C2×D18111yescharacteristic, complemented
279C2×M16—3621no—
280C2×M16—3621no—
281C2×M16—1811no—
282C2×M16—1811no—
283C2×M16—1811no—
284C2×M16—1811no—
285#108—1811no—
286#108—1811no—
287#108—1811no—
288#108C2×D18111yescharacteristic, complemented
289#101—1811no—
290#101—1811no—
291#100—1811no—
292#100—1811no—
293#100—1811no—
294#100—1811no—
295#100C2×D18111yescharacteristic, complemented
296#100C2×D18111yescharacteristic, complemented
297C24—3621no—
298#41—3621no—
299#41—3621no—
300#41—3621no—
301#41—3621no—
302#41—3621no—
303#41—3621no—
304#41—1811no—
305#41—1811no—
306#41—1811no—
307#41C22×D18111yescharacteristic
308C2×Q8—1811no—
309C2×Q8—1811no—
310C2×D4—3621no—
311C2×D4—3621no—
312C2×D4—3621no—
313C2×D4—3621no—
314C2×D4—3621no—
315C2×D4—3621no—
316C2×D4—3621no—
317C2×D4—1811no—
318C2×D4—1811no—
319C2×D4C22×D18111yescharacteristic
320C2×D4C22×D18111yescharacteristic
321C22×C4—3621no—
322C22×C4—3621no—
323C22×C4—1811no—
324Q16—3621no—
325Q16—3621no—
326SD16—21no—
327SD16—21no—
328SD16—3621no—
329SD16—3621no—
330SD16—3621no—
331SD16—3621no—
332SD16—3621no—
333SD16—3621no—
334SD16—3621no—
335SD16—3621no—
336SD16—3621no—
337SD16—3621no—
338D8—21no—
339D8—21no—
340D8—21no—
341D8—21no—
342D8—3621no—
343D8—3621no—
344D8—3621no—
345D8—3621no—
346D8—3621no—
347D8—3621no—
348M16—21no—
349M16—3621no—
350M16—3621no—
351M16—3621no—
352M16—3621no—
353M16—3621no—
354M16—3621no—
355M16—1811no—
356M16—1811no—
357M16—1811no—
358M16—1811no—
359M16—1811no—
360M16—1811no—
361M16—1811no—
362M16C22×D18111yescharacteristic
363M16C22×D18111yescharacteristic
364M16C22×D18111yescharacteristic
365C2×C8—21no—
366C2×C8—3621no—
367C2×C8—3621no—
368C2×C8—3621no—
369C2×C8—3621no—
370C2×C8—3621no—
371C2×C8—3621no—
372C2×C8—1811no—
373C2×C8C22×D18111yescharacteristic
374C23—21no—
375C23—21no—
376C23—3621no—
377C23—3621no—
378C23—14482no—
379C23—1811no—
380C23—7241no—
381C23—7241no—
382Q8—3621no—
383Q8—3621no—
384Q8—3621no—
385Q8—1811no—
386Q8D4×D18111yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—3621no—
393D4—3621no—
394D4—3621no—
395D4—3621no—
396D4—3621no—
397D4—3621no—
398D4—1811no—
399D4—7241no—
400D4—7241no—
401D4—7241no—
402D4—7241no—
403D4D4×D18111yescharacteristic
404C2×C4—21no—
405C2×C4—3621no—
406C2×C4—3621no—
407C2×C4—3621no—
408C2×C4—3621no—
409C2×C4—3621no—
410C2×C4—1811no—
411C2×C4—1811no—
412C2×C4—1811no—
413C2×C4—1811no—
414C2×C4—1811no—
415C2×C4—7241no—
416C2×C4—7241no—
417C2×C4C23×D18111yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—3621no—
422C8—3621no—
423C8—3621no—
424C8—1811no—
425C8—1811no—
426C8D4×D18111yescharacteristic
427C8D4×D18111yescharacteristic
428C22—21no—
429C22—41no—
430C22—41no—
431C22—81no—
432C22—81no—
433C22—3621no—
434C22—14481no—
435C22—14481no—
436C22—14482no—
437C22—14482no—
438C22—1811no—
439C22—1811no—
440C22—7241no—
441C22—7241no—
442C22#41×D18111yescharacteristic
443C4—21no—
444C4—3621no—
445C4—1811no—
446C4—1811no—
447C4—7241no—
448C4C2×D4×D18111yescharacteristic
449C4C2×D4×D18111yescharacteristic
450C2—21no—
451C2—41no—
452C2—81no—
453C2—81no—
454C2—3621no—
455C2—3621no—
456C2—14481no—
457C2—7241no—
458C2#121×D18111yescharacteristic
subgroupG/H
centerC2#121×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#472×C181C2
socleC2×C181C2×#121
hypercenter#472D181
hypocenterC181#3068
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #26042075C2semidirect
2. C2×#19153840C2semidirect
3. AGL(1,ℤ/8)×D181C2semidirect
4. #26042044C2semidirect
5. #26041548C2semidirect
6. #26041547C2semidirect
7. #26041546C2semidirect
8. #26041543C2semidirect
9. #26041542C2semidirect
10. #26041541C2semidirect
11. #26041540C2semidirect
12. #26041326C2semidirect
13. #26041260C2semidirect
14. #100×D181C2semidirect
15. #472×C181C2semidirect
19. #19153849C22semidirect
20. #19153846C22semidirect
25. #19153842C22semidirect
35. C2×#15668438C22semidirect
36. C2×#15668437C22semidirect
38. C2×#15668434C22semidirect
41. #19153824C22semidirect
42. #19153823C22semidirect
44. #19153822C22semidirect
51. #19153752C22semidirect
52. #19153751C22semidirect
53. #19153751C22semidirect
54. #19153734C22semidirect
55. #19153734C22semidirect
56. #19153733C22semidirect
57. #19153733C22semidirect
58. #19153732C22semidirect
59. #19153732C22semidirect
61. AGL(1,ℤ/8)×C181C22semidirect
62. C2×D8×C181C22semidirect
63. #131×C181C22semidirect
64. #108×C181C22semidirect
65. #100×C181C22semidirect
66. #100×C181C22semidirect
229. C181#3068semidirect
238. #472D181semidirect
264. AGL(1,ℤ/8)C2×D181semidirect
272. C2×D8C2×D181semidirect
278. #131C2×D181semidirect
288. #108C2×D181semidirect
295. #100C2×D181semidirect
296. #100C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420752
C2×#191538402
AGL(1,ℤ/8)×D1812
#260420442
#260415482
#260415472
#260415462
#260415432
#260415422
#260415412
#260415402
#260413262
#260412602
#100×D1812
#472×C1812
#3068181