Finite Group Databasebeta

← #471×D191 #37582215

Subgroups — #471×D191 #37582215

Order 24448.

468 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#137×D191C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D191C211yesmaximal, characteristic
3C2×SD16×D191C211yesmaximal, characteristic, complemented
4#131×D191C211yesmaximal, characteristic, complemented
5#27132929C211yesmaximal, characteristic, complemented
6#27132928C211yesmaximal, characteristic, complemented
7#27132927C211yesmaximal, characteristic, complemented
8#27132926C211yesmaximal, characteristic, complemented
9#27132925C211yesmaximal, characteristic, complemented
10#27132924C211yesmaximal, characteristic, complemented
11#27132923C211yesmaximal, characteristic, complemented
12#108×D191C211yesmaximal, characteristic, complemented
13#101×D191C211yesmaximal, characteristic, complemented
14#100×D191C211yesmaximal, characteristic, complemented
15#471×C191C211yesmaximal, characteristic, complemented
16#41×D191C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic
18C2×D4×D191C2211yescharacteristic
19#20259355C2211yescharacteristic, complemented
20#20259354—21no—
21#20259353C2211yescharacteristic, complemented
22#20259352C2211yescharacteristic, complemented
23#20259351—21no—
24#20259350C2211yescharacteristic
25#20259349—21no—
26#20259348C2211yescharacteristic
27#20259347C2211yescharacteristic
28#20259346—21no—
29Q16×D191—21no—
30C2×#15690276C2211yescharacteristic, complemented
31C2×#15690275C2211yescharacteristic, complemented
32C2×#15690274C2211yescharacteristic, complemented
33SD16×D191—21no—
34SD16×D191—21no—
35SD16×D191—21no—
36SD16×D191—21no—
37SD16×D191—21no—
38SD16×D191—21no—
39D8×D191—21no—
40#20259330—21no—
41#20259330C2211yescharacteristic, complemented
42#20259329C2211yescharacteristic, complemented
43#20259328—21no—
44#20259328C2211yescharacteristic, complemented
45M16×D191—21no—
46M16×D191C2211yescharacteristic
47M16×D191C2211yescharacteristic
48M16×D191C2211yescharacteristic
49C2×C8×D191—21no—
50C2×C8×D191C2211yescharacteristic
51#20259258C2211yescharacteristic, complemented
52#20259257C2211yescharacteristic, complemented
53#20259257C2211yescharacteristic, complemented
54#20259241C2211yescharacteristic, complemented
55#20259240C2211yescharacteristic, complemented
56#20259240C2211yescharacteristic, complemented
57#20259239C2211yescharacteristic, complemented
58#20259238C2211yescharacteristic, complemented
59#20259238C2211yescharacteristic, complemented
60#137×C191C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C191C2211yescharacteristic
62C2×SD16×C191C2211yescharacteristic, complemented
63#131×C191C2211yescharacteristic, complemented
64#108×C191C2211yescharacteristic, complemented
65#101×C191C2211yescharacteristic, complemented
66#100×C191C2211yescharacteristic, complemented
67C23×D191—21no—
68#15690288D411yescharacteristic
69#15690287—21no—
70#15690287C2311yescharacteristic
71#15690286—21no—
72#15690286D411yescharacteristic
73C2×Dic382—21no—
74C2×Dic382C2311yescharacteristic
75Q8×D191—21no—
76Q8×D191—21no—
77Q8×D191—21no—
78Q8×D191—21no—
79Q8×D191D411yescharacteristic
80C2×D764C2311yescharacteristic
81C2×C22⋊D191—21no—
82D4×D191—21no—
83D4×D191—21no—
84D4×D191—21no—
85D4×D191—21no—
86D4×D191—21no—
87D4×D191D411yescharacteristic
88C2×C4×D191—21no—
89C2×C4×D191—21no—
90C2×C4×D191C2311yescharacteristic
91Dic764—21no—
92#15690277—21no—
93#15690277—21no—
94#15690276—21no—
95#15690276—21no—
96#15690276—21no—
97#15690276—21no—
98#15690275—21no—
99#15690275—21no—
100#15690275—21no—
101#15690275—21no—
102#15690274—21no—
103#15690274—21no—
104#15690274—21no—
105#15690274—21no—
106D1528—21no—
107#15690272—21no—
108#15690272—21no—
109#15690271—21no—
110#15690271C2311yescharacteristic
111#15690271C2311yescharacteristic
112#15690271C2311yescharacteristic
113#15690270—21no—
114#15690270—21no—
115#15690270—21no—
116#15690270—21no—
117#15690270—21no—
118C2×C191⋊C8—21no—
119C2×C191⋊C8C2311yescharacteristic
120C8×D191—21no—
121C8×D191—21no—
122C8×D191—21no—
123C8×D191—21no—
124C8×D191—21no—
125C8×D191D411yescharacteristic
126C8×D191D411yescharacteristic
127C8×D191D411yescharacteristic
128C8×D191D411yescharacteristic
129#41×C191C2311yescharacteristic
130C2×Q8×C191C2311yescharacteristic
131C2×D4×C191C2311yescharacteristic
132Q16×C191—21no—
133SD16×C191—21no—
134SD16×C191—21no—
135SD16×C191—21no—
136SD16×C191—21no—
137D8×C191—21no—
138M16×C191—21no—
139M16×C191C2311yescharacteristic
140M16×C191C2311yescharacteristic
141M16×C191C2311yescharacteristic
142C2×C8×C191—21no—
143C2×C8×C191C2311yescharacteristic
144C22×D191—21no—
145C22×D191—21no—
146C22×D191—41no—
147C22×D191—41no—
148C22×D191—82no—
149C22×D191#4111yescharacteristic
150Dic382—21no—
151Dic382—21no—
152Dic382—21no—
153Dic382—41no—
154Dic382—41no—
155Dic382C2×D411yescharacteristic
156D764—21no—
157D764—21no—
158D764—21no—
159D764C2×D411yescharacteristic
160C22⋊D191—21no—
161C22⋊D191—21no—
162C22⋊D191—21no—
163C22⋊D191—41no—
164C22⋊D191—41no—
165C2×Dic191—21no—
166C2×Dic191—21no—
167C2×Dic191#4111yescharacteristic
168C4×D191—21no—
169C4×D191—21no—
170C4×D191—21no—
171C4×D191—41no—
172C4×D191—41no—
173C4×D191C2×D411yescharacteristic
174C4×D191C2×D411yescharacteristic
175C4×D191C2×D411yescharacteristic
176C4×D191C2×D411yescharacteristic
177C191⋊C8—21no—
178C191⋊C8—21no—
179C191⋊C8—21no—
180C191⋊C8C2×D411yescharacteristic
181C191⋊C8C2×D411yescharacteristic
182C23×C191—21no—
183Q8×C191—21no—
184Q8×C191—21no—
185Q8×C191C2×D411yescharacteristic
186D4×C191—21no—
187D4×C191—21no—
188D4×C191—21no—
189D4×C191C2×D411yescharacteristic
190C2×C4×C191—21no—
191C2×C4×C191—21no—
192C2×C4×C191C2411yescharacteristic
193C8×C191—21no—
194C8×C191—21no—
195C8×C191—21no—
196C8×C191C2×D411yescharacteristic
197C8×C191C2×D411yescharacteristic
198C22×C191—21no—
199C22×C191—41no—
200C22×C191—81no—
201C22×C191C2×#4111yescharacteristic
202C2×D191—22no—
203C2×D191—21no—
204C2×D191—42no—
205C2×D191—41no—
206C2×D191—82no—
207C2×D191—82no—
208C2×D191—81no—
209C2×D191#12111yescharacteristic
210C2×D191#12111yescharacteristic
211C4×C191—21no—
212C4×C191—41no—
213C4×C191C22×D411yescharacteristic
214C4×C191C22×D411yescharacteristic
215Dic191—21no—
216Dic191—41no—
217Dic191#12111yescharacteristic
218Dic191#12111yescharacteristic
219C2×C191—21no—
220C2×C191—41no—
221C2×C191—81no—
222C2×C191C2×#12111yescharacteristic
223D191—21no—
224D191—41no—
225D191—81no—
226D191#47112yesdirect product factor
227C191C2×#47111yescharacteristic, complemented, Sylow
228C2×#471—1911nomaximal, Sylow
229C2×#137—1911no—
230C2×AGL(1,ℤ/8)—1911no—
231C22×SD16—1911no—
232C2×#131—1911no—
233#471—1911no—
234#471—1911no—
235#471—1911no—
236#471—1911no—
237#471—1911no—
238#471—1911no—
239#471—1911no—
240#471D19111yescharacteristic, direct product factor
241C2×#108—1911no—
242C2×#101—1911no—
243C2×#100—1911no—
244C2×#41—1911no—
245C22×Q8—1911no—
246C22×D4—1911no—
247#137—3821no—
248#137—3821no—
249#137—1911no—
250#137—1911no—
251#137—1911no—
252#137C2×D19111yescharacteristic, complemented
253AGL(1,ℤ/8)—3821no—
254AGL(1,ℤ/8)—3821no—
255AGL(1,ℤ/8)—1911no—
256AGL(1,ℤ/8)—1911no—
257AGL(1,ℤ/8)—1911no—
258AGL(1,ℤ/8)C2×D19111yescharacteristic
259C2×Q16—3821no—
260C2×SD16—3821no—
261C2×SD16—3821no—
262C2×SD16—3821no—
263C2×SD16—3821no—
264C2×SD16—3821no—
265C2×SD16—3821no—
266C2×SD16—1911no—
267C2×SD16—1911no—
268C2×SD16—1911no—
269C2×SD16C2×D19111yescharacteristic, complemented
270C2×D8—3821no—
271#131—3821no—
272#131—3821no—
273#131—1911no—
274#131—1911no—
275#131—1911no—
276#131C2×D19111yescharacteristic, complemented
277C2×M16—3821no—
278C2×M16—1911no—
279C2×M16—1911no—
280C2×M16—1911no—
281C22×C8—3821no—
282C22×C8—1911no—
283#108—1911no—
284#108—1911no—
285#108—1911no—
286#108C2×D19111yescharacteristic, complemented
287#101—1911no—
288#101—1911no—
289#101—1911no—
290#101C2×D19111yescharacteristic, complemented
291#100—1911no—
292#100—1911no—
293#100—1911no—
294#100C2×D19111yescharacteristic, complemented
295C24—3821no—
296#41—3821no—
297#41—3821no—
298#41—1911no—
299#41—1911no—
300#41—1911no—
301#41C22×D19111yescharacteristic
302C2×Q8—3821no—
303C2×Q8—3821no—
304C2×Q8—3821no—
305C2×Q8—3821no—
306C2×Q8—3821no—
307C2×Q8—1911no—
308C2×Q8—1911no—
309C2×Q8C22×D19111yescharacteristic
310C2×D4—3821no—
311C2×D4—3821no—
312C2×D4—3821no—
313C2×D4—3821no—
314C2×D4—3821no—
315C2×D4—3821no—
316C2×D4—1911no—
317C2×D4—1911no—
318C2×D4C22×D19111yescharacteristic
319C22×C4—3821no—
320C22×C4—3821no—
321C22×C4—1911no—
322Q16—21no—
323Q16—3821no—
324Q16—3821no—
325Q16—3821no—
326SD16—21no—
327SD16—21no—
328SD16—21no—
329SD16—21no—
330SD16—3821no—
331SD16—3821no—
332SD16—3821no—
333SD16—3821no—
334SD16—3821no—
335SD16—3821no—
336SD16—3821no—
337SD16—3821no—
338SD16—3821no—
339SD16—3821no—
340SD16—3821no—
341SD16—3821no—
342D8—21no—
343D8—3821no—
344D8—3821no—
345D8—3821no—
346M16—21no—
347M16—3821no—
348M16—3821no—
349M16—3821no—
350M16—3821no—
351M16—3821no—
352M16—3821no—
353M16—1911no—
354M16—1911no—
355M16—1911no—
356M16C22×D19111yescharacteristic
357M16C22×D19111yescharacteristic
358M16C22×D19111yescharacteristic
359C2×C8—21no—
360C2×C8—3821no—
361C2×C8—3821no—
362C2×C8—3821no—
363C2×C8—3821no—
364C2×C8—3821no—
365C2×C8—3821no—
366C2×C8—1911no—
367C2×C8—1911no—
368C2×C8—1911no—
369C2×C8—1911no—
370C2×C8—1911no—
371C2×C8C22×D19111yescharacteristic
372C23—21no—
373C23—3821no—
374C23—3821no—
375C23—1911no—
376C23—15282no—
377C23—7641no—
378C23—7641no—
379Q8—21no—
380Q8—21no—
381Q8—3821no—
382Q8—3821no—
383Q8—3821no—
384Q8—1911no—
385Q8—7641no—
386Q8—7641no—
387Q8D4×D19111yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—3821no—
392D4—3821no—
393D4—3821no—
394D4—3821no—
395D4—3821no—
396D4—3821no—
397D4—1911no—
398D4—7641no—
399D4—7641no—
400D4D4×D19111yescharacteristic
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—3821no—
404C2×C4—3821no—
405C2×C4—3821no—
406C2×C4—3821no—
407C2×C4—3821no—
408C2×C4—1911no—
409C2×C4—1911no—
410C2×C4—1911no—
411C2×C4—1911no—
412C2×C4—1911no—
413C2×C4—7641no—
414C2×C4—7641no—
415C2×C4C23×D19111yescharacteristic
416C8—21no—
417C8—21no—
418C8—21no—
419C8—3821no—
420C8—3821no—
421C8—3821no—
422C8—1911no—
423C8—1911no—
424C8D4×D19111yescharacteristic
425C8D4×D19111yescharacteristic
426C22—21no—
427C22—41no—
428C22—81no—
429C22—3822no—
430C22—3821no—
431C22—1911no—
432C22—1911no—
433C22—15281no—
434C22—15282no—
435C22—15282no—
436C22—7642no—
437C22—7641no—
438C22#41×D19111yescharacteristic
439C4—21no—
440C4—41no—
441C4—3821no—
442C4—1911no—
443C4—1911no—
444C4—7641no—
445C4C2×D4×D19111yescharacteristic
446C4C2×D4×D19111yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—3821no—
451C2—1912no—
452C2—15281no—
453C2—7641no—
454C2#121×D19111yescharacteristic
subgroupG/H
centerC2#121×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#471×C191C2
socleC2×C191C2×#121
prime radicalAGL(1,ℤ/8)×D191C2
hypercenter#471D191
hypocenterC191C2×#471
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #137×D191C2semidirect
3. C2×SD16×D191C2semidirect
4. #131×D191C2semidirect
5. #27132929C2semidirect
6. #27132928C2semidirect
7. #27132927C2semidirect
8. #27132926C2semidirect
9. #27132925C2semidirect
10. #27132924C2semidirect
11. #27132923C2semidirect
12. #108×D191C2semidirect
13. #101×D191C2semidirect
14. #100×D191C2semidirect
15. #471×C191C2semidirect
19. #20259355C22semidirect
21. #20259353C22semidirect
22. #20259352C22semidirect
30. C2×#15690276C22semidirect
31. C2×#15690275C22semidirect
32. C2×#15690274C22semidirect
41. #20259330C22semidirect
42. #20259329C22semidirect
44. #20259328C22semidirect
51. #20259258C22semidirect
52. #20259257C22semidirect
53. #20259257C22semidirect
54. #20259241C22semidirect
55. #20259240C22semidirect
56. #20259240C22semidirect
57. #20259239C22semidirect
58. #20259238C22semidirect
59. #20259238C22semidirect
60. #137×C191C22semidirect
62. C2×SD16×C191C22semidirect
63. #131×C191C22semidirect
64. #108×C191C22semidirect
65. #101×C191C22semidirect
66. #100×C191C22semidirect
226. D191#471direct (N × H)
227. C191C2×#471semidirect
240. #471D191direct (N × H)
252. #137C2×D191semidirect
269. C2×SD16C2×D191semidirect
276. #131C2×D191semidirect
286. #108C2×D191semidirect
290. #101C2×D191semidirect
294. #100C2×D191semidirect

Maximal subgroups

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

subgroupindex
#137×D1912
AGL(1,ℤ/8)×D1912
C2×SD16×D1912
#131×D1912
#271329292
#271329282
#271329272
#271329262
#271329252
#271329242
#271329232
#108×D1912
#101×D1912
#100×D1912
#471×C1912
C2×#471191